Research Article Stability and Dynamical Analysis of a...

Research ArticleStability and Dynamical Analysis of a Biological System

Xinhong Pan12 Min Zhao23 Yapei Wang12 Hengguo Yu12

Zengling Ma23 and Qi Wang23

1 School of Mathematics and Information Science Wenzhou University Wenzhou Zhejiang 325035 China2 Zhejiang Provincial Key Laboratory for Water Environment and Marine Biological Resources ProtectionWenzhou University Wenzhou Zhejiang 325035 China

3 School of Life and Environmental Science Wenzhou University Wenzhou Zhejiang 325035 China

Correspondence should be addressed to Min Zhao zmcntomcom

Received 2 April 2014 Revised 6 May 2014 Accepted 20 May 2014 Published 22 July 2014

Academic Editor Imran Naeem

Copyright copy 2014 Xinhong Pan et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This study considers the spatiotemporal dynamics of a reaction-diffusion phytoplankton-zooplankton system with a double Alleeeffect on prey under a homogeneous boundary condition The qualitative properties are analyzed including the local stability ofall equilibria and the global asymptotic property of the unique positive equilibrium We also discuss the Hopf bifurcation and thesteady state bifurcation of the system These results are expected to help understand the complexity of the Allee effect and theinteraction between phytoplankton and zooplankton

1 Introduction

The upper layer of the ocean contains large volumes ofdrifting plankton which can be divided into phytoplanktonand zooplankton Phytoplankton is the autotrophic com-ponent of the plankton community which is consumedby zooplankton most of which are too small to observeindividually with the naked eye Zooplankton which areheterotrophic organisms in oceans are also mostly invisibleto the naked eye Therefore it is difficult and expensiveto quantify plankton directly Plankton not only play animportant role in the marine system because they are at thebottom level of the food chain that supports commercialfisheries but also play important roles in the cycling of manychemical elements such as carbon which may affect climatechange [1] Furthermore when plankton such as blue algaeand dinoflagellates are present in large concentrations thewater appears to be discolored or murky which is knownas a red tide and this can result in the death of marine andcoastal species of fish mammals and other organisms [1]Thus analyzing the dynamics of plankton using mathemat-ical models is beneficial for understanding the features ofplankton populations which have enormous economic andecological value

However the mechanism that leads to the occurrence ofred tides is still an unsolved issue Many models and theorieshave been proposed by mathematicians and ecologists toexplain this phenomenon but a general and correct expla-nation still remains a distant goal [2ndash5]

The popular mathematical model called a Gause-typepredator-prey model is used to consider the phytoplankton-zooplankton interaction in the following form

119889119901

119889119905= 119891 (119901119870) 119901 minus 119892 (119901) 119911

119889119911

119889119905= 119890119892 (119901) 119911 minus 120583119911

(1)

where 119901 = 119901(119905) and 119911 = 119911(119905) are the population densitiesor biomass of phytoplankton and zooplankton at time 119905respectively119891(119901119870) describes the intrinsic per capita growthrate of the phytoplankton which may be a logistic growthfunction exponential growth function or other functionswhere 119870 is known as the environmental carrying capacity119892(119901) is the per unit-predator consumption rate of preywhich is commonly called the functional response Someconventional forms of functional response include Hollingtypes I II III and IV and Ivlev type [6ndash8] 119890 (0 lt 119890 lt 1)

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014 Article ID 952840 11 pageshttpdxdoiorg1011552014952840

2 Abstract and Applied Analysis

is the conversion coefficient and 120583 represents the per capitapredator mortality which is assumed to have a linear formalthough other forms are possible [9 10]The global dynamicsof model (1) with a logistic growth rate have been studiedduring the last three decades based on theoretical analysis andnumerical simulations and many results have been reported[11ndash15]

In recent years the Allee effect has been the focus ofincreasing interest and it is recognized to be an importantphenomenon in many fields of ecology and conservationbiology by more and more people [16ndash21] The Allee effectis named after WC Allee [22] and it describes a positivecorrelation between the density or number of populationand individual fitness of population [16] Standard populationmodels assume that the fitness of population increases as thepopulation density or size declines [11ndash15 23 24] whereasAllee effect states that when a population is below a criticaldensity or size the population cannot sustain itself and thisleads to extinction Thus the Allee effect increases the likeli-hood of extinction [25] Stephens et al distinguished betweena component Allee effect and a demographic Allee effect [17]However conservation biologists are usually more interestedin the demographic Allee effect because it ultimately governsthe probability of the extinction or recovery of populationswith low abundances [16]

Very recent ecological research has shown that two ormore Allee effects can act on a single population simultane-ously which is known as the multiple (double) Allee effect[26 27]

There are many ways of describing the Allee effect [28]including the following differential equation

119889119901

119889119905= 119903 (1 minus

119901

119870)(1 minus

119898 + 119899

119901 + 119899)119901 (2)

where 119903 describes the growth rate and 119899 is an auxiliaryparameter where 119899 gt 0 119898 + 119899 gt 0 Indeed it is consideredthat 119901 represents the size of a fertile population and 119899 is thenonfertile population such as juvenile or oldest individuals[29] In this case 119898 (minus119870 lt 119898 lt 119870) is called theAllee threshold because when the population density orsize is below this threshold the population is destined forextinctionWhen119898 gt 0 (2) describes a strong Allee effect [830 31] In this case the population growth rate decreases if thepopulation size is below the threshold 119898 and the populationgoes to extinction [29] In addition (2) describes a weakAlleeeffect [29 31ndash33] for119898 lt 0

It is obvious that (2) is equal to

119889119901

119889119905=119903119901

119901 + 119899(1 minus

119901

119870) (119901 minus 119898) (3)

Gonzalez-Olivares et al [29] state that (3) describes a doubleAllee effect that is once in the factor 119898(119901) = 119901 minus 119898 and thesecond time in the term 119903(119901) = 119903119901(119901 + 119899) [34 35]

In a marine environment the plankton populations tendto move in horizontal and vertical directions due to thestrong water current This movement is usually modeled bya reaction-diffusion equation In this study we consider thefollowing reaction-diffusion model with constant diffusion

coefficient as well as a strong Allee effect in different spatiallocations within a fixed smooth bounded domainΩ isin 119877119899Weassume that the response function of the zooplankton followsthe law of mass action [15]

120597119901 (119909 119905)

120597119905= 119903 (1 minus

119901 (119909 119905)

119870)(1 minus

119898 + 119899

119901 (119909 119905) + 119899)119901 (119909 119905)

minus 119886119901 (119909 119905) 119911 (119909 119905) + 1198891Δ119901 (119909 119905) 119909 isin Ω 119905 gt 0

120597119911 (119909 119905)

120597119905= 119890119886119901 (119909 119905) 119911 (119909 119905) minus 120583119911 (119909 119905) + 119889

2Δ119911 (119909 119905)

119909 isin Ω 119905 gt 0

120597119901 (119909 119905)

120597119899=120597119911 (119909 119905)

120597119899= 0 119909 isin 120597Ω 119905 gt 0

119901 (119909 0) = 1199010(119909) ge 0 119911 (119909 0) = 119911

0(119909) ge 0 119909 isin Ω

(4)

where 119898 (0 lt 119898 lt 119870) is the Allee threshold 1198891 1198892are

the diffusion coefficients of phytoplankton and zooplanktonrespectively Δ is the Laplacian operator and 119909 is the spatialhabitat of two species andwe assume that the system is a closeecosystem and with a no-flux boundary condition

This paper is structured as follows In Section 2 weanalyze the basic dynamics of (4) including estimates of thesolution and the local and global stability of equilibria InSection 3 we provide the analysis of the Hopf bifurcation andthe steady state bifurcation A brief discussion and summaryare given in Section 4

2 Main Results

21 Basic Dynamics Suppose that 119899 119886 120583 1198891 1198892gt 0 0 lt 119890 lt

1 0 lt 119898 lt 119870 and Ω is a bounded domain then we obtainthe following results

Theorem 1 The system (4) has a unique solution and thesolution is bounded Furthermore the solution (119901 119911) of (4)satisfies

lim119905rarrinfin

sup119901 (119909 119905) le 119870

lim119905rarrinfin

supintΩ

119911 (119909 119905) 119889119909 le 119870[119890 +119890119903

119870119899120583(119870 minus 119898

2)

2

] |Ω|

(5)

if1198891= 1198892 lim119905rarrinfin

sup 119911(119909 119905) le 119870[119890+(119890119903119870119899120583)((119870minus119898)2)2]

Proof Let

119872(119901 119911) = 119903 (1 minus119901

119870)(1 minus

119898 + 119899

119901 + 119899)119901 minus 119886119901119911

119873 (119901 119911) = 119890119886119901119911 minus 120583119911

(6)

then 120597119872120597119911 = minus119886119901 le 0 120597119873120597119901 = 119890119886119911 ge 0 for 119901 ge0 119911 ge 0 which implies that (4) is a mixed quasi-monotone system [36] We define (119901(119909 119905) 119911(119909 119905)) = (0 0)

Abstract and Applied Analysis 3

14

12

1

08

06

04

02

00 10 20 30 40 50 60

(a)

1

09

08

07

06

05

04

03

02

01

00 10 20 30 40 50 60

(b)

Figure 1 Time evolution of (4) around (0 0) Phytoplankton is in (a) and zooplankton is in (b) The parameter values are 1198891= 0001

1198892= 0004 119890 = 05 119886 = 1 119899 = 05119898 = 05 120583 = 04 119903 = 2 and 119870 = 10

15

1

05

0200

150100

500 50

7090

110

(a)

3

25

2

15

1

05

0200

150

10050

0 5070

90110

(b)

Figure 2 Solution to (119901(119909 119905) 119911(119909 119905)) for (4) Phytoplankton is in (a) and zooplankton is in (b) The parameter values are the same as thosegiven in Figure 1

(119901(119909 119905) 119911(119909 119905)) = (119901lowast

(119905) 119911lowast

(119905)) where (119901lowast(119905) 119911lowast(119905)) is thesolution of the system

119889119901

119889119905= 119903 (1 minus

119901

119870)(1 minus

119898 + 119899

119901 + 119899)119901

119889119911

119889119905= 119890119886119901119911 minus 120583119911

119901 (0) = 119901lowast

119911 (0) = 119911lowast

(7)

where 119901lowast = supΩ1199010(119909) 119911lowast = sup

Ω1199110(119909) We find that

120597119901 (119909 119905)

120597119905minus Δ119901 (119909 119905) minus 119872 (119901 (119909 119905) 119911 (119909 119905)) = 0

120597119901 (119909 119905)

120597119905minus Δ119901 (119909 119905) minus 119872(119901 (119909 119905) 119911 (119909 119905)) = 0

(8)

Therefore

120597119901 (119909 119905)

120597119905minus Δ119901 (119909 119905) minus 119872 (119901 (119909 119905) 119911 (119909 119905))

ge

120597119901 (119909 119905)

120597119905minus Δ119901 (119909 119905) minus 119872(119901 (119909 119905) 119911 (119909 119905))

(9)

Similarly

120597119911 (119909 119905)

120597119905minus Δ119911 (119909 119905) minus 119873 (119901 (119909 119905) 119911 (119909 119905))

ge120597119911 (119909 119905)

120597119905minus Δ119911 (119909 119905) minus 119873 (119901 (119909 119905) 119911 (119909 119905))

(10)

This implies that (119901(119909 119905) 119911(119909 119905)) = (0 0) and (119901(119909 119905) 119911(119909 119905))= (119901lowast

(119905) 119911lowast

(119905)) are the lower solution and upper solution to(4) respectively (Figure 2) Therefore Theorem 533 in [36]


shows that system (4) has a unique solution ((119901(119909 119905) 119911(119909 119905))that satisfies

0 le 119901 (119909 119905) le 119901lowast

(119905) 0 le 119911 (119909 119905) le 119911lowast

(119905) (11)

According to the strong maximum principle and the bound-ary condition 119901(119909 119905) gt 0 119911(119909 119905) gt 0 for 119905 gt 0 and 119909 isin Ω

From the first equation of (7) we can see that 119901lowast(119905) rarr 0for 119901lowast lt 119898 and 119901lowast(119905) rarr 119870 for 119901lowast gt 119898 Thuslim119905rarrinfin

sup119901(119909 119905) le 119870 To estimate 119911(119909 119905) letintΩ

119901(119909 119905)119889119909 = 119875(119905) intΩ

119911(119909 119905)119889119909 = 119885(119905) then

119889119875

119889119905= intΩ

119901119905119889119909

= intΩ

[119903 (1 minus119901

119870)(1 minus

119898 + 119899

119901 + 119899)119901 minus 119886119901119911] 119889119909

+ intΩ

1198891Δ119901119889119909

(12)

119889119885

119889119905= intΩ

119890119886119901119911119889119909 minus 120583119885 + intΩ

1198892Δ119911119889119909 (13)

(12) lowast 119890 + (13) and with boundary conditions we obtain

(119890119875 + 119885)119905= 119890intΩ

119903 (1 minus119901

119870)119901 minus 119898

119901 + 119899119901119889119909 minus 120583119885

= 119890intΩ

119903 (1 minus119901

119870)119901 minus 119898

119901 + 119899119901119889119909 minus 120583 (119890119875 + 119885) minus 120583119890119875

le minus120583 (119890119875 + 119885) + [120583119890 +119890119903

119870119899(119870 minus 119898

2)

2

]119875

(14)

From the proof above we know that lim119905rarrinfin

sup119875(119905) le 119870 sdot|Ω| Thus for any 120576 gt 0 exist119879

1gt 0 when 119905 gt 119879

1

(119890119875 + 119885)119905

le minus120583 (119890119875 + 119885) + [120583119890 +119890119903

119870119899(119870 minus 119898

2)

2

] (119870 + 120576) |Ω|

(15)

By integrating (15) exist1198792gt 1198791 we have

intΩ

119911 (119909 119905) 119889119909

= 119885 (119905) lt 119890119875 (119905) + 119885 (119905)

le119870 + 120576

120583[120583119890 +

119890119903

119870119899(119870 minus 119898

2)

2

] |Ω| + 120576 for 119905 gt 1198792

(16)

This implies that lim119905rarrinfin

supintΩ

119911(119909 119905)119889119909 le 119870[119890 + (119890119903119870119899120583)

((119870 minus 119898)2)2

]|Ω|

If 1198891= 1198892 we can add the two equations in (4) and we

have

120597119908

120597119905= 119890119903 (1 minus

119901

119870)(1 minus

119898 + 119899

119901 + 119899)119901 minus 120583119911 + 119889

1Δ119908

119909 isin Ω 119905 gt 119879

120597119908

120597119899= 0 119909 isin 120597Ω 119905 gt 119879

119908 (119909 119879) = 119890119901 (119909 119879) + 119911 (119909 119879) 119909 isin Ω

(17)

Since

119890119903 (1 minus119901

119870)119901 minus 119898

119901 + 119899119901 minus 120583119911

= 119890119903 (1 minus119901

119870)119901 minus 119898

119901 + 119899119901 minus 120583119908 + 120583119890119901

le [119890119903

119870119899(119870 minus 119898

2)

2

+ 120583119890] 119901 minus 120583119908

le [119890119903

119870119899(119870 minus 119898

2)

2

+ 120583119890] (119870 + 120576) minus 120583119908

(18)

we know that the solution of the equation

120597Φ

120597119905= [119890119903

119870119899(119870 minus 119898

2)

2

+ 120583119890] (119870 + 120576) minus 120583V + 1198891ΔV

119909 isin Ω 119905 gt 119879

120597Φ

120597119899= 0 119909 isin 120597Ω 119905 gt 119879

(19)

is Φ(119909 119905) rarr (1120583)[(119890119903119870119899)((119870 minus 119898)2)2

+ 120583119890](119870 + 120576) for119905 rarr infin The comparison argument implies that lim

119905rarrinfinsup

119911(119909 119905) le lim119905rarrinfin

sup119908(119909 119905) le (1120583)[(119890119903119870119899)((119870 minus119898)2)2 +120583119890](119870 + 120576)

Theorem 2 If 1199010(119909) le 119898 then (119901(119909 119905) 119911(119909 119905)) rarr (0 0) or

(119901(119909 119905) 119911(119909 119905)) rarr (119898 0) as 119905 rarr infin

Proof From the proof of Theorem 1 if 1199010(119909) le 119901

lowast

lt 119898 then119901lowast

(119905) rarr 0 and consequently 119911lowast(119905) rarr 0 as 119905 rarr infin Thiscompletes the proof

From a biological viewpoint this implies that if theinitial population density is below the threshold 119898 thephytoplankton become extinct so the zooplankton wouldbecome extinct

22 Local and Global Stability of Equilibria System (4) hasfour nonnegative steady state solutions (0 0) (119898 0) (119870 0)and (119906 V

119906) where 119906 = 120583119890119886 V

119906= (119903119886)(1 minus (119906119870))((119906 minus

119898)(119906 + 119899))The local stability of the steady state solutions can be

analyzed as follows


Theorem 3 (1) (0 0) is locally asymptotically stable(2) (119870 0) is unstable(3) (119898 0) is locally asymptotically stable when 119890119886119870minus120583 lt 0

and is unstable for 119890119886119870 minus 120583 gt 0(4) (119906 V

119906) is locally asymptotically stable for 119906 lt 119906 lt 119870

and is unstable for 119906 lt 119906 lt 119870 where

119906 =2119899 minus radic41198992 + 4 (119898119899 + 119898119870 + 119899119870)

minus2 (20)

Proof The linearization of (4) at solution (119901 119911) can beexpressed as

120597119880

120597119905= (119863Δ + 119869

(119901119911))119880 (21)

where

119880 = (119901 (119909 119905) 119911 (119909 119905))119879

119863 = diag (1198891 1198892)

119869(119901119911)= (119860 (119901 119911) 119861 (119901 119911)

119862 (119901 119911) 119863 (119901 119911))

119860 (119901 119911) = 119903 (1 minus119901

119870)(1 minus

119898 + 119899

119901 + 119899) minus119903

119870119901(1 minus

119898 + 119899

119901 + 119899)

+ 119903 (1 minus119901

119870)119901119898 + 119899

(119901 + 119899)2minus 119886119911

119861 (119901 119911) = minus119886119901 119862 (119901 119911) = 119890119886119911 119863 (119901 119911) = 119890119886119901 minus 120583

(22)

According to Theorems 511 and 513 from [37] we knowthat if all the eigenvalues of the operator 119871 have negativereal parts then the solution (119901 119911) is asymptotically stable ifthere is at least one eigenvalue with a positive real part thenthe solution (119901 119911) is unstable if some eigenvalues have zeroreal parts then the stability cannot be determined using thismethod

Let 120582119894(119894 = 0 1 2 ) be the eigenvalues of minusΔ on Ω

under a homogeneous Neumann boundary condition and0 = 120582

0lt 1205821le 1205822le sdot sdot sdot lim

119894rarrinfin120582119894= infin Thus it is known

that 120585 is an eigenvalue of 119871 if and only if 120585 is the eigenvalue ofthe matrix 119869

119894= minus120582119894119863 + 119869(119901119911)

for some 119894 ge 0Consider the characteristic equation

det (120585119868 minus 119869119894) = 1205852

minus trace119869119894120585 + det 119869

119894 (23)

wheretrace119869

119894= minus120582119894(1198891+ 1198892) + 119860 (119901 119911) + 119863 (119901 119911)

det 119869119894= 119889111988921205822

119894minus (119860 (119901 119911) 119889

2+ 119863 (119901 119911) 119889

1) 120582119894+ det 119869

(119901119911)

(24)

(1) If (119901 119911) = (0 0) then 119869(00)= (119903sdot(minus(119898119899)) 0

0 minus120583)

trace119869119894= minus120582119894(1198891+ 1198892) minus (

119903119898

119899+ 120583) lt 0

det 119869119894= 119889111988921205822

119894+ (119903119898

1198991198892+ 1205831198891)120582119894+119903119898120583

119899gt 0

(25)

This implies that (0 0) is locally asymptotically stable

(2) If (119901 119911) = (119898 0) then 119869(1198980)

=

(119903sdot(1minus(119898119870))(119898(119898+119899)) minus119886119898

0 119890119886119898minus120583)

For 119894 = 0 one of the eigenvalues is 119903(1 minus (119898119870))(119898(119898 +119899)) gt 0 which implies that (119898 0) is an unstable point

(3) If (119901 119911) = (119870 0) then 119869(1198700)= (minus119903((119870minus119898)(119870+119899)) minus119886119870

0 119890119886119870minus120583)

When 119890119886119870 minus 120583 lt 0

trace119869119894= minus120582119894(1198891+ 1198892) minus 119903119870 minus 119898

119870 + 119899+ 119890119886119870 minus 120583 lt 0

det 119869119894= 119889111988921205822

119894+ (minus119903

119870 minus 119898

119870 + 1198991198892+ (119890119886119870 minus 120583) 119889

1)120582119894

minus 119903119870 minus 119898

119870 + 119899(119890119886119870 minus 120583) gt 0

(26)

This implies that (119870 0) is locally asymptotically stableWhen 119890119886119870 minus 120583 gt 0 for 119894 = 0 det 119869

119894= minus119903((119870 minus 119898)(119870 +

119899))(119890119886119870 minus 120583) lt 0 which implies that 119869119894has at least one

eigenvalue with positive real part This implies that (119870 0) isunstable

(4) If (119901 119911) = (119906 V119906) then 119869

(119906V119906)= (119860(119906) 119861(119906)

119862(119906) 0) where

119860 (119906) = 119903 (1 minus119906

119870)119906119898 + 119899

(119906 + 119899)2minus119903

119870119906119906 minus 119898

119906 + 119899 119861 (119906) = minus

120583

119890

119862 (119906) = 119890119903 (1 minus119906

119870)119906 minus 119898

119906 + 119899

(27)

and V gt 0 implies that119898 lt 119906 lt 119870 Let 119906 be the largest root of119860(119906) = 0

When 119906 lt 119906 lt 119870 then 119860(119906) lt 0 and

trace (119869119894) = minus120582

119894(1198891+ 1198892) + 119860 (119906) lt 0

det 119869119894= 119889111988921205822

119894minus 119860 (119906) 119889

2120582119894minus 119861 (119906) 119862 (119906) gt 0

(28)

This implies that (119906 V119906) is locally asymptotically stable

When 119898 lt 119906 lt 119906 then 119860(119906) gt 0 and trace (119869119894) = 119860(119906) gt

0 for 119894 = 0 which implies that 119869119894has at least one eigenvalue

with a positive real part This implies that (119906 V119906) is unstable

Theorem 4 If119870119898+119898119899+119870119899minus119899(120583119890119886) lt 0 then the positiveconstant steady state (119906 V

119906) is globally asymptotically stable

Proof Let us consider a Lyapunov function 119881 as

119881 = intΩ

119882(119901 (119909 119905) 119911 (119909 119905)) 119889119909 (29)

where

119882(119901 (119909 119905) 119911 (119909 119905)) = 1198881int

119901

119906

119901 minus 119906

119901119889119901 + 119888

2int

119911

V119906

119911 minus V119906

119911119889119911

(30)

and 119888119894gt 0 (119894 = 1 2) will be determined next


By taking the time derivative of 119881 we have

119889119881

119889119905= intΩ

(119882119901sdot 119901119905+119882119911sdot 119911119905) 119889119909

= intΩ

(1198881

119901 minus 119906

119901

120597119901

120597119905+ 1198882

119911 minus V119906

119911

120597119911

120597119905) 119889119909

= intΩ

(1198881(119901 minus 119906) [119903 (1 minus

119909

119870)119901 minus 119898

119901 + 119899minus 119886119911]

+ 1198882(119911 minus V

119906) 119890119886 (119901 minus 119906) ) 119889119909

+ intΩ

(1198881

119901 minus 119906

1199011198891Δ119901 + 119888

2

119911 minus V119906

1199111198892Δ119911)119889119909

= 1198681(119905) + 119868

2(119905)

(31)

Due to the Neumann boundary condition it can easily bederived that

1198682(119905) = minusint

Ω

(11988811198891

119906

11990121003816100381610038161003816nabla11990110038161003816100381610038162

+ 11988821198892

V119906

1199112|nabla119911|2

)119889119909 le 0 (32)

Further

1198681(119905)

= intΩ

(1198881(119901 minus 119906) [119903 (1 minus

119901

119870)119901 minus 119898

119901 + 119899minus 119903 (1 minus

119906

119870)119906 minus 119898

119906 + 119899

+119886V119906minus 119886119911] + 119888

2119890119886 (119901 minus 119906) (119911 minus V

119906) ) 119889119909

= intΩ

(1198881(119901 minus 119906)

2 119903

119870[minus119901119906 + 119870119898 minus 119899 (119901 + 119906) + 119899 (119898 + 119870)

(119901 + 119899) (119906 + 119899)]

+ (1198901198861198882minus 1198861198881) (119901 minus 119906) (119911 minus V

119906) ) 119889119909

(33)

If we choose 1198882gt 0 arbitrarily and 119888

1= 1198901198882 thenwe can obtain

119889119881

119889119905= 1198681(119905) + 119868

2(119905) le 119868

1(119905)

= intΩ

1198881

119903

119870[minus119901119906 + 119870119898 + 119899 (119898 + 119870) minus 119899 (119901 + 119906)

(119901 + 119899) (119906 + 119899)]

times (119901 minus 119906)2

119889119909

le intΩ

1198881

119903

119870

119870119898 + 119898119899 + 119870119899 minus 119899119906

119899 (119906 + 119899)(119901 minus 119906)

2

119889119909

(34)

Therefore if119870119898 + 119898119899 + 119870119899 minus 119899119906 lt 0 then 119889119881119889119905 le 0 and119889119881119889119905 = 0 if 119901 = 119906 119911 = V

119906 This completes the proof

3 Bifurcation Analysis

In this section we mainly analyze the stability of the steadystate (119906 V

119906) and take 119906 as the bifurcation parameter (or

equivalently take 119886 as a parameter) In particular we assumethat all of the eigenvalues of minusΔ are simple

We know from the proof ofTheorem 2 that the stability of(119906 V119906) is determined by the trace and determinant of 119869

119894 Let

119879 (119906 120582) = 119860 (119906) minus 120582 (1198891+ 1198892)

119863 (119906 120582) = 119889111988921205822

minus 1198892119860 (119906) 120582 minus 119861 (119906) 119862 (119906)

(35)

We refer to (119906 120582) isin 1198772+ 119879(119906 120582) = 0 as the Hopf bifurcation

curve and (119906 120582) isin 1198772+ 119863(119906 120582) = 0 as the steady state

bifurcation curve [38] (Figures 4 and 5)First 119879(119906 120582) = 0 is equal to 120582 = (119860(119906)(119889

1+1198892)) We can

summarize the properties of 119860(119906) as follows which are easyto prove so the proof is omitted

Lemma 5 Consider119860(119906) = (119903119906119870(119906+119899)2)[minus1199062minus2119899119906+119898119899+119898119870 + 119899119870] then 0 lt 119906lowast lt 119906 lt 119870 exists such that the followinghold

(1) If 119898119899 + 119898119870 + 119899119870 ge 0 then 119860(119906) gt 0 in (0 119906) and119860(0) = 119860(119906) = 0

(2) If 119898119899 + 119898119870 + 119899119870 lt 0 then exist119906119888isin (0 119906

lowast

) such that119860(119906) gt 0 in (119906

119888 119906) 119860(119906) lt 0 in (0 119906

119888) and 119860(0) =

119860(119906119888) = 119860(119906) = 0

(3) 1198601015840(119906) gt 0 in (max0 119906119888 119906lowast

) 1198601015840(119906) gt 0 in (119906lowast 119906)and 1198601015840(119906lowast) = 0

Second 119863(119906 120582) = 0 is equal to 120582plusmn(119906) = (119889

2119860(119906) plusmn

radic119889221198602(119906) + 4119889

11198892119861(119906)119862(119906))2119889

11198892 Let

119878 (119906) = 1198892

21198602

(119906) + 411988911198891119861 (119906) 119862 (119906) (36)

Since

119878 (119898) = 1198892

21198602

(119906) gt 0 119878 (119906) = 411988911198892119861 (119906) 119862 (119906) lt 0

(37)

according to the continuity of 119878(119906) there exists a 119906119904isin (119898 119906)

such that 119878(119906119904) = 0 Thus we can summarize the properties of

120582 as follows

Lemma 6 If 119906lowast le 119898 then 120582+(119906) is decreasing and 120582

minus(119906) is

increasing in 119906 isin (119898 120582119904)

Proof Differentiating119863(119906 120582) = 0 with respect to 119906 we have

211988911198892120582 (119906) 120582

1015840

(119906) minus 11988921198601015840

(119906) 120582 (119906)

minus 1198892119860 (119906) 120582

1015840

(119906) minus 119861 (119906) 1198621015840

(119906) = 0

(38)

Therefore 1205821015840(119906) = (119861(119906)1198621015840(119906) + 11988921198601015840

(119906)120582(119906))1198892(21198891120582(119906) minus

119860(119906)) From the definition of 120582plusmn(119906) when 119906 isin (119898 119906

119904)

1198892(21198891120582+(119906) minus 119860(119906)) gt 0 and 119889

2(21198891120582minus(119906) minus 119860(119906)) lt 0


However 119862(119906) = 119890119903(1 minus (119906119870))(1 minus ((119898 + 119899)(119906 + 119899)))1198621015840

(119906) = 119890119903(minus1199062

minus2119899119906+119898119899+119898119870+119899119870)119870(119906+119899)2 and 1198621015840(119906) =

0 imply that 119906 = = (2119899 plusmn radic41198992 + 4(119898119899 + 119898119870 + 119899119870)) minus 2and

119860 () = 119903 (1 minus

119870) 119898 + 119899

( + 119899)2minus119903

119870 minus 119898

+ 119899

=119903

119870( + 119899)2[minus2

minus 2119899 + 119898119899 + 119899119870 + 119898119870] = 0

(39)

Since 119860(119898) gt 0 and 119898 lt lt 119870 then = 119906 Therefore119861(119906) lt 0 1198621015840(119906) gt 0 and 120582

plusmn(119906) gt 0 when 119906 isin (119898 120582

119904) If

119906lowast

le 119898 then 1198601015840(119906) le 0 for 119906 isin (119898 120582119904) Thus 119861(119906)1198621015840(119906) +

11988921198601015840

(119906)120582plusmn(119906) lt 0 This implies that 1205821015840

+(119906) lt 0

31 Hopf Bifurcation Analysis In this section we mainlyanalyze the properties of the Hopf bifurcation for system (4)According to [38] we know that a Hopf bifurcation point 119906must satisfy the following conditions

(A) There exists 119894 (119894 = 0 1 2 ) such that

119879119894(119906) = 0 119863

119894(119906) gt 0 119879

119895(119906) = 0 119863

119895(119906) = 0

for 119895 = 119894(40)

and the unique pair of complex eigenvalues 120572(119906) plusmn119894120596(119906) 120572(119906) = 0 120596(119906) gt 0 exist and are continuouslydifferentiable in 119906 with 1205721015840(119906) = 0

Theorem 7 If 119906lowast le 119898 holds then 119896 isin 119873 exists such that thesystem (4) undergoes a Hopf bifurcation at 119906 = 119906

119894(0 le 119894 le 119896)

and a smooth curve of positive periodic orbits of (4) bifurcatesfrom (119906 119901 119911) = (119906

119894 119906119894 V119906119894

) The bifurcating periodic orbitsfrom 119906 = 119906 are spatially homogeneous and theHopf bifurcationat 119906 = 119906 is supercritical and backward if 119899 gt 119906

Proof It can be verified that 119879119894(119906) lt 0 and 119863

119894(119906) gt 0 for

119906 isin (119906119870) which implies that the potential Hopf bifurcationpoint must be in (119898 119906] However 119879

0(119906) = 0 119863

0(119906) gt 0 and

119879119895(119906) = minus(119889

1+ 1198892)120582119895lt 0 119863

119895(119906) gt 0 for 119895 ge 1 Therefore

1199060= 119906 is a Hopf bifurcation point If 119906lowast le 119898 holds we know

that119860(119906) decreases strictly in (119898 119906) For every 119894 gt 0 let 119906119894be

the solution of 119860(119906) = (1198891+ 1198892)120582119894 so we have

119898 lt 119906ℎlt 119906ℎminus1lt sdot sdot sdot lt 119906

1lt 119906 (41)

where 120582ℎis the largest eigenvalue for 120582

119894lt 119860(119906)(119889

1+ 1198892)

Therefore 119879119894(119906119894) = 0 and 119879

119895(119906119894) = 0 for 119895 = 119894 1 le 119894 le ℎ

Geometrically from 119879119894(119906) = 0 we can determine that 120582 is

a parabola of 119906 with 120582(119906) = 120582(119906119888) = 120582(0) = 0 in (119906 120582)

coordinate systemHowever119879(119898 120582) = 0 implies that 120582(119898) =119860(119898)(119889

1+ 1198892) = 119903(1 minus (119898119870))(119898(119898+ 119899)) gt 0119863(119898 120582) = 0

implies that 120582 = 0 or 120582(119898) = 119860(119898)1198891gt 119860(119898)(119889

1+ 1198892)

and Lemma 6 implies that 120582+(119906) is decreasing and 120582

minus(119906)

is increasing in 119906 isin (119898 120582119904) if 119906lowast le 119898 Thus the curves

119879(119898 120582) = 0 and 119863(119898 120582) = 0 have only one intersectionpoint which is noted as (119906

119897 120582119897) in119906 isin (119898 119906)Then119863

119894(119906119894) gt 0

if 119906119894gt 119906119887and 119863

119894(119906119894) lt 0 if 119906

119894lt 119906119887 1 le 119894 le ℎ According to

Theorem 21 in [13] a smooth curve of positive periodic orbitsof (4) bifurcates from (119906 119901 119911) = (119906

119894 119906119894 V119906119894

) According toTheorem 31 in [8]

119886 (119906) =11990611989110158401015840

(119906)

16(119891101584010158401015840

(119906)

11989110158401015840 (119906)+2

119906)

=minus119906 (119899 minus 119906) (2119899

2

+ 119898119899 + 119870119899 + 119870119898)

8(119906 + 119899)5

lt 0

(42)

if 119899 gt 119906 holds

32 Steady State Bifurcation Analysis In this section weconsider the steady state bifurcation of system (4) Thenonnegative steady state solutions of (4) satisfy the followingsystem

minus1198891Δ119901 = 119903 (1 minus

119901

119870)(1 minus

119898 + 119899

119901 + 119899)119901 minus 119886119901119911 119909 isin Ω

minus1198892Δ119911 = 119890119886119901119911 minus 120583119911 119909 isin Ω

120597119901

120597119899=120597119911

120597119899= 0 119909 isin 120597Ω

(43)

Apparently (43) has spatially homogeneous solutions (0 0)(119870 0) (119898 0) (119906 V

119906) First we discuss the nonnegative steady

state solutions of (4) Recall the maximum principle [13]

Lemma 8 Let Ω be a bounded Lipschitz domain in 119877119899 andlet 119892 isin 119862(Ω times 119877) If 119910 isin 11988212(Ω) is a weak solution of theinequalities

Δ119910 + 119892 (119909 119910 (119909)) ge 0 119894119899 Ω120597119910 (119909)

120597119899le 0 119900119899 120597Ω

(44)

and if there is a constant119872 such that 119892(119909 119910) lt 0 for 119910 gt 119872then 119910 le 119872 ae in Ω

From Lemma 8 it can easily be derived that all nontrivialsolutions of equation

119903 (1 minus119901

119870)(1 minus

119898 + 119899

119901 + 119899)119901 + 119889

1Δ119901 = 0 119909 isin Ω

120597119901

120597119899= 0 119909 isin 120597Ω

(45)

satisfy 0 le 119901(119909) le 119870

Theorem 9 The solutions of system (43) are in the form ofeither (119901(119909) 0) or (119901(119909) 119911(119909)) satisfying

0 lt 119901 (119909) lt 119870 0 lt 119911 (119909) lt119890119903

119899120583(119870 minus 119898

2)

2

minus1198891119870

2≜ 119871

(46)

Proof If 1199090isin Ω exists such that 119911(119909

0) = 0 from the strong

maximum principle and the boundary condition 120597119911120597119899 = 0


we can derive 119911(119909) equiv 0 in Ω so 119901(119909) satisfies (45) Similarlyif 1199090isin Ω exists such that 119901(119909

0) = 0 we can derive 119901(119909) equiv

0 and consequently 119911(119909) equiv 0 If for all 119909 isin Ω 119901(119909) gt 0and 119911(119909) gt 0 hold according to the above discussion andthe strong maximum principle we have 0 lt 119901(119909) lt 119870 for all119909 isin Ω After adding the two equations in (43) we have

minus (1198901198891Δ119901 + 119889

2Δ119911)

= 119890119903 (1 minus119901

119870)(1 minus

119898 + 119899

119901 + 119899)119901 minus 120583119911

= 119890119903 (1 minus119901

119870)(1 minus

119898 + 119899

119901 + 119899)119901 minus

120583

1198892

(1198891119901 + 1198892119911) +

1205831198891

1198892

119901

le [119890119903

119870119899(119870 minus 119898

2)

2

minus1205831198891

1198892

]119870 minus120583

1198892

(1198891119901 + 1198892119911)

(47)

The maximum principle and Green formula imply that

1198892119911 lt 1198891119901 + 1198892119911 lt1198892

120583[119890119903

119870119899(119870 minus 119898

2)

2

minus1205831198891

2]119870 (48)

Therefore we can show the nonexistence of positive steadystate solutions when the diffusion coefficients are large

Theorem 10 Let 1198631gt (1198702

+ 119898119870 minus 119898119899)1198701198992

1205821be a fixed

constantThen another constant1198632exists such that if 119889

1ge 1198631

and 1198892ge 1198632hold (43) has no nonconstant positive solution

Proof Assume that (119901(119909) 119911(119909)) is a positive solution of (43)For convenience we denote

1198911(119901 119911) = 119903 (1 minus

119901

119870)(1 minus

119898 + 119899

119901 + 119899)119901 minus 119886119901119911

1198912(119901 119911) = 119890119886119901119911 minus 120583119911

(49)

Let

119901 =1

|Ω|intΩ

119901 (119909) 119889119909

119911 =1

|Ω|intΩ

119911 (119909) 119889119909

(50)

then intΩ

(119901minus119901)119889119909 = 0 intΩ

(119911minus119911)119889119909 = 0 Bymultiplying (119901minus119901)by the first equation in (43) and then integrating on Ω wehave

intΩ

1198891

1003816100381610038161003816nabla11990110038161003816100381610038162

119889119909

= intΩ

1198911(119901 119911) (119901 minus 119901) 119889119909

= intΩ

(1198911(119901 119911) minus 119891

1(119901 119911)) (119901 minus 119901) 119889119909

= intΩ

((minus119901119901 (119901 + 119901) (119901 minus 119901)2

+ (119870 + 119898) 119901119901(119901 minus 119901)2

minus 119870119898119899 (119901 minus 119901))

times ((119901 + 119899) (119901 + 119899))minus1

minus119886119901 (119901 minus 119901) (119911 minus 119911) minus 119886119911(119911 minus 119911)2

) 119889119909

le intΩ

((119870 + 119898119870 minus 119898119899)119870

1198992(119901 minus 119901)

2

+1198861198701003816100381610038161003816119901 minus 119901

1003816100381610038161003816 |119911 minus 119911| ) 119889119909

(51)

Similarly

intΩ

1198892|nabla119911|2

119889119909

= intΩ

1198912(119901 119911) (119911 minus 119911) 119889119909

= intΩ

(1198912(119901 119911) minus 119891

2(119901 119911)) (119911 minus 119911) 119889119909

= intΩ

((119890119886119901 minus 120583) (119911 minus 119911)2

minus 119890119886119911 (119901 minus 119901) (119911 minus 119911)) 119889119909

le intΩ

((119890119886119870 minus 120583) (119911 minus 119911)2

+ 1198901198861198711003816100381610038161003816119901 minus 119901

1003816100381610038161003816 |119911 minus 119911|) 119889119909

(52)

where 119871 is defined inTheorem 9Therefore

intΩ

(1198891

1003816100381610038161003816nabla11990110038161003816100381610038162

+ 1198892|nabla119911|2

) 119889119909

le intΩ

((1198702

+ 119898119870 minus 119898119899)119870

1198992(119901 minus 119901)

2

+ (119886119870 + 119890119886119871)1003816100381610038161003816119901 minus 119901

1003816100381610038161003816 |119911 minus 119911|

+ (119890119886119870 minus 120583) (119911 minus 119911)2

)119889119909

le intΩ

(((1198702

+ 119898119870 minus 119898119899)119870

1198992+119886119870 + 119890119886119871

2120576) (119901 minus 119901)

2

+(119890119886119871 + 119886119870

2120576+ 119890119886119870 minus 120583) (119911 minus 119911)

2

)119889119909

(53)

Using the Poincare inequality we can obtain

intΩ

(11988911205821

1003816100381610038161003816119901 minus 11990110038161003816100381610038162

+ 11988921205821|119911 minus 119911|

2

) 119889119909

le intΩ

(((1198702

+ 119898119870 minus 119898119899)119870

1198992+119886119870 + 119890119886119871

2120576) (119901 minus 119901)

2

+ (119890119886119871 + 119886119870

2120576+ 119890119886119870 minus 120583) (119911 minus 119911)

2

)119889119909

(54)


Under the assumption 11988911205821gt (119870

2

+ 119898119870 minus 119898119899)1198701198992 a

sufficiently small 120576 gt 0 exists such that 11988911205821ge (1198702

+ 119898119870 minus

119898119899)1198701198992

+((119886119870+119890119886119871)2)120576 Let1198632= ((119890119886119871+119886119870)2120576+119890119886119870minus

120583)(11205821) then we can have 119901 = 119901 119911 = 119911 This completes the

proof

We assume that all of the eigenvalues of minusΔ are simpleand the corresponding eigenfunctions are denoted by 120601

119894(119909)

Reference [13] gives an example of 119899 = 1 with 120582119894= 1198942

1198972 and

120601119894(119909) = cos(119894119909119897) Let

119879119894(119906) = minus120582

119894(1198891+ 1198892) + 119860 (119906)

119863119894(119906) = 119889

111988921205822

119894minus 119860 (119906) 119889

2120582119894minus 119861 (119906) 119862 (119906)

(55)

According to [13] we know that a steady state bifurcationpoint 119906must satisfy the following conditions

(B) 119894 (119894 = 0 1 2 ) exists such that

119863119894(119906) = 0 119879

119894(119906) = 0 119863

119895(119906) = 0 119879

119895(119906) = 0

for 119895 = 119894

119889119863119894(119906)

119889119906= 0

(56)

Theorem 11 1198961015840 isin 119873 exists such that system (43) undergoes asteady state bifurcation at 119906 = 119906

119894(1 le 119894 le 119896

1015840

) if 119906lowast le 119898 holds

Proof Apparently (B) is not established for 119894 = 0 It is knownthat 119863

119894(119906) is a degree 3 polynomial of 119906 and there are at

most three 119906119894for 119863

119894(119906119894) = 119863(119906

119894 120582119894) = 0 In particular if

the parameters are selected such that 119906lowast le 119898 holds thenfor each 119894 gt 0 there exists a unique 119906

119894isin (119898 119906

119904) such that

119863119894(119906119894) = 119863(119906

119894 120582119894) = 0 Thus there is at most one bifurcation

point However 1198631015840119894(119906) = minus119860(119906)119889

2120582119894minus 119861(119906)119862

1015840

(119906) From theproof of Lemma 6 we know that 119861(119906)1198621015840(119906)+119889

21198601015840

(119906)120582plusmn(119906) lt

0 for 119906 isin (119898 120582119904) Therefore 119889119863

119894(119906)119889119906 = 0 holds if 119906lowast le

119898

4 Discussion

Reaction-diffusion phytoplankton-zooplankton models withAllee effects have been studied extensively in recent years Inthis study we rigorously considered a Gause-type predator-prey model with a double Allee effect on prey which wasformulated as (4) It is known that the predator-prey modelwith the most usual form of Allee effect has a unique limitcycle but the existence of two limit cycles was proved byGonzalez-Olivares et al [29] with a double Allee effectThus the double Allee effect produces different results withdifferent mathematical expressions

The paper [15] found that the system without Allee effectwas always stable and without fluctuations but in this paperthe results of the stability of the equilibrium and the bifur-cation analysis based on a rigorous theoretical analysis showthat this system has complex spatiotemporal dynamics for1199010(119909) le 119898 the phytoplankton is destined to become extinct

and leads to the extinction of zooplankton after considering

79998

79999

8

80001

80002

80003

80004

80005

80006

80007

80008

0 20 40 60 80 100 120

(a)

23076

23077

23077

23078

23078

23079

23079

2308

2308

0 20 40 60 80 100 120

(b)

Figure 3 Time evolution of (4) around the interior equilibrium(8 230769) Phytoplankton is in (a) and zooplankton is in (b) Theparameter values are 119889

1= 0001 119889

2= 0004 119890 = 05 119886 = 01 119899 = 5

119898 = 05 120583 = 04 119903 = 2 and 119870 = 10 The figure shows that bothtypes of plankton exhibit stable behavior

the strong Allee effect in phytoplankton extinction for bothspecies is always a locally stable equilibrium But for 119906 gt119898 which is the condition in which the interior equilibriumexists the interior equilibrium is globally stable in some caseand there always exist some other spatiotemporal patterns inother cases (Figure 3)

Overall our results indicate that the impact of the Alleeeffect increases the spatiotemporal complexity of the systemThe mathematical form which expresses the double Alleeeffect has a strong impact of the dynamics of system Thuswe think it is important for ecologists to be aware of thedifference of the selection on the forms of Allee effect

The limitations of our study are that we only considera simplest phytoplankton-zooplankton interaction and thespecial formalization to describe the Allee effect What ismore is that compared with the ODE dynamics the resultsshown here are still coarse Therefore further research is still


120582

u

D(u 120582) = 0

T(u 120582) = 0

900

800

700

600

500

400

300

200

100

0

0 1 2 3 4 5 6 7

Figure 4 The graph of 119879(119906 120582) = 0 and 119863(119906 120582) = 0 where 1198891=

0001 1198892= 0004 119890 = 05 119886 = 1 119899 = 5 119898 = 5 120583 = 04 119903 = 2 and

119870 = 10 In this case 119906lowast le 119898

120582

u

D(u 120582) = 0

T(u 120582) = 0

0 1 2 3 4

300

200

100

0


0001 1198892= 0004 119890 = 05 119886 = 1 119899 = 5 119898 = 1 120583 = 04 119903 = 2 and

119870 = 10 In this case 119906lowast gt 119898

needed to elaborate a general theory on the influence of thisecological phenomenon

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (Grant no 31170338) by the KeyProgram of Zhejiang Provincial Natural Science Foundationof China (Grant no LZ12C03001) and by the National KeyBasic Research Program of China (973 Program Grant no2012CB426510)

References

[1] P Shen and Z Peng Microbiology Higher Education PressBeijing China 2003

[2] E Beltrami and T O Carroll ldquoModeling the role of viraldiseases in recurrent phytoplankton bloomsrdquo Journal of Mathe-matical Biology vol 32 no 8 pp 857ndash863 1994

[3] A M Edwards and J Brindley ldquoZooplankton mortality and thedynamical behaviour of plankton population modelsrdquo Bulletinof Mathematical Biology vol 61 no 2 pp 303ndash339 1999

[4] A Zingone D Sarno and G Forlani ldquoSeasonal dynamics inthe abundance ofMicromonas pusilla (Prasinophyceae) and itsviruses in the Gulf of Naples (Mediterranean Sea)rdquo Journal ofPlankton Research vol 21 no 11 pp 2143ndash2159 1999

[5] J E Truscott and J Brindley ldquoOcean plankton populations asexcitable mediardquo Bulletin of Mathematical Biology vol 56 no5 pp 981ndash998 1994

[6] C S Hollling ldquoThe components of predation as revealed by astudy of small mammal predation of the European pine sawflyrdquoThe Canadian Entomologist vol 91 no 5 pp 293ndash329 1959

[7] S Ruan and D Xiao ldquoGlobal analysis in a predator-prey systemwith nonmonotonic functional responserdquo SIAM Journal onApplied Mathematics vol 61 no 4 pp 1445ndash1472 2001

[8] J Wang J Shi and J Wei ldquoPredator-prey system with strongAllee effect in preyrdquo Journal of Mathematical Biology vol 62no 3 pp 291ndash331 2011

[9] P J Pal T Saha M Sen and M Banerjee ldquoA delayed predator-prey model with strong Allee effect in prey population growthrdquoNonlinear Dynamics vol 68 no 1-2 pp 23ndash42 2012

[10] M Haque ldquoA detailed study of the Beddington-DeAngelispredator-prey modelrdquoMathematical Biosciences vol 234 no 1pp 1ndash16 2011

[11] J Huang G Lu and S Ruan ldquoExistence of traveling wavesolutions in a diffusive predator-prey modelrdquo Journal of Mathe-matical Biology vol 46 no 2 pp 132ndash152 2003

[12] A B Medvinsky S V Petrovskii I A Tikhonova H Malchowand B-L Li ldquoSpatiotemporal complexity of plankton and fishdynamicsrdquo SIAM Review vol 44 no 3 pp 311ndash370 2002

[13] F Yi JWei and J Shi ldquoBifurcation and spatiotemporal patternsin a homogeneous diffusive predator-prey systemrdquo Journal ofDifferential Equations vol 246 no 5 pp 1944ndash1977 2009

[14] E R Abraham ldquoThe generation of plankton patchiness byturbulent stirringrdquoNature vol 391 no 6667 pp 577ndash580 1998

[15] R Bhattacharyya and B Mukhopadhyay ldquoModeling fluctua-tions in aminimal planktonmodel role of spatial heterogeneityand stochasticityrdquo Advances in Complex Systems vol 10 no 2pp 197ndash216 2007

[16] J Gascoigne and R N Lipcius ldquoAllee effects inmarine systemsrdquoMarine Ecology Progress Series vol 269 pp 49ndash59 2004

[17] P A Stephens W J Sutherland and R P Freckleton ldquoWhat isthe Allee effectrdquo Oikos vol 87 no 1 pp 185ndash190 1999


[18] AM Kramer B Dennis A M Liebhold and J M Drake ldquoTheevidence for Allee effectsrdquo Population Ecology vol 51 no 3 pp341ndash354 2009

[19] W Z Lidicker Jr ldquoThe Allee effect its history and futureimportancerdquoThe Open Ecology Journal vol 3 pp 71ndash82 2010

[20] M R Owen andM A Lewis ldquoHow predation can slow stop orreverse a prey invasionrdquo Bulletin of Mathematical Biology vol63 no 4 pp 655ndash684 2001

[21] F Courchamp L Berec and J Gascoigne ldquoAllee effects inecology and conservationrdquo Environmental Conservation vol36 no 1 pp 80ndash85 2008

[22] W C Allee Animal Aggregations A Study in General SociologyUniversity of Chicago Press Chicago Ill USA AMS Press NewYork NY USA 1931

[23] S Ruan ldquoPersistence and coexistence in zooplankton-phytoplankton-nutrient models with instantaneous nutrientrecyclingrdquo Journal of Mathematical Biology vol 31 no 6 pp633ndash654 1993

[24] Y Zhu Y Cai S Yan and W Wang ldquoDynamical analysis of adelayed reaction-diffusion predator-prey systemrdquo Abstract andApplied Analysis vol 2012 Article ID 323186 23 pages 2012

[25] F Courchamp T Clutton-Brock and B Grenfell ldquoInversedensity dependence and the Allee effectrdquo Trends in Ecology andEvolution vol 14 no 10 pp 405ndash410 1999

[26] L Berec E Angulo and F Courchamp ldquoMultiple Allee effectsand population managementrdquo Trends in Ecology and Evolutionvol 22 no 4 pp 185ndash191 2007

[27] E Angulo G W Roemer L Berec J Gascoigne and FCourchamp ldquoDouble Allee effects and extinction in the islandfoxrdquo Conservation Biology vol 21 no 4 pp 1082ndash1091 2007

[28] D S Boukal and L Berec ldquoSingle-species models of the Alleeeffect extinction boundaries sex ratios and mate encountersrdquoJournal of Theoretical Biology vol 218 no 3 pp 375ndash394 2002

[29] E Gonzalez-Olivares B Gonzalez-Yanez J M Lorca A Rojas-Palma and J D Flores ldquoConsequences of double Allee effect onthe number of limit cycles in a predator-preymodelrdquoComputersamp Mathematics with Applications vol 62 no 9 pp 3449ndash34632011

[30] G A K van Voorn L Hemerik M P Boer and B W KooildquoHeteroclinic orbits indicate overexploitation in predator-preysystems with a strong Allee effectrdquo Mathematical Biosciencesvol 209 no 2 pp 451ndash469 2007

[31] M-H Wang and M Kot ldquoSpeeds of invasion in a model withstrong or weak Allee effectsrdquoMathematical Biosciences vol 171no 1 pp 83ndash97 2001

[32] E Gonzalez-Olivares and A Rojas-Palma ldquoMultiple limitcycles in a Gause type predator-prey model with Holling typeIII functional response and Allee effect on preyrdquo Bulletin ofMathematical Biology vol 73 no 6 pp 1378ndash1397 2011

[33] M Liermann and R Hilborn ldquoDepensation evidence modelsand implicationsrdquo Fish and Fisheries vol 2 no 1 pp 33ndash582001

[34] S-R Zhou Y-F Liu and G Wang ldquoThe stability of predator-prey systems subject to the Allee effectsrdquoTheoretical PopulationBiology vol 67 no 1 pp 23ndash31 2005

[35] J Zu andMMimura ldquoThe impact of Allee effect on a predator-prey system with Holling type II functional responserdquo AppliedMathematics and Computation vol 217 no 7 pp 3542ndash35562010

[36] Q Ye and Z Li Introduction to Reaction-Diffusion EquationsScience Press Beijing China 1994

[37] D Henry Geometric Theory of Semilinear Parabolic Equationsvol 840 of Lecture Notes in Mathematics Springer New YorkNY USA 1981

[38] JWang J Shi and JWei ldquoDynamics and pattern formation in adiffusive predator-prey system with strong Allee effect in preyrdquoJournal of Differential Equations vol 251 no 4-5 pp 1276ndash13042011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


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OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


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Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


is the conversion coefficient and 120583 represents the per capitapredator mortality which is assumed to have a linear formalthough other forms are possible [9 10]The global dynamicsof model (1) with a logistic growth rate have been studiedduring the last three decades based on theoretical analysis andnumerical simulations and many results have been reported[11ndash15]

In recent years the Allee effect has been the focus ofincreasing interest and it is recognized to be an importantphenomenon in many fields of ecology and conservationbiology by more and more people [16ndash21] The Allee effectis named after WC Allee [22] and it describes a positivecorrelation between the density or number of populationand individual fitness of population [16] Standard populationmodels assume that the fitness of population increases as thepopulation density or size declines [11ndash15 23 24] whereasAllee effect states that when a population is below a criticaldensity or size the population cannot sustain itself and thisleads to extinction Thus the Allee effect increases the likeli-hood of extinction [25] Stephens et al distinguished betweena component Allee effect and a demographic Allee effect [17]However conservation biologists are usually more interestedin the demographic Allee effect because it ultimately governsthe probability of the extinction or recovery of populationswith low abundances [16]

Very recent ecological research has shown that two ormore Allee effects can act on a single population simultane-ously which is known as the multiple (double) Allee effect[26 27]

There are many ways of describing the Allee effect [28]including the following differential equation

119889119901

119889119905= 119903 (1 minus

119901

119870)(1 minus

119898 + 119899

119901 + 119899)119901 (2)

where 119903 describes the growth rate and 119899 is an auxiliaryparameter where 119899 gt 0 119898 + 119899 gt 0 Indeed it is consideredthat 119901 represents the size of a fertile population and 119899 is thenonfertile population such as juvenile or oldest individuals[29] In this case 119898 (minus119870 lt 119898 lt 119870) is called theAllee threshold because when the population density orsize is below this threshold the population is destined forextinctionWhen119898 gt 0 (2) describes a strong Allee effect [830 31] In this case the population growth rate decreases if thepopulation size is below the threshold 119898 and the populationgoes to extinction [29] In addition (2) describes a weakAlleeeffect [29 31ndash33] for119898 lt 0

It is obvious that (2) is equal to

119889119901

119889119905=119903119901

119901 + 119899(1 minus

119901

119870) (119901 minus 119898) (3)

Gonzalez-Olivares et al [29] state that (3) describes a doubleAllee effect that is once in the factor 119898(119901) = 119901 minus 119898 and thesecond time in the term 119903(119901) = 119903119901(119901 + 119899) [34 35]

In a marine environment the plankton populations tendto move in horizontal and vertical directions due to thestrong water current This movement is usually modeled bya reaction-diffusion equation In this study we consider thefollowing reaction-diffusion model with constant diffusion

coefficient as well as a strong Allee effect in different spatiallocations within a fixed smooth bounded domainΩ isin 119877119899Weassume that the response function of the zooplankton followsthe law of mass action [15]

120597119901 (119909 119905)

120597119905= 119903 (1 minus

119901 (119909 119905)

119870)(1 minus

119898 + 119899

119901 (119909 119905) + 119899)119901 (119909 119905)

minus 119886119901 (119909 119905) 119911 (119909 119905) + 1198891Δ119901 (119909 119905) 119909 isin Ω 119905 gt 0

120597119911 (119909 119905)

120597119905= 119890119886119901 (119909 119905) 119911 (119909 119905) minus 120583119911 (119909 119905) + 119889

2Δ119911 (119909 119905)

119909 isin Ω 119905 gt 0

120597119901 (119909 119905)

120597119899=120597119911 (119909 119905)

120597119899= 0 119909 isin 120597Ω 119905 gt 0

119901 (119909 0) = 1199010(119909) ge 0 119911 (119909 0) = 119911

0(119909) ge 0 119909 isin Ω

(4)

where 119898 (0 lt 119898 lt 119870) is the Allee threshold 1198891 1198892are

the diffusion coefficients of phytoplankton and zooplanktonrespectively Δ is the Laplacian operator and 119909 is the spatialhabitat of two species andwe assume that the system is a closeecosystem and with a no-flux boundary condition

This paper is structured as follows In Section 2 weanalyze the basic dynamics of (4) including estimates of thesolution and the local and global stability of equilibria InSection 3 we provide the analysis of the Hopf bifurcation andthe steady state bifurcation A brief discussion and summaryare given in Section 4

2 Main Results

21 Basic Dynamics Suppose that 119899 119886 120583 1198891 1198892gt 0 0 lt 119890 lt

1 0 lt 119898 lt 119870 and Ω is a bounded domain then we obtainthe following results

Theorem 1 The system (4) has a unique solution and thesolution is bounded Furthermore the solution (119901 119911) of (4)satisfies

lim119905rarrinfin

sup119901 (119909 119905) le 119870

lim119905rarrinfin

supintΩ

119911 (119909 119905) 119889119909 le 119870[119890 +119890119903

119870119899120583(119870 minus 119898

2)

2

] |Ω|

(5)

if1198891= 1198892 lim119905rarrinfin

sup 119911(119909 119905) le 119870[119890+(119890119903119870119899120583)((119870minus119898)2)2]

Proof Let

119872(119901 119911) = 119903 (1 minus119901

119870)(1 minus

119898 + 119899

119901 + 119899)119901 minus 119886119901119911

119873 (119901 119911) = 119890119886119901119911 minus 120583119911

(6)

then 120597119872120597119911 = minus119886119901 le 0 120597119873120597119901 = 119890119886119911 ge 0 for 119901 ge0 119911 ge 0 which implies that (4) is a mixed quasi-monotone system [36] We define (119901(119909 119905) 119911(119909 119905)) = (0 0)


14

12

1

08

06

04

02

00 10 20 30 40 50 60

(a)

1

09

08

07

06

05

04

03

02

01

00 10 20 30 40 50 60

(b)


1198892= 0004 119890 = 05 119886 = 1 119899 = 05119898 = 05 120583 = 04 119903 = 2 and 119870 = 10

15

1

05

0200

150100

500 50

7090

110

(a)

3

25

2

15

1

05

0200

150

10050

0 5070

90110

(b)


(119901(119909 119905) 119911(119909 119905)) = (119901lowast

(119905) 119911lowast


119889119901

119889119905= 119903 (1 minus

119901

119870)(1 minus

119898 + 119899

119901 + 119899)119901

119889119911

119889119905= 119890119886119901119911 minus 120583119911

119901 (0) = 119901lowast

119911 (0) = 119911lowast

(7)


Ω1199110(119909) We find that

120597119901 (119909 119905)

120597119905minus Δ119901 (119909 119905) minus 119872 (119901 (119909 119905) 119911 (119909 119905)) = 0

120597119901 (119909 119905)

120597119905minus Δ119901 (119909 119905) minus 119872(119901 (119909 119905) 119911 (119909 119905)) = 0

(8)

Therefore

120597119901 (119909 119905)

120597119905minus Δ119901 (119909 119905) minus 119872 (119901 (119909 119905) 119911 (119909 119905))

ge

120597119901 (119909 119905)

120597119905minus Δ119901 (119909 119905) minus 119872(119901 (119909 119905) 119911 (119909 119905))

(9)

Similarly

120597119911 (119909 119905)

120597119905minus Δ119911 (119909 119905) minus 119873 (119901 (119909 119905) 119911 (119909 119905))

ge120597119911 (119909 119905)

120597119905minus Δ119911 (119909 119905) minus 119873 (119901 (119909 119905) 119911 (119909 119905))

(10)


(119905) 119911lowast




0 le 119901 (119909 119905) le 119901lowast

(119905) 0 le 119911 (119909 119905) le 119911lowast

(119905) (11)




119901(119909 119905)119889119909 = 119875(119905) intΩ

119911(119909 119905)119889119909 = 119885(119905) then

119889119875

119889119905= intΩ

119901119905119889119909

= intΩ

[119903 (1 minus119901

119870)(1 minus

119898 + 119899

119901 + 119899)119901 minus 119886119901119911] 119889119909

+ intΩ

1198891Δ119901119889119909

(12)

119889119885

119889119905= intΩ

119890119886119901119911119889119909 minus 120583119885 + intΩ

1198892Δ119911119889119909 (13)


(119890119875 + 119885)119905= 119890intΩ

119903 (1 minus119901

119870)119901 minus 119898

119901 + 119899119901119889119909 minus 120583119885

= 119890intΩ

119903 (1 minus119901

119870)119901 minus 119898

119901 + 119899119901119889119909 minus 120583 (119890119875 + 119885) minus 120583119890119875

le minus120583 (119890119875 + 119885) + [120583119890 +119890119903

119870119899(119870 minus 119898

2)

2

]119875

(14)



1gt 0 when 119905 gt 119879

1

(119890119875 + 119885)119905

le minus120583 (119890119875 + 119885) + [120583119890 +119890119903

119870119899(119870 minus 119898

2)

2

] (119870 + 120576) |Ω|

(15)


intΩ

119911 (119909 119905) 119889119909

= 119885 (119905) lt 119890119875 (119905) + 119885 (119905)

le119870 + 120576

120583[120583119890 +

119890119903

119870119899(119870 minus 119898

2)

2

] |Ω| + 120576 for 119905 gt 1198792

(16)


supintΩ

119911(119909 119905)119889119909 le 119870[119890 + (119890119903119870119899120583)

((119870 minus 119898)2)2

]|Ω|


have

120597119908

120597119905= 119890119903 (1 minus

119901

119870)(1 minus

119898 + 119899

119901 + 119899)119901 minus 120583119911 + 119889

1Δ119908

119909 isin Ω 119905 gt 119879

120597119908

120597119899= 0 119909 isin 120597Ω 119905 gt 119879

119908 (119909 119879) = 119890119901 (119909 119879) + 119911 (119909 119879) 119909 isin Ω

(17)

Since

119890119903 (1 minus119901

119870)119901 minus 119898

119901 + 119899119901 minus 120583119911

= 119890119903 (1 minus119901

119870)119901 minus 119898

119901 + 119899119901 minus 120583119908 + 120583119890119901

le [119890119903

119870119899(119870 minus 119898

2)

2

+ 120583119890] 119901 minus 120583119908

le [119890119903

119870119899(119870 minus 119898

2)

2

+ 120583119890] (119870 + 120576) minus 120583119908

(18)


120597Φ

120597119905= [119890119903

119870119899(119870 minus 119898

2)

2

+ 120583119890] (119870 + 120576) minus 120583V + 1198891ΔV

119909 isin Ω 119905 gt 119879

120597Φ

120597119899= 0 119909 isin 120597Ω 119905 gt 119879

(19)

is Φ(119909 119905) rarr (1120583)[(119890119903119870119899)((119870 minus 119898)2)2


119905rarrinfinsup

119911(119909 119905) le lim119905rarrinfin

sup119908(119909 119905) le (1120583)[(119890119903119870119899)((119870 minus119898)2)2 +120583119890](119870 + 120576)




lowast





119906) where 119906 = 120583119890119886 V

119906= (119903119886)(1 minus (119906119870))((119906 minus


analyzed as follows






119906 =2119899 minus radic41198992 + 4 (119898119899 + 119898119870 + 119899119870)

minus2 (20)


120597119880

120597119905= (119863Δ + 119869

(119901119911))119880 (21)

where

119880 = (119901 (119909 119905) 119911 (119909 119905))119879

119863 = diag (1198891 1198892)

119869(119901119911)= (119860 (119901 119911) 119861 (119901 119911)

119862 (119901 119911) 119863 (119901 119911))

119860 (119901 119911) = 119903 (1 minus119901

119870)(1 minus

119898 + 119899

119901 + 119899) minus119903

119870119901(1 minus

119898 + 119899

119901 + 119899)

+ 119903 (1 minus119901

119870)119901119898 + 119899

(119901 + 119899)2minus 119886119911

119861 (119901 119911) = minus119886119901 119862 (119901 119911) = 119890119886119911 119863 (119901 119911) = 119890119886119901 minus 120583

(22)







119894= minus120582119894119863 + 119869(119901119911)


det (120585119868 minus 119869119894) = 1205852

minus trace119869119894120585 + det 119869

119894 (23)

wheretrace119869

119894= minus120582119894(1198891+ 1198892) + 119860 (119901 119911) + 119863 (119901 119911)

det 119869119894= 119889111988921205822

119894minus (119860 (119901 119911) 119889

2+ 119863 (119901 119911) 119889

1) 120582119894+ det 119869

(119901119911)

(24)


0 minus120583)


119903119898

119899+ 120583) lt 0

det 119869119894= 119889111988921205822

119894+ (119903119898

1198991198892+ 1205831198891)120582119894+119903119898120583

119899gt 0

(25)


(2) If (119901 119911) = (119898 0) then 119869(1198980)

=

(119903sdot(1minus(119898119870))(119898(119898+119899)) minus119886119898

0 119890119886119898minus120583)



0 119890119886119870minus120583)

When 119890119886119870 minus 120583 lt 0


119870 + 119899+ 119890119886119870 minus 120583 lt 0

det 119869119894= 119889111988921205822

119894+ (minus119903

119870 minus 119898

119870 + 1198991198892+ (119890119886119870 minus 120583) 119889

1)120582119894

minus 119903119870 minus 119898

119870 + 119899(119890119886119870 minus 120583) gt 0

(26)


119894= minus119903((119870 minus 119898)(119870 +



(4) If (119901 119911) = (119906 V119906) then 119869

(119906V119906)= (119860(119906) 119861(119906)

119862(119906) 0) where

119860 (119906) = 119903 (1 minus119906

119870)119906119898 + 119899

(119906 + 119899)2minus119903

119870119906119906 minus 119898

119906 + 119899 119861 (119906) = minus

120583

119890

119862 (119906) = 119890119903 (1 minus119906

119870)119906 minus 119898

119906 + 119899

(27)



trace (119869119894) = minus120582

119894(1198891+ 1198892) + 119860 (119906) lt 0

det 119869119894= 119889111988921205822

119894minus 119860 (119906) 119889

2120582119894minus 119861 (119906) 119862 (119906) gt 0

(28)








119881 = intΩ

119882(119901 (119909 119905) 119911 (119909 119905)) 119889119909 (29)

where

119882(119901 (119909 119905) 119911 (119909 119905)) = 1198881int

119901

119906

119901 minus 119906

119901119889119901 + 119888

2int

119911

V119906

119911 minus V119906

119911119889119911

(30)




119889119881

119889119905= intΩ

(119882119901sdot 119901119905+119882119911sdot 119911119905) 119889119909

= intΩ

(1198881

119901 minus 119906

119901

120597119901

120597119905+ 1198882

119911 minus V119906

119911

120597119911

120597119905) 119889119909

= intΩ

(1198881(119901 minus 119906) [119903 (1 minus

119909

119870)119901 minus 119898

119901 + 119899minus 119886119911]

+ 1198882(119911 minus V

119906) 119890119886 (119901 minus 119906) ) 119889119909

+ intΩ

(1198881

119901 minus 119906

1199011198891Δ119901 + 119888

2

119911 minus V119906

1199111198892Δ119911)119889119909

= 1198681(119905) + 119868

2(119905)

(31)


1198682(119905) = minusint

Ω

(11988811198891

119906

11990121003816100381610038161003816nabla11990110038161003816100381610038162

+ 11988821198892

V119906

1199112|nabla119911|2

)119889119909 le 0 (32)

Further

1198681(119905)

= intΩ

(1198881(119901 minus 119906) [119903 (1 minus

119901

119870)119901 minus 119898

119901 + 119899minus 119903 (1 minus

119906

119870)119906 minus 119898

119906 + 119899

+119886V119906minus 119886119911] + 119888

2119890119886 (119901 minus 119906) (119911 minus V

119906) ) 119889119909

= intΩ

(1198881(119901 minus 119906)

2 119903

119870[minus119901119906 + 119870119898 minus 119899 (119901 + 119906) + 119899 (119898 + 119870)

(119901 + 119899) (119906 + 119899)]


119906) ) 119889119909

(33)



119889119881

119889119905= 1198681(119905) + 119868

2(119905) le 119868

1(119905)

= intΩ

1198881

119903

119870[minus119901119906 + 119870119898 + 119899 (119898 + 119870) minus 119899 (119901 + 119906)

(119901 + 119899) (119906 + 119899)]

times (119901 minus 119906)2

119889119909

le intΩ

1198881

119903

119870

119870119898 + 119898119899 + 119870119899 minus 119899119906

119899 (119906 + 119899)(119901 minus 119906)

2

119889119909

(34)








119894 Let

119879 (119906 120582) = 119860 (119906) minus 120582 (1198891+ 1198892)

119863 (119906 120582) = 119889111988921205822

minus 1198892119860 (119906) 120582 minus 119861 (119906) 119862 (119906)

(35)




1+1198892)) We can





lowast

) such that119860(119906) gt 0 in (119906

119888 119906) 119860(119906) lt 0 in (0 119906

119888) and 119860(0) =

119860(119906119888) = 119860(119906) = 0

(3) 1198601015840(119906) gt 0 in (max0 119906119888 119906lowast



2119860(119906) plusmn

radic119889221198602(119906) + 4119889

11198892119861(119906)119862(119906))2119889

11198892 Let

119878 (119906) = 1198892

21198602

(119906) + 411988911198891119861 (119906) 119862 (119906) (36)

Since

119878 (119898) = 1198892

21198602

(119906) gt 0 119878 (119906) = 411988911198892119861 (119906) 119862 (119906) lt 0

(37)



120582 as follows


minus(119906) is



211988911198892120582 (119906) 120582

1015840

(119906) minus 11988921198601015840

(119906) 120582 (119906)

minus 1198892119860 (119906) 120582

1015840

(119906) minus 119861 (119906) 1198621015840

(119906) = 0

(38)

Therefore 1205821015840(119906) = (119861(119906)1198621015840(119906) + 11988921198601015840

(119906)120582(119906))1198892(21198891120582(119906) minus


119904)

1198892(21198891120582+(119906) minus 119860(119906)) gt 0 and 119889

2(21198891120582minus(119906) minus 119860(119906)) lt 0



(119906) = 119890119903(minus1199062

minus2119899119906+119898119899+119898119870+119899119870)119870(119906+119899)2 and 1198621015840(119906) =


119860 () = 119903 (1 minus

119870) 119898 + 119899

( + 119899)2minus119903

119870 minus 119898

+ 119899

=119903

119870( + 119899)2[minus2

minus 2119899 + 119898119899 + 119899119870 + 119898119870] = 0

(39)



119904) If

119906lowast


11988921198601015840


+(119906) lt 0



119879119894(119906) = 0 119863

119894(119906) gt 0 119879

119895(119906) = 0 119863

119895(119906) = 0

for 119895 = 119894(40)



119894(0 le 119894 le 119896)


119894 119906119894 V119906119894



119894(119906) gt 0 for


0(119906) = 0 119863

0(119906) gt 0 and

119879119895(119906) = minus(119889

1+ 1198892)120582119895lt 0 119863






1lt 119906 (41)


119894lt 119860(119906)(119889

1+ 1198892)

Therefore 119879119894(119906119894) = 0 and 119879

119895(119906119894) = 0 for 119895 = 119894 1 le 119894 le ℎ




1+ 1198892) = 119903(1 minus (119898119870))(119898(119898+ 119899)) gt 0119863(119898 120582) = 0


1+ 1198892)


minus(119906)



119897 120582119897) in119906 isin (119898 119906)Then119863

119894(119906119894) gt 0

if 119906119894gt 119906119887and 119863

119894(119906119894) lt 0 if 119906



119894 119906119894 V119906119894


119886 (119906) =11990611989110158401015840

(119906)

16(119891101584010158401015840

(119906)

11989110158401015840 (119906)+2

119906)

=minus119906 (119899 minus 119906) (2119899

2

+ 119898119899 + 119870119899 + 119870119898)

8(119906 + 119899)5

lt 0

(42)

if 119899 gt 119906 holds


minus1198891Δ119901 = 119903 (1 minus

119901

119870)(1 minus

119898 + 119899

119901 + 119899)119901 minus 119886119901119911 119909 isin Ω


120597119901

120597119899=120597119911

120597119899= 0 119909 isin 120597Ω

(43)





Δ119910 + 119892 (119909 119910 (119909)) ge 0 119894119899 Ω120597119910 (119909)

120597119899le 0 119900119899 120597Ω

(44)



119903 (1 minus119901

119870)(1 minus

119898 + 119899

119901 + 119899)119901 + 119889

1Δ119901 = 0 119909 isin Ω

120597119901

120597119899= 0 119909 isin 120597Ω

(45)

satisfy 0 le 119901(119909) le 119870


0 lt 119901 (119909) lt 119870 0 lt 119911 (119909) lt119890119903

119899120583(119870 minus 119898

2)

2

minus1198891119870

2≜ 119871

(46)








minus (1198901198891Δ119901 + 119889

2Δ119911)

= 119890119903 (1 minus119901

119870)(1 minus

119898 + 119899

119901 + 119899)119901 minus 120583119911

= 119890119903 (1 minus119901

119870)(1 minus

119898 + 119899

119901 + 119899)119901 minus

120583

1198892

(1198891119901 + 1198892119911) +

1205831198891

1198892

119901

le [119890119903

119870119899(119870 minus 119898

2)

2

minus1205831198891

1198892

]119870 minus120583

1198892

(1198891119901 + 1198892119911)

(47)


1198892119911 lt 1198891119901 + 1198892119911 lt1198892

120583[119890119903

119870119899(119870 minus 119898

2)

2

minus1205831198891

2]119870 (48)


Theorem 10 Let 1198631gt (1198702

+ 119898119870 minus 119898119899)1198701198992

1205821be a fixed


1ge 1198631



1198911(119901 119911) = 119903 (1 minus

119901

119870)(1 minus

119898 + 119899

119901 + 119899)119901 minus 119886119901119911

1198912(119901 119911) = 119890119886119901119911 minus 120583119911

(49)

Let

119901 =1

|Ω|intΩ

119901 (119909) 119889119909

119911 =1

|Ω|intΩ

119911 (119909) 119889119909

(50)

then intΩ

(119901minus119901)119889119909 = 0 intΩ


intΩ

1198891

1003816100381610038161003816nabla11990110038161003816100381610038162

119889119909

= intΩ

1198911(119901 119911) (119901 minus 119901) 119889119909

= intΩ

(1198911(119901 119911) minus 119891

1(119901 119911)) (119901 minus 119901) 119889119909

= intΩ

((minus119901119901 (119901 + 119901) (119901 minus 119901)2

+ (119870 + 119898) 119901119901(119901 minus 119901)2

minus 119870119898119899 (119901 minus 119901))

times ((119901 + 119899) (119901 + 119899))minus1


) 119889119909

le intΩ

((119870 + 119898119870 minus 119898119899)119870

1198992(119901 minus 119901)

2

+1198861198701003816100381610038161003816119901 minus 119901

1003816100381610038161003816 |119911 minus 119911| ) 119889119909

(51)

Similarly

intΩ

1198892|nabla119911|2

119889119909

= intΩ

1198912(119901 119911) (119911 minus 119911) 119889119909

= intΩ

(1198912(119901 119911) minus 119891

2(119901 119911)) (119911 minus 119911) 119889119909

= intΩ

((119890119886119901 minus 120583) (119911 minus 119911)2


le intΩ

((119890119886119870 minus 120583) (119911 minus 119911)2

+ 1198901198861198711003816100381610038161003816119901 minus 119901

1003816100381610038161003816 |119911 minus 119911|) 119889119909

(52)


intΩ

(1198891

1003816100381610038161003816nabla11990110038161003816100381610038162

+ 1198892|nabla119911|2

) 119889119909

le intΩ

((1198702

+ 119898119870 minus 119898119899)119870

1198992(119901 minus 119901)

2

+ (119886119870 + 119890119886119871)1003816100381610038161003816119901 minus 119901

1003816100381610038161003816 |119911 minus 119911|

+ (119890119886119870 minus 120583) (119911 minus 119911)2

)119889119909

le intΩ

(((1198702

+ 119898119870 minus 119898119899)119870

1198992+119886119870 + 119890119886119871

2120576) (119901 minus 119901)

2

+(119890119886119871 + 119886119870

2120576+ 119890119886119870 minus 120583) (119911 minus 119911)

2

)119889119909

(53)


intΩ

(11988911205821

1003816100381610038161003816119901 minus 11990110038161003816100381610038162

+ 11988921205821|119911 minus 119911|

2

) 119889119909

le intΩ

(((1198702

+ 119898119870 minus 119898119899)119870

1198992+119886119870 + 119890119886119871

2120576) (119901 minus 119901)

2

+ (119890119886119871 + 119886119870

2120576+ 119890119886119870 minus 120583) (119911 minus 119911)

2

)119889119909

(54)



2

+ 119898119870 minus 119898119899)1198701198992 a


+ 119898119870 minus

119898119899)1198701198992

+((119886119870+119890119886119871)2)120576 Let1198632= ((119890119886119871+119886119870)2120576+119890119886119870minus


proof


119894(119909)


1198972 and

120601119894(119909) = cos(119894119909119897) Let

119879119894(119906) = minus120582

119894(1198891+ 1198892) + 119860 (119906)

119863119894(119906) = 119889

111988921205822

119894minus 119860 (119906) 119889

2120582119894minus 119861 (119906) 119862 (119906)

(55)



119863119894(119906) = 0 119879

119894(119906) = 0 119863

119895(119906) = 0 119879

119895(119906) = 0

for 119895 = 119894

119889119863119894(119906)

119889119906= 0

(56)


119894(1 le 119894 le 119896

1015840




most three 119906119894for 119863

119894(119906119894) = 119863(119906

119894 120582119894) = 0 In particular if


119894isin (119898 119906

119904) such that

119863119894(119906119894) = 119863(119906



2120582119894minus 119861(119906)119862

1015840


21198601015840

(119906)120582plusmn(119906) lt



119898

4 Discussion




79998

79999

8

80001

80002

80003

80004

80005

80006

80007

80008

0 20 40 60 80 100 120

(a)

23076

23077

23077

23078

23078

23079

23079

2308

2308

0 20 40 60 80 100 120

(b)


1= 0001 119889

2= 0004 119890 = 05 119886 = 01 119899 = 5






120582

u

D(u 120582) = 0

T(u 120582) = 0

900

800

700

600

500

400

300

200

100

0

0 1 2 3 4 5 6 7


0001 1198892= 0004 119890 = 05 119886 = 1 119899 = 5 119898 = 5 120583 = 04 119903 = 2 and


120582

u

D(u 120582) = 0

T(u 120582) = 0

0 1 2 3 4

300

200

100

0


0001 1198892= 0004 119890 = 05 119886 = 1 119899 = 5 119898 = 1 120583 = 04 119903 = 2 and





Acknowledgments


References















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










14

12

1

08

06

04

02

00 10 20 30 40 50 60

(a)

1

09

08

07

06

05

04

03

02

01

00 10 20 30 40 50 60

(b)


1198892= 0004 119890 = 05 119886 = 1 119899 = 05119898 = 05 120583 = 04 119903 = 2 and 119870 = 10

15

1

05

0200

150100

500 50

7090

110

(a)

3

25

2

15

1

05

0200

150

10050

0 5070

90110

(b)


(119901(119909 119905) 119911(119909 119905)) = (119901lowast

(119905) 119911lowast


119889119901

119889119905= 119903 (1 minus

119901

119870)(1 minus

119898 + 119899

119901 + 119899)119901

119889119911

119889119905= 119890119886119901119911 minus 120583119911

119901 (0) = 119901lowast

119911 (0) = 119911lowast

(7)


Ω1199110(119909) We find that

120597119901 (119909 119905)

120597119905minus Δ119901 (119909 119905) minus 119872 (119901 (119909 119905) 119911 (119909 119905)) = 0

120597119901 (119909 119905)

120597119905minus Δ119901 (119909 119905) minus 119872(119901 (119909 119905) 119911 (119909 119905)) = 0

(8)

Therefore

120597119901 (119909 119905)

120597119905minus Δ119901 (119909 119905) minus 119872 (119901 (119909 119905) 119911 (119909 119905))

ge

120597119901 (119909 119905)

120597119905minus Δ119901 (119909 119905) minus 119872(119901 (119909 119905) 119911 (119909 119905))

(9)

Similarly

120597119911 (119909 119905)

120597119905minus Δ119911 (119909 119905) minus 119873 (119901 (119909 119905) 119911 (119909 119905))

ge120597119911 (119909 119905)

120597119905minus Δ119911 (119909 119905) minus 119873 (119901 (119909 119905) 119911 (119909 119905))

(10)


(119905) 119911lowast




0 le 119901 (119909 119905) le 119901lowast

(119905) 0 le 119911 (119909 119905) le 119911lowast

(119905) (11)




119901(119909 119905)119889119909 = 119875(119905) intΩ

119911(119909 119905)119889119909 = 119885(119905) then

119889119875

119889119905= intΩ

119901119905119889119909

= intΩ

[119903 (1 minus119901

119870)(1 minus

119898 + 119899

119901 + 119899)119901 minus 119886119901119911] 119889119909

+ intΩ

1198891Δ119901119889119909

(12)

119889119885

119889119905= intΩ

119890119886119901119911119889119909 minus 120583119885 + intΩ

1198892Δ119911119889119909 (13)


(119890119875 + 119885)119905= 119890intΩ

119903 (1 minus119901

119870)119901 minus 119898

119901 + 119899119901119889119909 minus 120583119885

= 119890intΩ

119903 (1 minus119901

119870)119901 minus 119898

119901 + 119899119901119889119909 minus 120583 (119890119875 + 119885) minus 120583119890119875

le minus120583 (119890119875 + 119885) + [120583119890 +119890119903

119870119899(119870 minus 119898

2)

2

]119875

(14)



1gt 0 when 119905 gt 119879

1

(119890119875 + 119885)119905

le minus120583 (119890119875 + 119885) + [120583119890 +119890119903

119870119899(119870 minus 119898

2)

2

] (119870 + 120576) |Ω|

(15)


intΩ

119911 (119909 119905) 119889119909

= 119885 (119905) lt 119890119875 (119905) + 119885 (119905)

le119870 + 120576

120583[120583119890 +

119890119903

119870119899(119870 minus 119898

2)

2

] |Ω| + 120576 for 119905 gt 1198792

(16)


supintΩ

119911(119909 119905)119889119909 le 119870[119890 + (119890119903119870119899120583)

((119870 minus 119898)2)2

]|Ω|


have

120597119908

120597119905= 119890119903 (1 minus

119901

119870)(1 minus

119898 + 119899

119901 + 119899)119901 minus 120583119911 + 119889

1Δ119908

119909 isin Ω 119905 gt 119879

120597119908

120597119899= 0 119909 isin 120597Ω 119905 gt 119879

119908 (119909 119879) = 119890119901 (119909 119879) + 119911 (119909 119879) 119909 isin Ω

(17)

Since

119890119903 (1 minus119901

119870)119901 minus 119898

119901 + 119899119901 minus 120583119911

= 119890119903 (1 minus119901

119870)119901 minus 119898

119901 + 119899119901 minus 120583119908 + 120583119890119901

le [119890119903

119870119899(119870 minus 119898

2)

2

+ 120583119890] 119901 minus 120583119908

le [119890119903

119870119899(119870 minus 119898

2)

2

+ 120583119890] (119870 + 120576) minus 120583119908

(18)


120597Φ

120597119905= [119890119903

119870119899(119870 minus 119898

2)

2

+ 120583119890] (119870 + 120576) minus 120583V + 1198891ΔV

119909 isin Ω 119905 gt 119879

120597Φ

120597119899= 0 119909 isin 120597Ω 119905 gt 119879

(19)

is Φ(119909 119905) rarr (1120583)[(119890119903119870119899)((119870 minus 119898)2)2


119905rarrinfinsup

119911(119909 119905) le lim119905rarrinfin

sup119908(119909 119905) le (1120583)[(119890119903119870119899)((119870 minus119898)2)2 +120583119890](119870 + 120576)




lowast





119906) where 119906 = 120583119890119886 V

119906= (119903119886)(1 minus (119906119870))((119906 minus


analyzed as follows






119906 =2119899 minus radic41198992 + 4 (119898119899 + 119898119870 + 119899119870)

minus2 (20)


120597119880

120597119905= (119863Δ + 119869

(119901119911))119880 (21)

where

119880 = (119901 (119909 119905) 119911 (119909 119905))119879

119863 = diag (1198891 1198892)

119869(119901119911)= (119860 (119901 119911) 119861 (119901 119911)

119862 (119901 119911) 119863 (119901 119911))

119860 (119901 119911) = 119903 (1 minus119901

119870)(1 minus

119898 + 119899

119901 + 119899) minus119903

119870119901(1 minus

119898 + 119899

119901 + 119899)

+ 119903 (1 minus119901

119870)119901119898 + 119899

(119901 + 119899)2minus 119886119911

119861 (119901 119911) = minus119886119901 119862 (119901 119911) = 119890119886119911 119863 (119901 119911) = 119890119886119901 minus 120583

(22)







119894= minus120582119894119863 + 119869(119901119911)


det (120585119868 minus 119869119894) = 1205852

minus trace119869119894120585 + det 119869

119894 (23)

wheretrace119869

119894= minus120582119894(1198891+ 1198892) + 119860 (119901 119911) + 119863 (119901 119911)

det 119869119894= 119889111988921205822

119894minus (119860 (119901 119911) 119889

2+ 119863 (119901 119911) 119889

1) 120582119894+ det 119869

(119901119911)

(24)


0 minus120583)


119903119898

119899+ 120583) lt 0

det 119869119894= 119889111988921205822

119894+ (119903119898

1198991198892+ 1205831198891)120582119894+119903119898120583

119899gt 0

(25)


(2) If (119901 119911) = (119898 0) then 119869(1198980)

=

(119903sdot(1minus(119898119870))(119898(119898+119899)) minus119886119898

0 119890119886119898minus120583)



0 119890119886119870minus120583)

When 119890119886119870 minus 120583 lt 0


119870 + 119899+ 119890119886119870 minus 120583 lt 0

det 119869119894= 119889111988921205822

119894+ (minus119903

119870 minus 119898

119870 + 1198991198892+ (119890119886119870 minus 120583) 119889

1)120582119894

minus 119903119870 minus 119898

119870 + 119899(119890119886119870 minus 120583) gt 0

(26)


119894= minus119903((119870 minus 119898)(119870 +



(4) If (119901 119911) = (119906 V119906) then 119869

(119906V119906)= (119860(119906) 119861(119906)

119862(119906) 0) where

119860 (119906) = 119903 (1 minus119906

119870)119906119898 + 119899

(119906 + 119899)2minus119903

119870119906119906 minus 119898

119906 + 119899 119861 (119906) = minus

120583

119890

119862 (119906) = 119890119903 (1 minus119906

119870)119906 minus 119898

119906 + 119899

(27)



trace (119869119894) = minus120582

119894(1198891+ 1198892) + 119860 (119906) lt 0

det 119869119894= 119889111988921205822

119894minus 119860 (119906) 119889

2120582119894minus 119861 (119906) 119862 (119906) gt 0

(28)








119881 = intΩ

119882(119901 (119909 119905) 119911 (119909 119905)) 119889119909 (29)

where

119882(119901 (119909 119905) 119911 (119909 119905)) = 1198881int

119901

119906

119901 minus 119906

119901119889119901 + 119888

2int

119911

V119906

119911 minus V119906

119911119889119911

(30)




119889119881

119889119905= intΩ

(119882119901sdot 119901119905+119882119911sdot 119911119905) 119889119909

= intΩ

(1198881

119901 minus 119906

119901

120597119901

120597119905+ 1198882

119911 minus V119906

119911

120597119911

120597119905) 119889119909

= intΩ

(1198881(119901 minus 119906) [119903 (1 minus

119909

119870)119901 minus 119898

119901 + 119899minus 119886119911]

+ 1198882(119911 minus V

119906) 119890119886 (119901 minus 119906) ) 119889119909

+ intΩ

(1198881

119901 minus 119906

1199011198891Δ119901 + 119888

2

119911 minus V119906

1199111198892Δ119911)119889119909

= 1198681(119905) + 119868

2(119905)

(31)


1198682(119905) = minusint

Ω

(11988811198891

119906

11990121003816100381610038161003816nabla11990110038161003816100381610038162

+ 11988821198892

V119906

1199112|nabla119911|2

)119889119909 le 0 (32)

Further

1198681(119905)

= intΩ

(1198881(119901 minus 119906) [119903 (1 minus

119901

119870)119901 minus 119898

119901 + 119899minus 119903 (1 minus

119906

119870)119906 minus 119898

119906 + 119899

+119886V119906minus 119886119911] + 119888

2119890119886 (119901 minus 119906) (119911 minus V

119906) ) 119889119909

= intΩ

(1198881(119901 minus 119906)

2 119903

119870[minus119901119906 + 119870119898 minus 119899 (119901 + 119906) + 119899 (119898 + 119870)

(119901 + 119899) (119906 + 119899)]


119906) ) 119889119909

(33)



119889119881

119889119905= 1198681(119905) + 119868

2(119905) le 119868

1(119905)

= intΩ

1198881

119903

119870[minus119901119906 + 119870119898 + 119899 (119898 + 119870) minus 119899 (119901 + 119906)

(119901 + 119899) (119906 + 119899)]

times (119901 minus 119906)2

119889119909

le intΩ

1198881

119903

119870

119870119898 + 119898119899 + 119870119899 minus 119899119906

119899 (119906 + 119899)(119901 minus 119906)

2

119889119909

(34)








119894 Let

119879 (119906 120582) = 119860 (119906) minus 120582 (1198891+ 1198892)

119863 (119906 120582) = 119889111988921205822

minus 1198892119860 (119906) 120582 minus 119861 (119906) 119862 (119906)

(35)




1+1198892)) We can





lowast

) such that119860(119906) gt 0 in (119906

119888 119906) 119860(119906) lt 0 in (0 119906

119888) and 119860(0) =

119860(119906119888) = 119860(119906) = 0

(3) 1198601015840(119906) gt 0 in (max0 119906119888 119906lowast



2119860(119906) plusmn

radic119889221198602(119906) + 4119889

11198892119861(119906)119862(119906))2119889

11198892 Let

119878 (119906) = 1198892

21198602

(119906) + 411988911198891119861 (119906) 119862 (119906) (36)

Since

119878 (119898) = 1198892

21198602

(119906) gt 0 119878 (119906) = 411988911198892119861 (119906) 119862 (119906) lt 0

(37)



120582 as follows


minus(119906) is



211988911198892120582 (119906) 120582

1015840

(119906) minus 11988921198601015840

(119906) 120582 (119906)

minus 1198892119860 (119906) 120582

1015840

(119906) minus 119861 (119906) 1198621015840

(119906) = 0

(38)

Therefore 1205821015840(119906) = (119861(119906)1198621015840(119906) + 11988921198601015840

(119906)120582(119906))1198892(21198891120582(119906) minus


119904)

1198892(21198891120582+(119906) minus 119860(119906)) gt 0 and 119889

2(21198891120582minus(119906) minus 119860(119906)) lt 0



(119906) = 119890119903(minus1199062

minus2119899119906+119898119899+119898119870+119899119870)119870(119906+119899)2 and 1198621015840(119906) =


119860 () = 119903 (1 minus

119870) 119898 + 119899

( + 119899)2minus119903

119870 minus 119898

+ 119899

=119903

119870( + 119899)2[minus2

minus 2119899 + 119898119899 + 119899119870 + 119898119870] = 0

(39)



119904) If

119906lowast


11988921198601015840


+(119906) lt 0



119879119894(119906) = 0 119863

119894(119906) gt 0 119879

119895(119906) = 0 119863

119895(119906) = 0

for 119895 = 119894(40)



119894(0 le 119894 le 119896)


119894 119906119894 V119906119894



119894(119906) gt 0 for


0(119906) = 0 119863

0(119906) gt 0 and

119879119895(119906) = minus(119889

1+ 1198892)120582119895lt 0 119863






1lt 119906 (41)


119894lt 119860(119906)(119889

1+ 1198892)

Therefore 119879119894(119906119894) = 0 and 119879

119895(119906119894) = 0 for 119895 = 119894 1 le 119894 le ℎ




1+ 1198892) = 119903(1 minus (119898119870))(119898(119898+ 119899)) gt 0119863(119898 120582) = 0


1+ 1198892)


minus(119906)



119897 120582119897) in119906 isin (119898 119906)Then119863

119894(119906119894) gt 0

if 119906119894gt 119906119887and 119863

119894(119906119894) lt 0 if 119906



119894 119906119894 V119906119894


119886 (119906) =11990611989110158401015840

(119906)

16(119891101584010158401015840

(119906)

11989110158401015840 (119906)+2

119906)

=minus119906 (119899 minus 119906) (2119899

2

+ 119898119899 + 119870119899 + 119870119898)

8(119906 + 119899)5

lt 0

(42)

if 119899 gt 119906 holds


minus1198891Δ119901 = 119903 (1 minus

119901

119870)(1 minus

119898 + 119899

119901 + 119899)119901 minus 119886119901119911 119909 isin Ω


120597119901

120597119899=120597119911

120597119899= 0 119909 isin 120597Ω

(43)





Δ119910 + 119892 (119909 119910 (119909)) ge 0 119894119899 Ω120597119910 (119909)

120597119899le 0 119900119899 120597Ω

(44)



119903 (1 minus119901

119870)(1 minus

119898 + 119899

119901 + 119899)119901 + 119889

1Δ119901 = 0 119909 isin Ω

120597119901

120597119899= 0 119909 isin 120597Ω

(45)

satisfy 0 le 119901(119909) le 119870


0 lt 119901 (119909) lt 119870 0 lt 119911 (119909) lt119890119903

119899120583(119870 minus 119898

2)

2

minus1198891119870

2≜ 119871

(46)








minus (1198901198891Δ119901 + 119889

2Δ119911)

= 119890119903 (1 minus119901

119870)(1 minus

119898 + 119899

119901 + 119899)119901 minus 120583119911

= 119890119903 (1 minus119901

119870)(1 minus

119898 + 119899

119901 + 119899)119901 minus

120583

1198892

(1198891119901 + 1198892119911) +

1205831198891

1198892

119901

le [119890119903

119870119899(119870 minus 119898

2)

2

minus1205831198891

1198892

]119870 minus120583

1198892

(1198891119901 + 1198892119911)

(47)


1198892119911 lt 1198891119901 + 1198892119911 lt1198892

120583[119890119903

119870119899(119870 minus 119898

2)

2

minus1205831198891

2]119870 (48)


Theorem 10 Let 1198631gt (1198702

+ 119898119870 minus 119898119899)1198701198992

1205821be a fixed


1ge 1198631



1198911(119901 119911) = 119903 (1 minus

119901

119870)(1 minus

119898 + 119899

119901 + 119899)119901 minus 119886119901119911

1198912(119901 119911) = 119890119886119901119911 minus 120583119911

(49)

Let

119901 =1

|Ω|intΩ

119901 (119909) 119889119909

119911 =1

|Ω|intΩ

119911 (119909) 119889119909

(50)

then intΩ

(119901minus119901)119889119909 = 0 intΩ


intΩ

1198891

1003816100381610038161003816nabla11990110038161003816100381610038162

119889119909

= intΩ

1198911(119901 119911) (119901 minus 119901) 119889119909

= intΩ

(1198911(119901 119911) minus 119891

1(119901 119911)) (119901 minus 119901) 119889119909

= intΩ

((minus119901119901 (119901 + 119901) (119901 minus 119901)2

+ (119870 + 119898) 119901119901(119901 minus 119901)2

minus 119870119898119899 (119901 minus 119901))

times ((119901 + 119899) (119901 + 119899))minus1


) 119889119909

le intΩ

((119870 + 119898119870 minus 119898119899)119870

1198992(119901 minus 119901)

2

+1198861198701003816100381610038161003816119901 minus 119901

1003816100381610038161003816 |119911 minus 119911| ) 119889119909

(51)

Similarly

intΩ

1198892|nabla119911|2

119889119909

= intΩ

1198912(119901 119911) (119911 minus 119911) 119889119909

= intΩ

(1198912(119901 119911) minus 119891

2(119901 119911)) (119911 minus 119911) 119889119909

= intΩ

((119890119886119901 minus 120583) (119911 minus 119911)2


le intΩ

((119890119886119870 minus 120583) (119911 minus 119911)2

+ 1198901198861198711003816100381610038161003816119901 minus 119901

1003816100381610038161003816 |119911 minus 119911|) 119889119909

(52)


intΩ

(1198891

1003816100381610038161003816nabla11990110038161003816100381610038162

+ 1198892|nabla119911|2

) 119889119909

le intΩ

((1198702

+ 119898119870 minus 119898119899)119870

1198992(119901 minus 119901)

2

+ (119886119870 + 119890119886119871)1003816100381610038161003816119901 minus 119901

1003816100381610038161003816 |119911 minus 119911|

+ (119890119886119870 minus 120583) (119911 minus 119911)2

)119889119909

le intΩ

(((1198702

+ 119898119870 minus 119898119899)119870

1198992+119886119870 + 119890119886119871

2120576) (119901 minus 119901)

2

+(119890119886119871 + 119886119870

2120576+ 119890119886119870 minus 120583) (119911 minus 119911)

2

)119889119909

(53)


intΩ

(11988911205821

1003816100381610038161003816119901 minus 11990110038161003816100381610038162

+ 11988921205821|119911 minus 119911|

2

) 119889119909

le intΩ

(((1198702

+ 119898119870 minus 119898119899)119870

1198992+119886119870 + 119890119886119871

2120576) (119901 minus 119901)

2

+ (119890119886119871 + 119886119870

2120576+ 119890119886119870 minus 120583) (119911 minus 119911)

2

)119889119909

(54)



2

+ 119898119870 minus 119898119899)1198701198992 a


+ 119898119870 minus

119898119899)1198701198992

+((119886119870+119890119886119871)2)120576 Let1198632= ((119890119886119871+119886119870)2120576+119890119886119870minus


proof


119894(119909)


1198972 and

120601119894(119909) = cos(119894119909119897) Let

119879119894(119906) = minus120582

119894(1198891+ 1198892) + 119860 (119906)

119863119894(119906) = 119889

111988921205822

119894minus 119860 (119906) 119889

2120582119894minus 119861 (119906) 119862 (119906)

(55)



119863119894(119906) = 0 119879

119894(119906) = 0 119863

119895(119906) = 0 119879

119895(119906) = 0

for 119895 = 119894

119889119863119894(119906)

119889119906= 0

(56)


119894(1 le 119894 le 119896

1015840




most three 119906119894for 119863

119894(119906119894) = 119863(119906

119894 120582119894) = 0 In particular if


119894isin (119898 119906

119904) such that

119863119894(119906119894) = 119863(119906



2120582119894minus 119861(119906)119862

1015840


21198601015840

(119906)120582plusmn(119906) lt



119898

4 Discussion




79998

79999

8

80001

80002

80003

80004

80005

80006

80007

80008

0 20 40 60 80 100 120

(a)

23076

23077

23077

23078

23078

23079

23079

2308

2308

0 20 40 60 80 100 120

(b)


1= 0001 119889

2= 0004 119890 = 05 119886 = 01 119899 = 5






120582

u

D(u 120582) = 0

T(u 120582) = 0

900

800

700

600

500

400

300

200

100

0

0 1 2 3 4 5 6 7


0001 1198892= 0004 119890 = 05 119886 = 1 119899 = 5 119898 = 5 120583 = 04 119903 = 2 and


120582

u

D(u 120582) = 0

T(u 120582) = 0

0 1 2 3 4

300

200

100

0


0001 1198892= 0004 119890 = 05 119886 = 1 119899 = 5 119898 = 1 120583 = 04 119903 = 2 and





Acknowledgments


References















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











0 le 119901 (119909 119905) le 119901lowast

(119905) 0 le 119911 (119909 119905) le 119911lowast

(119905) (11)




119901(119909 119905)119889119909 = 119875(119905) intΩ

119911(119909 119905)119889119909 = 119885(119905) then

119889119875

119889119905= intΩ

119901119905119889119909

= intΩ

[119903 (1 minus119901

119870)(1 minus

119898 + 119899

119901 + 119899)119901 minus 119886119901119911] 119889119909

+ intΩ

1198891Δ119901119889119909

(12)

119889119885

119889119905= intΩ

119890119886119901119911119889119909 minus 120583119885 + intΩ

1198892Δ119911119889119909 (13)


(119890119875 + 119885)119905= 119890intΩ

119903 (1 minus119901

119870)119901 minus 119898

119901 + 119899119901119889119909 minus 120583119885

= 119890intΩ

119903 (1 minus119901

119870)119901 minus 119898

119901 + 119899119901119889119909 minus 120583 (119890119875 + 119885) minus 120583119890119875

le minus120583 (119890119875 + 119885) + [120583119890 +119890119903

119870119899(119870 minus 119898

2)

2

]119875

(14)



1gt 0 when 119905 gt 119879

1

(119890119875 + 119885)119905

le minus120583 (119890119875 + 119885) + [120583119890 +119890119903

119870119899(119870 minus 119898

2)

2

] (119870 + 120576) |Ω|

(15)


intΩ

119911 (119909 119905) 119889119909

= 119885 (119905) lt 119890119875 (119905) + 119885 (119905)

le119870 + 120576

120583[120583119890 +

119890119903

119870119899(119870 minus 119898

2)

2

] |Ω| + 120576 for 119905 gt 1198792

(16)


supintΩ

119911(119909 119905)119889119909 le 119870[119890 + (119890119903119870119899120583)

((119870 minus 119898)2)2

]|Ω|


have

120597119908

120597119905= 119890119903 (1 minus

119901

119870)(1 minus

119898 + 119899

119901 + 119899)119901 minus 120583119911 + 119889

1Δ119908

119909 isin Ω 119905 gt 119879

120597119908

120597119899= 0 119909 isin 120597Ω 119905 gt 119879

119908 (119909 119879) = 119890119901 (119909 119879) + 119911 (119909 119879) 119909 isin Ω

(17)

Since

119890119903 (1 minus119901

119870)119901 minus 119898

119901 + 119899119901 minus 120583119911

= 119890119903 (1 minus119901

119870)119901 minus 119898

119901 + 119899119901 minus 120583119908 + 120583119890119901

le [119890119903

119870119899(119870 minus 119898

2)

2

+ 120583119890] 119901 minus 120583119908

le [119890119903

119870119899(119870 minus 119898

2)

2

+ 120583119890] (119870 + 120576) minus 120583119908

(18)


120597Φ

120597119905= [119890119903

119870119899(119870 minus 119898

2)

2

+ 120583119890] (119870 + 120576) minus 120583V + 1198891ΔV

119909 isin Ω 119905 gt 119879

120597Φ

120597119899= 0 119909 isin 120597Ω 119905 gt 119879

(19)

is Φ(119909 119905) rarr (1120583)[(119890119903119870119899)((119870 minus 119898)2)2


119905rarrinfinsup

119911(119909 119905) le lim119905rarrinfin

sup119908(119909 119905) le (1120583)[(119890119903119870119899)((119870 minus119898)2)2 +120583119890](119870 + 120576)




lowast





119906) where 119906 = 120583119890119886 V

119906= (119903119886)(1 minus (119906119870))((119906 minus


analyzed as follows






119906 =2119899 minus radic41198992 + 4 (119898119899 + 119898119870 + 119899119870)

minus2 (20)


120597119880

120597119905= (119863Δ + 119869

(119901119911))119880 (21)

where

119880 = (119901 (119909 119905) 119911 (119909 119905))119879

119863 = diag (1198891 1198892)

119869(119901119911)= (119860 (119901 119911) 119861 (119901 119911)

119862 (119901 119911) 119863 (119901 119911))

119860 (119901 119911) = 119903 (1 minus119901

119870)(1 minus

119898 + 119899

119901 + 119899) minus119903

119870119901(1 minus

119898 + 119899

119901 + 119899)

+ 119903 (1 minus119901

119870)119901119898 + 119899

(119901 + 119899)2minus 119886119911

119861 (119901 119911) = minus119886119901 119862 (119901 119911) = 119890119886119911 119863 (119901 119911) = 119890119886119901 minus 120583

(22)







119894= minus120582119894119863 + 119869(119901119911)


det (120585119868 minus 119869119894) = 1205852

minus trace119869119894120585 + det 119869

119894 (23)

wheretrace119869

119894= minus120582119894(1198891+ 1198892) + 119860 (119901 119911) + 119863 (119901 119911)

det 119869119894= 119889111988921205822

119894minus (119860 (119901 119911) 119889

2+ 119863 (119901 119911) 119889

1) 120582119894+ det 119869

(119901119911)

(24)


0 minus120583)


119903119898

119899+ 120583) lt 0

det 119869119894= 119889111988921205822

119894+ (119903119898

1198991198892+ 1205831198891)120582119894+119903119898120583

119899gt 0

(25)


(2) If (119901 119911) = (119898 0) then 119869(1198980)

=

(119903sdot(1minus(119898119870))(119898(119898+119899)) minus119886119898

0 119890119886119898minus120583)



0 119890119886119870minus120583)

When 119890119886119870 minus 120583 lt 0


119870 + 119899+ 119890119886119870 minus 120583 lt 0

det 119869119894= 119889111988921205822

119894+ (minus119903

119870 minus 119898

119870 + 1198991198892+ (119890119886119870 minus 120583) 119889

1)120582119894

minus 119903119870 minus 119898

119870 + 119899(119890119886119870 minus 120583) gt 0

(26)


119894= minus119903((119870 minus 119898)(119870 +



(4) If (119901 119911) = (119906 V119906) then 119869

(119906V119906)= (119860(119906) 119861(119906)

119862(119906) 0) where

119860 (119906) = 119903 (1 minus119906

119870)119906119898 + 119899

(119906 + 119899)2minus119903

119870119906119906 minus 119898

119906 + 119899 119861 (119906) = minus

120583

119890

119862 (119906) = 119890119903 (1 minus119906

119870)119906 minus 119898

119906 + 119899

(27)



trace (119869119894) = minus120582

119894(1198891+ 1198892) + 119860 (119906) lt 0

det 119869119894= 119889111988921205822

119894minus 119860 (119906) 119889

2120582119894minus 119861 (119906) 119862 (119906) gt 0

(28)








119881 = intΩ

119882(119901 (119909 119905) 119911 (119909 119905)) 119889119909 (29)

where

119882(119901 (119909 119905) 119911 (119909 119905)) = 1198881int

119901

119906

119901 minus 119906

119901119889119901 + 119888

2int

119911

V119906

119911 minus V119906

119911119889119911

(30)




119889119881

119889119905= intΩ

(119882119901sdot 119901119905+119882119911sdot 119911119905) 119889119909

= intΩ

(1198881

119901 minus 119906

119901

120597119901

120597119905+ 1198882

119911 minus V119906

119911

120597119911

120597119905) 119889119909

= intΩ

(1198881(119901 minus 119906) [119903 (1 minus

119909

119870)119901 minus 119898

119901 + 119899minus 119886119911]

+ 1198882(119911 minus V

119906) 119890119886 (119901 minus 119906) ) 119889119909

+ intΩ

(1198881

119901 minus 119906

1199011198891Δ119901 + 119888

2

119911 minus V119906

1199111198892Δ119911)119889119909

= 1198681(119905) + 119868

2(119905)

(31)


1198682(119905) = minusint

Ω

(11988811198891

119906

11990121003816100381610038161003816nabla11990110038161003816100381610038162

+ 11988821198892

V119906

1199112|nabla119911|2

)119889119909 le 0 (32)

Further

1198681(119905)

= intΩ

(1198881(119901 minus 119906) [119903 (1 minus

119901

119870)119901 minus 119898

119901 + 119899minus 119903 (1 minus

119906

119870)119906 minus 119898

119906 + 119899

+119886V119906minus 119886119911] + 119888

2119890119886 (119901 minus 119906) (119911 minus V

119906) ) 119889119909

= intΩ

(1198881(119901 minus 119906)

2 119903

119870[minus119901119906 + 119870119898 minus 119899 (119901 + 119906) + 119899 (119898 + 119870)

(119901 + 119899) (119906 + 119899)]


119906) ) 119889119909

(33)



119889119881

119889119905= 1198681(119905) + 119868

2(119905) le 119868

1(119905)

= intΩ

1198881

119903

119870[minus119901119906 + 119870119898 + 119899 (119898 + 119870) minus 119899 (119901 + 119906)

(119901 + 119899) (119906 + 119899)]

times (119901 minus 119906)2

119889119909

le intΩ

1198881

119903

119870

119870119898 + 119898119899 + 119870119899 minus 119899119906

119899 (119906 + 119899)(119901 minus 119906)

2

119889119909

(34)








119894 Let

119879 (119906 120582) = 119860 (119906) minus 120582 (1198891+ 1198892)

119863 (119906 120582) = 119889111988921205822

minus 1198892119860 (119906) 120582 minus 119861 (119906) 119862 (119906)

(35)




1+1198892)) We can





lowast

) such that119860(119906) gt 0 in (119906

119888 119906) 119860(119906) lt 0 in (0 119906

119888) and 119860(0) =

119860(119906119888) = 119860(119906) = 0

(3) 1198601015840(119906) gt 0 in (max0 119906119888 119906lowast



2119860(119906) plusmn

radic119889221198602(119906) + 4119889

11198892119861(119906)119862(119906))2119889

11198892 Let

119878 (119906) = 1198892

21198602

(119906) + 411988911198891119861 (119906) 119862 (119906) (36)

Since

119878 (119898) = 1198892

21198602

(119906) gt 0 119878 (119906) = 411988911198892119861 (119906) 119862 (119906) lt 0

(37)



120582 as follows


minus(119906) is



211988911198892120582 (119906) 120582

1015840

(119906) minus 11988921198601015840

(119906) 120582 (119906)

minus 1198892119860 (119906) 120582

1015840

(119906) minus 119861 (119906) 1198621015840

(119906) = 0

(38)

Therefore 1205821015840(119906) = (119861(119906)1198621015840(119906) + 11988921198601015840

(119906)120582(119906))1198892(21198891120582(119906) minus


119904)

1198892(21198891120582+(119906) minus 119860(119906)) gt 0 and 119889

2(21198891120582minus(119906) minus 119860(119906)) lt 0



(119906) = 119890119903(minus1199062

minus2119899119906+119898119899+119898119870+119899119870)119870(119906+119899)2 and 1198621015840(119906) =


119860 () = 119903 (1 minus

119870) 119898 + 119899

( + 119899)2minus119903

119870 minus 119898

+ 119899

=119903

119870( + 119899)2[minus2

minus 2119899 + 119898119899 + 119899119870 + 119898119870] = 0

(39)



119904) If

119906lowast


11988921198601015840


+(119906) lt 0



119879119894(119906) = 0 119863

119894(119906) gt 0 119879

119895(119906) = 0 119863

119895(119906) = 0

for 119895 = 119894(40)



119894(0 le 119894 le 119896)


119894 119906119894 V119906119894



119894(119906) gt 0 for


0(119906) = 0 119863

0(119906) gt 0 and

119879119895(119906) = minus(119889

1+ 1198892)120582119895lt 0 119863






1lt 119906 (41)


119894lt 119860(119906)(119889

1+ 1198892)

Therefore 119879119894(119906119894) = 0 and 119879

119895(119906119894) = 0 for 119895 = 119894 1 le 119894 le ℎ




1+ 1198892) = 119903(1 minus (119898119870))(119898(119898+ 119899)) gt 0119863(119898 120582) = 0


1+ 1198892)


minus(119906)



119897 120582119897) in119906 isin (119898 119906)Then119863

119894(119906119894) gt 0

if 119906119894gt 119906119887and 119863

119894(119906119894) lt 0 if 119906



119894 119906119894 V119906119894


119886 (119906) =11990611989110158401015840

(119906)

16(119891101584010158401015840

(119906)

11989110158401015840 (119906)+2

119906)

=minus119906 (119899 minus 119906) (2119899

2

+ 119898119899 + 119870119899 + 119870119898)

8(119906 + 119899)5

lt 0

(42)

if 119899 gt 119906 holds


minus1198891Δ119901 = 119903 (1 minus

119901

119870)(1 minus

119898 + 119899

119901 + 119899)119901 minus 119886119901119911 119909 isin Ω


120597119901

120597119899=120597119911

120597119899= 0 119909 isin 120597Ω

(43)





Δ119910 + 119892 (119909 119910 (119909)) ge 0 119894119899 Ω120597119910 (119909)

120597119899le 0 119900119899 120597Ω

(44)



119903 (1 minus119901

119870)(1 minus

119898 + 119899

119901 + 119899)119901 + 119889

1Δ119901 = 0 119909 isin Ω

120597119901

120597119899= 0 119909 isin 120597Ω

(45)

satisfy 0 le 119901(119909) le 119870


0 lt 119901 (119909) lt 119870 0 lt 119911 (119909) lt119890119903

119899120583(119870 minus 119898

2)

2

minus1198891119870

2≜ 119871

(46)








minus (1198901198891Δ119901 + 119889

2Δ119911)

= 119890119903 (1 minus119901

119870)(1 minus

119898 + 119899

119901 + 119899)119901 minus 120583119911

= 119890119903 (1 minus119901

119870)(1 minus

119898 + 119899

119901 + 119899)119901 minus

120583

1198892

(1198891119901 + 1198892119911) +

1205831198891

1198892

119901

le [119890119903

119870119899(119870 minus 119898

2)

2

minus1205831198891

1198892

]119870 minus120583

1198892

(1198891119901 + 1198892119911)

(47)


1198892119911 lt 1198891119901 + 1198892119911 lt1198892

120583[119890119903

119870119899(119870 minus 119898

2)

2

minus1205831198891

2]119870 (48)


Theorem 10 Let 1198631gt (1198702

+ 119898119870 minus 119898119899)1198701198992

1205821be a fixed


1ge 1198631



1198911(119901 119911) = 119903 (1 minus

119901

119870)(1 minus

119898 + 119899

119901 + 119899)119901 minus 119886119901119911

1198912(119901 119911) = 119890119886119901119911 minus 120583119911

(49)

Let

119901 =1

|Ω|intΩ

119901 (119909) 119889119909

119911 =1

|Ω|intΩ

119911 (119909) 119889119909

(50)

then intΩ

(119901minus119901)119889119909 = 0 intΩ


intΩ

1198891

1003816100381610038161003816nabla11990110038161003816100381610038162

119889119909

= intΩ

1198911(119901 119911) (119901 minus 119901) 119889119909

= intΩ

(1198911(119901 119911) minus 119891

1(119901 119911)) (119901 minus 119901) 119889119909

= intΩ

((minus119901119901 (119901 + 119901) (119901 minus 119901)2

+ (119870 + 119898) 119901119901(119901 minus 119901)2

minus 119870119898119899 (119901 minus 119901))

times ((119901 + 119899) (119901 + 119899))minus1


) 119889119909

le intΩ

((119870 + 119898119870 minus 119898119899)119870

1198992(119901 minus 119901)

2

+1198861198701003816100381610038161003816119901 minus 119901

1003816100381610038161003816 |119911 minus 119911| ) 119889119909

(51)

Similarly

intΩ

1198892|nabla119911|2

119889119909

= intΩ

1198912(119901 119911) (119911 minus 119911) 119889119909

= intΩ

(1198912(119901 119911) minus 119891

2(119901 119911)) (119911 minus 119911) 119889119909

= intΩ

((119890119886119901 minus 120583) (119911 minus 119911)2


le intΩ

((119890119886119870 minus 120583) (119911 minus 119911)2

+ 1198901198861198711003816100381610038161003816119901 minus 119901

1003816100381610038161003816 |119911 minus 119911|) 119889119909

(52)


intΩ

(1198891

1003816100381610038161003816nabla11990110038161003816100381610038162

+ 1198892|nabla119911|2

) 119889119909

le intΩ

((1198702

+ 119898119870 minus 119898119899)119870

1198992(119901 minus 119901)

2

+ (119886119870 + 119890119886119871)1003816100381610038161003816119901 minus 119901

1003816100381610038161003816 |119911 minus 119911|

+ (119890119886119870 minus 120583) (119911 minus 119911)2

)119889119909

le intΩ

(((1198702

+ 119898119870 minus 119898119899)119870

1198992+119886119870 + 119890119886119871

2120576) (119901 minus 119901)

2

+(119890119886119871 + 119886119870

2120576+ 119890119886119870 minus 120583) (119911 minus 119911)

2

)119889119909

(53)


intΩ

(11988911205821

1003816100381610038161003816119901 minus 11990110038161003816100381610038162

+ 11988921205821|119911 minus 119911|

2

) 119889119909

le intΩ

(((1198702

+ 119898119870 minus 119898119899)119870

1198992+119886119870 + 119890119886119871

2120576) (119901 minus 119901)

2

+ (119890119886119871 + 119886119870

2120576+ 119890119886119870 minus 120583) (119911 minus 119911)

2

)119889119909

(54)



2

+ 119898119870 minus 119898119899)1198701198992 a


+ 119898119870 minus

119898119899)1198701198992

+((119886119870+119890119886119871)2)120576 Let1198632= ((119890119886119871+119886119870)2120576+119890119886119870minus


proof


119894(119909)


1198972 and

120601119894(119909) = cos(119894119909119897) Let

119879119894(119906) = minus120582

119894(1198891+ 1198892) + 119860 (119906)

119863119894(119906) = 119889

111988921205822

119894minus 119860 (119906) 119889

2120582119894minus 119861 (119906) 119862 (119906)

(55)



119863119894(119906) = 0 119879

119894(119906) = 0 119863

119895(119906) = 0 119879

119895(119906) = 0

for 119895 = 119894

119889119863119894(119906)

119889119906= 0

(56)


119894(1 le 119894 le 119896

1015840




most three 119906119894for 119863

119894(119906119894) = 119863(119906

119894 120582119894) = 0 In particular if


119894isin (119898 119906

119904) such that

119863119894(119906119894) = 119863(119906



2120582119894minus 119861(119906)119862

1015840


21198601015840

(119906)120582plusmn(119906) lt



119898

4 Discussion




79998

79999

8

80001

80002

80003

80004

80005

80006

80007

80008

0 20 40 60 80 100 120

(a)

23076

23077

23077

23078

23078

23079

23079

2308

2308

0 20 40 60 80 100 120

(b)


1= 0001 119889

2= 0004 119890 = 05 119886 = 01 119899 = 5






120582

u

D(u 120582) = 0

T(u 120582) = 0

900

800

700

600

500

400

300

200

100

0

0 1 2 3 4 5 6 7


0001 1198892= 0004 119890 = 05 119886 = 1 119899 = 5 119898 = 5 120583 = 04 119903 = 2 and


120582

u

D(u 120582) = 0

T(u 120582) = 0

0 1 2 3 4

300

200

100

0


0001 1198892= 0004 119890 = 05 119886 = 1 119899 = 5 119898 = 1 120583 = 04 119903 = 2 and





Acknowledgments


References















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














119906 =2119899 minus radic41198992 + 4 (119898119899 + 119898119870 + 119899119870)

minus2 (20)


120597119880

120597119905= (119863Δ + 119869

(119901119911))119880 (21)

where

119880 = (119901 (119909 119905) 119911 (119909 119905))119879

119863 = diag (1198891 1198892)

119869(119901119911)= (119860 (119901 119911) 119861 (119901 119911)

119862 (119901 119911) 119863 (119901 119911))

119860 (119901 119911) = 119903 (1 minus119901

119870)(1 minus

119898 + 119899

119901 + 119899) minus119903

119870119901(1 minus

119898 + 119899

119901 + 119899)

+ 119903 (1 minus119901

119870)119901119898 + 119899

(119901 + 119899)2minus 119886119911

119861 (119901 119911) = minus119886119901 119862 (119901 119911) = 119890119886119911 119863 (119901 119911) = 119890119886119901 minus 120583

(22)







119894= minus120582119894119863 + 119869(119901119911)


det (120585119868 minus 119869119894) = 1205852

minus trace119869119894120585 + det 119869

119894 (23)

wheretrace119869

119894= minus120582119894(1198891+ 1198892) + 119860 (119901 119911) + 119863 (119901 119911)

det 119869119894= 119889111988921205822

119894minus (119860 (119901 119911) 119889

2+ 119863 (119901 119911) 119889

1) 120582119894+ det 119869

(119901119911)

(24)


0 minus120583)


119903119898

119899+ 120583) lt 0

det 119869119894= 119889111988921205822

119894+ (119903119898

1198991198892+ 1205831198891)120582119894+119903119898120583

119899gt 0

(25)


(2) If (119901 119911) = (119898 0) then 119869(1198980)

=

(119903sdot(1minus(119898119870))(119898(119898+119899)) minus119886119898

0 119890119886119898minus120583)



0 119890119886119870minus120583)

When 119890119886119870 minus 120583 lt 0


119870 + 119899+ 119890119886119870 minus 120583 lt 0

det 119869119894= 119889111988921205822

119894+ (minus119903

119870 minus 119898

119870 + 1198991198892+ (119890119886119870 minus 120583) 119889

1)120582119894

minus 119903119870 minus 119898

119870 + 119899(119890119886119870 minus 120583) gt 0

(26)


119894= minus119903((119870 minus 119898)(119870 +



(4) If (119901 119911) = (119906 V119906) then 119869

(119906V119906)= (119860(119906) 119861(119906)

119862(119906) 0) where

119860 (119906) = 119903 (1 minus119906

119870)119906119898 + 119899

(119906 + 119899)2minus119903

119870119906119906 minus 119898

119906 + 119899 119861 (119906) = minus

120583

119890

119862 (119906) = 119890119903 (1 minus119906

119870)119906 minus 119898

119906 + 119899

(27)



trace (119869119894) = minus120582

119894(1198891+ 1198892) + 119860 (119906) lt 0

det 119869119894= 119889111988921205822

119894minus 119860 (119906) 119889

2120582119894minus 119861 (119906) 119862 (119906) gt 0

(28)








119881 = intΩ

119882(119901 (119909 119905) 119911 (119909 119905)) 119889119909 (29)

where

119882(119901 (119909 119905) 119911 (119909 119905)) = 1198881int

119901

119906

119901 minus 119906

119901119889119901 + 119888

2int

119911

V119906

119911 minus V119906

119911119889119911

(30)




119889119881

119889119905= intΩ

(119882119901sdot 119901119905+119882119911sdot 119911119905) 119889119909

= intΩ

(1198881

119901 minus 119906

119901

120597119901

120597119905+ 1198882

119911 minus V119906

119911

120597119911

120597119905) 119889119909

= intΩ

(1198881(119901 minus 119906) [119903 (1 minus

119909

119870)119901 minus 119898

119901 + 119899minus 119886119911]

+ 1198882(119911 minus V

119906) 119890119886 (119901 minus 119906) ) 119889119909

+ intΩ

(1198881

119901 minus 119906

1199011198891Δ119901 + 119888

2

119911 minus V119906

1199111198892Δ119911)119889119909

= 1198681(119905) + 119868

2(119905)

(31)


1198682(119905) = minusint

Ω

(11988811198891

119906

11990121003816100381610038161003816nabla11990110038161003816100381610038162

+ 11988821198892

V119906

1199112|nabla119911|2

)119889119909 le 0 (32)

Further

1198681(119905)

= intΩ

(1198881(119901 minus 119906) [119903 (1 minus

119901

119870)119901 minus 119898

119901 + 119899minus 119903 (1 minus

119906

119870)119906 minus 119898

119906 + 119899

+119886V119906minus 119886119911] + 119888

2119890119886 (119901 minus 119906) (119911 minus V

119906) ) 119889119909

= intΩ

(1198881(119901 minus 119906)

2 119903

119870[minus119901119906 + 119870119898 minus 119899 (119901 + 119906) + 119899 (119898 + 119870)

(119901 + 119899) (119906 + 119899)]


119906) ) 119889119909

(33)



119889119881

119889119905= 1198681(119905) + 119868

2(119905) le 119868

1(119905)

= intΩ

1198881

119903

119870[minus119901119906 + 119870119898 + 119899 (119898 + 119870) minus 119899 (119901 + 119906)

(119901 + 119899) (119906 + 119899)]

times (119901 minus 119906)2

119889119909

le intΩ

1198881

119903

119870

119870119898 + 119898119899 + 119870119899 minus 119899119906

119899 (119906 + 119899)(119901 minus 119906)

2

119889119909

(34)








119894 Let

119879 (119906 120582) = 119860 (119906) minus 120582 (1198891+ 1198892)

119863 (119906 120582) = 119889111988921205822

minus 1198892119860 (119906) 120582 minus 119861 (119906) 119862 (119906)

(35)




1+1198892)) We can





lowast

) such that119860(119906) gt 0 in (119906

119888 119906) 119860(119906) lt 0 in (0 119906

119888) and 119860(0) =

119860(119906119888) = 119860(119906) = 0

(3) 1198601015840(119906) gt 0 in (max0 119906119888 119906lowast



2119860(119906) plusmn

radic119889221198602(119906) + 4119889

11198892119861(119906)119862(119906))2119889

11198892 Let

119878 (119906) = 1198892

21198602

(119906) + 411988911198891119861 (119906) 119862 (119906) (36)

Since

119878 (119898) = 1198892

21198602

(119906) gt 0 119878 (119906) = 411988911198892119861 (119906) 119862 (119906) lt 0

(37)



120582 as follows


minus(119906) is



211988911198892120582 (119906) 120582

1015840

(119906) minus 11988921198601015840

(119906) 120582 (119906)

minus 1198892119860 (119906) 120582

1015840

(119906) minus 119861 (119906) 1198621015840

(119906) = 0

(38)

Therefore 1205821015840(119906) = (119861(119906)1198621015840(119906) + 11988921198601015840

(119906)120582(119906))1198892(21198891120582(119906) minus


119904)

1198892(21198891120582+(119906) minus 119860(119906)) gt 0 and 119889

2(21198891120582minus(119906) minus 119860(119906)) lt 0



(119906) = 119890119903(minus1199062

minus2119899119906+119898119899+119898119870+119899119870)119870(119906+119899)2 and 1198621015840(119906) =


119860 () = 119903 (1 minus

119870) 119898 + 119899

( + 119899)2minus119903

119870 minus 119898

+ 119899

=119903

119870( + 119899)2[minus2

minus 2119899 + 119898119899 + 119899119870 + 119898119870] = 0

(39)



119904) If

119906lowast


11988921198601015840


+(119906) lt 0



119879119894(119906) = 0 119863

119894(119906) gt 0 119879

119895(119906) = 0 119863

119895(119906) = 0

for 119895 = 119894(40)



119894(0 le 119894 le 119896)


119894 119906119894 V119906119894



119894(119906) gt 0 for


0(119906) = 0 119863

0(119906) gt 0 and

119879119895(119906) = minus(119889

1+ 1198892)120582119895lt 0 119863






1lt 119906 (41)


119894lt 119860(119906)(119889

1+ 1198892)

Therefore 119879119894(119906119894) = 0 and 119879

119895(119906119894) = 0 for 119895 = 119894 1 le 119894 le ℎ




1+ 1198892) = 119903(1 minus (119898119870))(119898(119898+ 119899)) gt 0119863(119898 120582) = 0


1+ 1198892)


minus(119906)



119897 120582119897) in119906 isin (119898 119906)Then119863

119894(119906119894) gt 0

if 119906119894gt 119906119887and 119863

119894(119906119894) lt 0 if 119906



119894 119906119894 V119906119894


119886 (119906) =11990611989110158401015840

(119906)

16(119891101584010158401015840

(119906)

11989110158401015840 (119906)+2

119906)

=minus119906 (119899 minus 119906) (2119899

2

+ 119898119899 + 119870119899 + 119870119898)

8(119906 + 119899)5

lt 0

(42)

if 119899 gt 119906 holds


minus1198891Δ119901 = 119903 (1 minus

119901

119870)(1 minus

119898 + 119899

119901 + 119899)119901 minus 119886119901119911 119909 isin Ω


120597119901

120597119899=120597119911

120597119899= 0 119909 isin 120597Ω

(43)





Δ119910 + 119892 (119909 119910 (119909)) ge 0 119894119899 Ω120597119910 (119909)

120597119899le 0 119900119899 120597Ω

(44)



119903 (1 minus119901

119870)(1 minus

119898 + 119899

119901 + 119899)119901 + 119889

1Δ119901 = 0 119909 isin Ω

120597119901

120597119899= 0 119909 isin 120597Ω

(45)

satisfy 0 le 119901(119909) le 119870


0 lt 119901 (119909) lt 119870 0 lt 119911 (119909) lt119890119903

119899120583(119870 minus 119898

2)

2

minus1198891119870

2≜ 119871

(46)








minus (1198901198891Δ119901 + 119889

2Δ119911)

= 119890119903 (1 minus119901

119870)(1 minus

119898 + 119899

119901 + 119899)119901 minus 120583119911

= 119890119903 (1 minus119901

119870)(1 minus

119898 + 119899

119901 + 119899)119901 minus

120583

1198892

(1198891119901 + 1198892119911) +

1205831198891

1198892

119901

le [119890119903

119870119899(119870 minus 119898

2)

2

minus1205831198891

1198892

]119870 minus120583

1198892

(1198891119901 + 1198892119911)

(47)


1198892119911 lt 1198891119901 + 1198892119911 lt1198892

120583[119890119903

119870119899(119870 minus 119898

2)

2

minus1205831198891

2]119870 (48)


Theorem 10 Let 1198631gt (1198702

+ 119898119870 minus 119898119899)1198701198992

1205821be a fixed


1ge 1198631



1198911(119901 119911) = 119903 (1 minus

119901

119870)(1 minus

119898 + 119899

119901 + 119899)119901 minus 119886119901119911

1198912(119901 119911) = 119890119886119901119911 minus 120583119911

(49)

Let

119901 =1

|Ω|intΩ

119901 (119909) 119889119909

119911 =1

|Ω|intΩ

119911 (119909) 119889119909

(50)

then intΩ

(119901minus119901)119889119909 = 0 intΩ


intΩ

1198891

1003816100381610038161003816nabla11990110038161003816100381610038162

119889119909

= intΩ

1198911(119901 119911) (119901 minus 119901) 119889119909

= intΩ

(1198911(119901 119911) minus 119891

1(119901 119911)) (119901 minus 119901) 119889119909

= intΩ

((minus119901119901 (119901 + 119901) (119901 minus 119901)2

+ (119870 + 119898) 119901119901(119901 minus 119901)2

minus 119870119898119899 (119901 minus 119901))

times ((119901 + 119899) (119901 + 119899))minus1


) 119889119909

le intΩ

((119870 + 119898119870 minus 119898119899)119870

1198992(119901 minus 119901)

2

+1198861198701003816100381610038161003816119901 minus 119901

1003816100381610038161003816 |119911 minus 119911| ) 119889119909

(51)

Similarly

intΩ

1198892|nabla119911|2

119889119909

= intΩ

1198912(119901 119911) (119911 minus 119911) 119889119909

= intΩ

(1198912(119901 119911) minus 119891

2(119901 119911)) (119911 minus 119911) 119889119909

= intΩ

((119890119886119901 minus 120583) (119911 minus 119911)2


le intΩ

((119890119886119870 minus 120583) (119911 minus 119911)2

+ 1198901198861198711003816100381610038161003816119901 minus 119901

1003816100381610038161003816 |119911 minus 119911|) 119889119909

(52)


intΩ

(1198891

1003816100381610038161003816nabla11990110038161003816100381610038162

+ 1198892|nabla119911|2

) 119889119909

le intΩ

((1198702

+ 119898119870 minus 119898119899)119870

1198992(119901 minus 119901)

2

+ (119886119870 + 119890119886119871)1003816100381610038161003816119901 minus 119901

1003816100381610038161003816 |119911 minus 119911|

+ (119890119886119870 minus 120583) (119911 minus 119911)2

)119889119909

le intΩ

(((1198702

+ 119898119870 minus 119898119899)119870

1198992+119886119870 + 119890119886119871

2120576) (119901 minus 119901)

2

+(119890119886119871 + 119886119870

2120576+ 119890119886119870 minus 120583) (119911 minus 119911)

2

)119889119909

(53)


intΩ

(11988911205821

1003816100381610038161003816119901 minus 11990110038161003816100381610038162

+ 11988921205821|119911 minus 119911|

2

) 119889119909

le intΩ

(((1198702

+ 119898119870 minus 119898119899)119870

1198992+119886119870 + 119890119886119871

2120576) (119901 minus 119901)

2

+ (119890119886119871 + 119886119870

2120576+ 119890119886119870 minus 120583) (119911 minus 119911)

2

)119889119909

(54)



2

+ 119898119870 minus 119898119899)1198701198992 a


+ 119898119870 minus

119898119899)1198701198992

+((119886119870+119890119886119871)2)120576 Let1198632= ((119890119886119871+119886119870)2120576+119890119886119870minus


proof


119894(119909)


1198972 and

120601119894(119909) = cos(119894119909119897) Let

119879119894(119906) = minus120582

119894(1198891+ 1198892) + 119860 (119906)

119863119894(119906) = 119889

111988921205822

119894minus 119860 (119906) 119889

2120582119894minus 119861 (119906) 119862 (119906)

(55)



119863119894(119906) = 0 119879

119894(119906) = 0 119863

119895(119906) = 0 119879

119895(119906) = 0

for 119895 = 119894

119889119863119894(119906)

119889119906= 0

(56)


119894(1 le 119894 le 119896

1015840




most three 119906119894for 119863

119894(119906119894) = 119863(119906

119894 120582119894) = 0 In particular if


119894isin (119898 119906

119904) such that

119863119894(119906119894) = 119863(119906



2120582119894minus 119861(119906)119862

1015840


21198601015840

(119906)120582plusmn(119906) lt



119898

4 Discussion




79998

79999

8

80001

80002

80003

80004

80005

80006

80007

80008

0 20 40 60 80 100 120

(a)

23076

23077

23077

23078

23078

23079

23079

2308

2308

0 20 40 60 80 100 120

(b)


1= 0001 119889

2= 0004 119890 = 05 119886 = 01 119899 = 5






120582

u

D(u 120582) = 0

T(u 120582) = 0

900

800

700

600

500

400

300

200

100

0

0 1 2 3 4 5 6 7


0001 1198892= 0004 119890 = 05 119886 = 1 119899 = 5 119898 = 5 120583 = 04 119903 = 2 and


120582

u

D(u 120582) = 0

T(u 120582) = 0

0 1 2 3 4

300

200

100

0


0001 1198892= 0004 119890 = 05 119886 = 1 119899 = 5 119898 = 1 120583 = 04 119903 = 2 and





Acknowledgments


References















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119889119881

119889119905= intΩ

(119882119901sdot 119901119905+119882119911sdot 119911119905) 119889119909

= intΩ

(1198881

119901 minus 119906

119901

120597119901

120597119905+ 1198882

119911 minus V119906

119911

120597119911

120597119905) 119889119909

= intΩ

(1198881(119901 minus 119906) [119903 (1 minus

119909

119870)119901 minus 119898

119901 + 119899minus 119886119911]

+ 1198882(119911 minus V

119906) 119890119886 (119901 minus 119906) ) 119889119909

+ intΩ

(1198881

119901 minus 119906

1199011198891Δ119901 + 119888

2

119911 minus V119906

1199111198892Δ119911)119889119909

= 1198681(119905) + 119868

2(119905)

(31)


1198682(119905) = minusint

Ω

(11988811198891

119906

11990121003816100381610038161003816nabla11990110038161003816100381610038162

+ 11988821198892

V119906

1199112|nabla119911|2

)119889119909 le 0 (32)

Further

1198681(119905)

= intΩ

(1198881(119901 minus 119906) [119903 (1 minus

119901

119870)119901 minus 119898

119901 + 119899minus 119903 (1 minus

119906

119870)119906 minus 119898

119906 + 119899

+119886V119906minus 119886119911] + 119888

2119890119886 (119901 minus 119906) (119911 minus V

119906) ) 119889119909

= intΩ

(1198881(119901 minus 119906)

2 119903

119870[minus119901119906 + 119870119898 minus 119899 (119901 + 119906) + 119899 (119898 + 119870)

(119901 + 119899) (119906 + 119899)]


119906) ) 119889119909

(33)



119889119881

119889119905= 1198681(119905) + 119868

2(119905) le 119868

1(119905)

= intΩ

1198881

119903

119870[minus119901119906 + 119870119898 + 119899 (119898 + 119870) minus 119899 (119901 + 119906)

(119901 + 119899) (119906 + 119899)]

times (119901 minus 119906)2

119889119909

le intΩ

1198881

119903

119870

119870119898 + 119898119899 + 119870119899 minus 119899119906

119899 (119906 + 119899)(119901 minus 119906)

2

119889119909

(34)








119894 Let

119879 (119906 120582) = 119860 (119906) minus 120582 (1198891+ 1198892)

119863 (119906 120582) = 119889111988921205822

minus 1198892119860 (119906) 120582 minus 119861 (119906) 119862 (119906)

(35)




1+1198892)) We can





lowast

) such that119860(119906) gt 0 in (119906

119888 119906) 119860(119906) lt 0 in (0 119906

119888) and 119860(0) =

119860(119906119888) = 119860(119906) = 0

(3) 1198601015840(119906) gt 0 in (max0 119906119888 119906lowast



2119860(119906) plusmn

radic119889221198602(119906) + 4119889

11198892119861(119906)119862(119906))2119889

11198892 Let

119878 (119906) = 1198892

21198602

(119906) + 411988911198891119861 (119906) 119862 (119906) (36)

Since

119878 (119898) = 1198892

21198602

(119906) gt 0 119878 (119906) = 411988911198892119861 (119906) 119862 (119906) lt 0

(37)



120582 as follows


minus(119906) is



211988911198892120582 (119906) 120582

1015840

(119906) minus 11988921198601015840

(119906) 120582 (119906)

minus 1198892119860 (119906) 120582

1015840

(119906) minus 119861 (119906) 1198621015840

(119906) = 0

(38)

Therefore 1205821015840(119906) = (119861(119906)1198621015840(119906) + 11988921198601015840

(119906)120582(119906))1198892(21198891120582(119906) minus


119904)

1198892(21198891120582+(119906) minus 119860(119906)) gt 0 and 119889

2(21198891120582minus(119906) minus 119860(119906)) lt 0



(119906) = 119890119903(minus1199062

minus2119899119906+119898119899+119898119870+119899119870)119870(119906+119899)2 and 1198621015840(119906) =


119860 () = 119903 (1 minus

119870) 119898 + 119899

( + 119899)2minus119903

119870 minus 119898

+ 119899

=119903

119870( + 119899)2[minus2

minus 2119899 + 119898119899 + 119899119870 + 119898119870] = 0

(39)



119904) If

119906lowast


11988921198601015840


+(119906) lt 0



119879119894(119906) = 0 119863

119894(119906) gt 0 119879

119895(119906) = 0 119863

119895(119906) = 0

for 119895 = 119894(40)



119894(0 le 119894 le 119896)


119894 119906119894 V119906119894



119894(119906) gt 0 for


0(119906) = 0 119863

0(119906) gt 0 and

119879119895(119906) = minus(119889

1+ 1198892)120582119895lt 0 119863






1lt 119906 (41)


119894lt 119860(119906)(119889

1+ 1198892)

Therefore 119879119894(119906119894) = 0 and 119879

119895(119906119894) = 0 for 119895 = 119894 1 le 119894 le ℎ




1+ 1198892) = 119903(1 minus (119898119870))(119898(119898+ 119899)) gt 0119863(119898 120582) = 0


1+ 1198892)


minus(119906)



119897 120582119897) in119906 isin (119898 119906)Then119863

119894(119906119894) gt 0

if 119906119894gt 119906119887and 119863

119894(119906119894) lt 0 if 119906



119894 119906119894 V119906119894


119886 (119906) =11990611989110158401015840

(119906)

16(119891101584010158401015840

(119906)

11989110158401015840 (119906)+2

119906)

=minus119906 (119899 minus 119906) (2119899

2

+ 119898119899 + 119870119899 + 119870119898)

8(119906 + 119899)5

lt 0

(42)

if 119899 gt 119906 holds


minus1198891Δ119901 = 119903 (1 minus

119901

119870)(1 minus

119898 + 119899

119901 + 119899)119901 minus 119886119901119911 119909 isin Ω


120597119901

120597119899=120597119911

120597119899= 0 119909 isin 120597Ω

(43)





Δ119910 + 119892 (119909 119910 (119909)) ge 0 119894119899 Ω120597119910 (119909)

120597119899le 0 119900119899 120597Ω

(44)



119903 (1 minus119901

119870)(1 minus

119898 + 119899

119901 + 119899)119901 + 119889

1Δ119901 = 0 119909 isin Ω

120597119901

120597119899= 0 119909 isin 120597Ω

(45)

satisfy 0 le 119901(119909) le 119870


0 lt 119901 (119909) lt 119870 0 lt 119911 (119909) lt119890119903

119899120583(119870 minus 119898

2)

2

minus1198891119870

2≜ 119871

(46)








minus (1198901198891Δ119901 + 119889

2Δ119911)

= 119890119903 (1 minus119901

119870)(1 minus

119898 + 119899

119901 + 119899)119901 minus 120583119911

= 119890119903 (1 minus119901

119870)(1 minus

119898 + 119899

119901 + 119899)119901 minus

120583

1198892

(1198891119901 + 1198892119911) +

1205831198891

1198892

119901

le [119890119903

119870119899(119870 minus 119898

2)

2

minus1205831198891

1198892

]119870 minus120583

1198892

(1198891119901 + 1198892119911)

(47)


1198892119911 lt 1198891119901 + 1198892119911 lt1198892

120583[119890119903

119870119899(119870 minus 119898

2)

2

minus1205831198891

2]119870 (48)


Theorem 10 Let 1198631gt (1198702

+ 119898119870 minus 119898119899)1198701198992

1205821be a fixed


1ge 1198631



1198911(119901 119911) = 119903 (1 minus

119901

119870)(1 minus

119898 + 119899

119901 + 119899)119901 minus 119886119901119911

1198912(119901 119911) = 119890119886119901119911 minus 120583119911

(49)

Let

119901 =1

|Ω|intΩ

119901 (119909) 119889119909

119911 =1

|Ω|intΩ

119911 (119909) 119889119909

(50)

then intΩ

(119901minus119901)119889119909 = 0 intΩ


intΩ

1198891

1003816100381610038161003816nabla11990110038161003816100381610038162

119889119909

= intΩ

1198911(119901 119911) (119901 minus 119901) 119889119909

= intΩ

(1198911(119901 119911) minus 119891

1(119901 119911)) (119901 minus 119901) 119889119909

= intΩ

((minus119901119901 (119901 + 119901) (119901 minus 119901)2

+ (119870 + 119898) 119901119901(119901 minus 119901)2

minus 119870119898119899 (119901 minus 119901))

times ((119901 + 119899) (119901 + 119899))minus1


) 119889119909

le intΩ

((119870 + 119898119870 minus 119898119899)119870

1198992(119901 minus 119901)

2

+1198861198701003816100381610038161003816119901 minus 119901

1003816100381610038161003816 |119911 minus 119911| ) 119889119909

(51)

Similarly

intΩ

1198892|nabla119911|2

119889119909

= intΩ

1198912(119901 119911) (119911 minus 119911) 119889119909

= intΩ

(1198912(119901 119911) minus 119891

2(119901 119911)) (119911 minus 119911) 119889119909

= intΩ

((119890119886119901 minus 120583) (119911 minus 119911)2


le intΩ

((119890119886119870 minus 120583) (119911 minus 119911)2

+ 1198901198861198711003816100381610038161003816119901 minus 119901

1003816100381610038161003816 |119911 minus 119911|) 119889119909

(52)


intΩ

(1198891

1003816100381610038161003816nabla11990110038161003816100381610038162

+ 1198892|nabla119911|2

) 119889119909

le intΩ

((1198702

+ 119898119870 minus 119898119899)119870

1198992(119901 minus 119901)

2

+ (119886119870 + 119890119886119871)1003816100381610038161003816119901 minus 119901

1003816100381610038161003816 |119911 minus 119911|

+ (119890119886119870 minus 120583) (119911 minus 119911)2

)119889119909

le intΩ

(((1198702

+ 119898119870 minus 119898119899)119870

1198992+119886119870 + 119890119886119871

2120576) (119901 minus 119901)

2

+(119890119886119871 + 119886119870

2120576+ 119890119886119870 minus 120583) (119911 minus 119911)

2

)119889119909

(53)


intΩ

(11988911205821

1003816100381610038161003816119901 minus 11990110038161003816100381610038162

+ 11988921205821|119911 minus 119911|

2

) 119889119909

le intΩ

(((1198702

+ 119898119870 minus 119898119899)119870

1198992+119886119870 + 119890119886119871

2120576) (119901 minus 119901)

2

+ (119890119886119871 + 119886119870

2120576+ 119890119886119870 minus 120583) (119911 minus 119911)

2

)119889119909

(54)



2

+ 119898119870 minus 119898119899)1198701198992 a


+ 119898119870 minus

119898119899)1198701198992

+((119886119870+119890119886119871)2)120576 Let1198632= ((119890119886119871+119886119870)2120576+119890119886119870minus


proof


119894(119909)


1198972 and

120601119894(119909) = cos(119894119909119897) Let

119879119894(119906) = minus120582

119894(1198891+ 1198892) + 119860 (119906)

119863119894(119906) = 119889

111988921205822

119894minus 119860 (119906) 119889

2120582119894minus 119861 (119906) 119862 (119906)

(55)



119863119894(119906) = 0 119879

119894(119906) = 0 119863

119895(119906) = 0 119879

119895(119906) = 0

for 119895 = 119894

119889119863119894(119906)

119889119906= 0

(56)


119894(1 le 119894 le 119896

1015840




most three 119906119894for 119863

119894(119906119894) = 119863(119906

119894 120582119894) = 0 In particular if


119894isin (119898 119906

119904) such that

119863119894(119906119894) = 119863(119906



2120582119894minus 119861(119906)119862

1015840


21198601015840

(119906)120582plusmn(119906) lt



119898

4 Discussion




79998

79999

8

80001

80002

80003

80004

80005

80006

80007

80008

0 20 40 60 80 100 120

(a)

23076

23077

23077

23078

23078

23079

23079

2308

2308

0 20 40 60 80 100 120

(b)


1= 0001 119889

2= 0004 119890 = 05 119886 = 01 119899 = 5






120582

u

D(u 120582) = 0

T(u 120582) = 0

900

800

700

600

500

400

300

200

100

0

0 1 2 3 4 5 6 7


0001 1198892= 0004 119890 = 05 119886 = 1 119899 = 5 119898 = 5 120583 = 04 119903 = 2 and


120582

u

D(u 120582) = 0

T(u 120582) = 0

0 1 2 3 4

300

200

100

0


0001 1198892= 0004 119890 = 05 119886 = 1 119899 = 5 119898 = 1 120583 = 04 119903 = 2 and





Acknowledgments


References















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











(119906) = 119890119903(minus1199062

minus2119899119906+119898119899+119898119870+119899119870)119870(119906+119899)2 and 1198621015840(119906) =


119860 () = 119903 (1 minus

119870) 119898 + 119899

( + 119899)2minus119903

119870 minus 119898

+ 119899

=119903

119870( + 119899)2[minus2

minus 2119899 + 119898119899 + 119899119870 + 119898119870] = 0

(39)



119904) If

119906lowast


11988921198601015840


+(119906) lt 0



119879119894(119906) = 0 119863

119894(119906) gt 0 119879

119895(119906) = 0 119863

119895(119906) = 0

for 119895 = 119894(40)



119894(0 le 119894 le 119896)


119894 119906119894 V119906119894



119894(119906) gt 0 for


0(119906) = 0 119863

0(119906) gt 0 and

119879119895(119906) = minus(119889

1+ 1198892)120582119895lt 0 119863






1lt 119906 (41)


119894lt 119860(119906)(119889

1+ 1198892)

Therefore 119879119894(119906119894) = 0 and 119879

119895(119906119894) = 0 for 119895 = 119894 1 le 119894 le ℎ




1+ 1198892) = 119903(1 minus (119898119870))(119898(119898+ 119899)) gt 0119863(119898 120582) = 0


1+ 1198892)


minus(119906)



119897 120582119897) in119906 isin (119898 119906)Then119863

119894(119906119894) gt 0

if 119906119894gt 119906119887and 119863

119894(119906119894) lt 0 if 119906



119894 119906119894 V119906119894


119886 (119906) =11990611989110158401015840

(119906)

16(119891101584010158401015840

(119906)

11989110158401015840 (119906)+2

119906)

=minus119906 (119899 minus 119906) (2119899

2

+ 119898119899 + 119870119899 + 119870119898)

8(119906 + 119899)5

lt 0

(42)

if 119899 gt 119906 holds


minus1198891Δ119901 = 119903 (1 minus

119901

119870)(1 minus

119898 + 119899

119901 + 119899)119901 minus 119886119901119911 119909 isin Ω


120597119901

120597119899=120597119911

120597119899= 0 119909 isin 120597Ω

(43)





Δ119910 + 119892 (119909 119910 (119909)) ge 0 119894119899 Ω120597119910 (119909)

120597119899le 0 119900119899 120597Ω

(44)



119903 (1 minus119901

119870)(1 minus

119898 + 119899

119901 + 119899)119901 + 119889

1Δ119901 = 0 119909 isin Ω

120597119901

120597119899= 0 119909 isin 120597Ω

(45)

satisfy 0 le 119901(119909) le 119870


0 lt 119901 (119909) lt 119870 0 lt 119911 (119909) lt119890119903

119899120583(119870 minus 119898

2)

2

minus1198891119870

2≜ 119871

(46)








minus (1198901198891Δ119901 + 119889

2Δ119911)

= 119890119903 (1 minus119901

119870)(1 minus

119898 + 119899

119901 + 119899)119901 minus 120583119911

= 119890119903 (1 minus119901

119870)(1 minus

119898 + 119899

119901 + 119899)119901 minus

120583

1198892

(1198891119901 + 1198892119911) +

1205831198891

1198892

119901

le [119890119903

119870119899(119870 minus 119898

2)

2

minus1205831198891

1198892

]119870 minus120583

1198892

(1198891119901 + 1198892119911)

(47)


1198892119911 lt 1198891119901 + 1198892119911 lt1198892

120583[119890119903

119870119899(119870 minus 119898

2)

2

minus1205831198891

2]119870 (48)


Theorem 10 Let 1198631gt (1198702

+ 119898119870 minus 119898119899)1198701198992

1205821be a fixed


1ge 1198631



1198911(119901 119911) = 119903 (1 minus

119901

119870)(1 minus

119898 + 119899

119901 + 119899)119901 minus 119886119901119911

1198912(119901 119911) = 119890119886119901119911 minus 120583119911

(49)

Let

119901 =1

|Ω|intΩ

119901 (119909) 119889119909

119911 =1

|Ω|intΩ

119911 (119909) 119889119909

(50)

then intΩ

(119901minus119901)119889119909 = 0 intΩ


intΩ

1198891

1003816100381610038161003816nabla11990110038161003816100381610038162

119889119909

= intΩ

1198911(119901 119911) (119901 minus 119901) 119889119909

= intΩ

(1198911(119901 119911) minus 119891

1(119901 119911)) (119901 minus 119901) 119889119909

= intΩ

((minus119901119901 (119901 + 119901) (119901 minus 119901)2

+ (119870 + 119898) 119901119901(119901 minus 119901)2

minus 119870119898119899 (119901 minus 119901))

times ((119901 + 119899) (119901 + 119899))minus1


) 119889119909

le intΩ

((119870 + 119898119870 minus 119898119899)119870

1198992(119901 minus 119901)

2

+1198861198701003816100381610038161003816119901 minus 119901

1003816100381610038161003816 |119911 minus 119911| ) 119889119909

(51)

Similarly

intΩ

1198892|nabla119911|2

119889119909

= intΩ

1198912(119901 119911) (119911 minus 119911) 119889119909

= intΩ

(1198912(119901 119911) minus 119891

2(119901 119911)) (119911 minus 119911) 119889119909

= intΩ

((119890119886119901 minus 120583) (119911 minus 119911)2


le intΩ

((119890119886119870 minus 120583) (119911 minus 119911)2

+ 1198901198861198711003816100381610038161003816119901 minus 119901

1003816100381610038161003816 |119911 minus 119911|) 119889119909

(52)


intΩ

(1198891

1003816100381610038161003816nabla11990110038161003816100381610038162

+ 1198892|nabla119911|2

) 119889119909

le intΩ

((1198702

+ 119898119870 minus 119898119899)119870

1198992(119901 minus 119901)

2

+ (119886119870 + 119890119886119871)1003816100381610038161003816119901 minus 119901

1003816100381610038161003816 |119911 minus 119911|

+ (119890119886119870 minus 120583) (119911 minus 119911)2

)119889119909

le intΩ

(((1198702

+ 119898119870 minus 119898119899)119870

1198992+119886119870 + 119890119886119871

2120576) (119901 minus 119901)

2

+(119890119886119871 + 119886119870

2120576+ 119890119886119870 minus 120583) (119911 minus 119911)

2

)119889119909

(53)


intΩ

(11988911205821

1003816100381610038161003816119901 minus 11990110038161003816100381610038162

+ 11988921205821|119911 minus 119911|

2

) 119889119909

le intΩ

(((1198702

+ 119898119870 minus 119898119899)119870

1198992+119886119870 + 119890119886119871

2120576) (119901 minus 119901)

2

+ (119890119886119871 + 119886119870

2120576+ 119890119886119870 minus 120583) (119911 minus 119911)

2

)119889119909

(54)



2

+ 119898119870 minus 119898119899)1198701198992 a


+ 119898119870 minus

119898119899)1198701198992

+((119886119870+119890119886119871)2)120576 Let1198632= ((119890119886119871+119886119870)2120576+119890119886119870minus


proof


119894(119909)


1198972 and

120601119894(119909) = cos(119894119909119897) Let

119879119894(119906) = minus120582

119894(1198891+ 1198892) + 119860 (119906)

119863119894(119906) = 119889

111988921205822

119894minus 119860 (119906) 119889

2120582119894minus 119861 (119906) 119862 (119906)

(55)



119863119894(119906) = 0 119879

119894(119906) = 0 119863

119895(119906) = 0 119879

119895(119906) = 0

for 119895 = 119894

119889119863119894(119906)

119889119906= 0

(56)


119894(1 le 119894 le 119896

1015840




most three 119906119894for 119863

119894(119906119894) = 119863(119906

119894 120582119894) = 0 In particular if


119894isin (119898 119906

119904) such that

119863119894(119906119894) = 119863(119906



2120582119894minus 119861(119906)119862

1015840


21198601015840

(119906)120582plusmn(119906) lt



119898

4 Discussion




79998

79999

8

80001

80002

80003

80004

80005

80006

80007

80008

0 20 40 60 80 100 120

(a)

23076

23077

23077

23078

23078

23079

23079

2308

2308

0 20 40 60 80 100 120

(b)


1= 0001 119889

2= 0004 119890 = 05 119886 = 01 119899 = 5






120582

u

D(u 120582) = 0

T(u 120582) = 0

900

800

700

600

500

400

300

200

100

0

0 1 2 3 4 5 6 7


0001 1198892= 0004 119890 = 05 119886 = 1 119899 = 5 119898 = 5 120583 = 04 119903 = 2 and


120582

u

D(u 120582) = 0

T(u 120582) = 0

0 1 2 3 4

300

200

100

0


0001 1198892= 0004 119890 = 05 119886 = 1 119899 = 5 119898 = 1 120583 = 04 119903 = 2 and





Acknowledgments


References















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













minus (1198901198891Δ119901 + 119889

2Δ119911)

= 119890119903 (1 minus119901

119870)(1 minus

119898 + 119899

119901 + 119899)119901 minus 120583119911

= 119890119903 (1 minus119901

119870)(1 minus

119898 + 119899

119901 + 119899)119901 minus

120583

1198892

(1198891119901 + 1198892119911) +

1205831198891

1198892

119901

le [119890119903

119870119899(119870 minus 119898

2)

2

minus1205831198891

1198892

]119870 minus120583

1198892

(1198891119901 + 1198892119911)

(47)


1198892119911 lt 1198891119901 + 1198892119911 lt1198892

120583[119890119903

119870119899(119870 minus 119898

2)

2

minus1205831198891

2]119870 (48)


Theorem 10 Let 1198631gt (1198702

+ 119898119870 minus 119898119899)1198701198992

1205821be a fixed


1ge 1198631



1198911(119901 119911) = 119903 (1 minus

119901

119870)(1 minus

119898 + 119899

119901 + 119899)119901 minus 119886119901119911

1198912(119901 119911) = 119890119886119901119911 minus 120583119911

(49)

Let

119901 =1

|Ω|intΩ

119901 (119909) 119889119909

119911 =1

|Ω|intΩ

119911 (119909) 119889119909

(50)

then intΩ

(119901minus119901)119889119909 = 0 intΩ


intΩ

1198891

1003816100381610038161003816nabla11990110038161003816100381610038162

119889119909

= intΩ

1198911(119901 119911) (119901 minus 119901) 119889119909

= intΩ

(1198911(119901 119911) minus 119891

1(119901 119911)) (119901 minus 119901) 119889119909

= intΩ

((minus119901119901 (119901 + 119901) (119901 minus 119901)2

+ (119870 + 119898) 119901119901(119901 minus 119901)2

minus 119870119898119899 (119901 minus 119901))

times ((119901 + 119899) (119901 + 119899))minus1


) 119889119909

le intΩ

((119870 + 119898119870 minus 119898119899)119870

1198992(119901 minus 119901)

2

+1198861198701003816100381610038161003816119901 minus 119901

1003816100381610038161003816 |119911 minus 119911| ) 119889119909

(51)

Similarly

intΩ

1198892|nabla119911|2

119889119909

= intΩ

1198912(119901 119911) (119911 minus 119911) 119889119909

= intΩ

(1198912(119901 119911) minus 119891

2(119901 119911)) (119911 minus 119911) 119889119909

= intΩ

((119890119886119901 minus 120583) (119911 minus 119911)2


le intΩ

((119890119886119870 minus 120583) (119911 minus 119911)2

+ 1198901198861198711003816100381610038161003816119901 minus 119901

1003816100381610038161003816 |119911 minus 119911|) 119889119909

(52)


intΩ

(1198891

1003816100381610038161003816nabla11990110038161003816100381610038162

+ 1198892|nabla119911|2

) 119889119909

le intΩ

((1198702

+ 119898119870 minus 119898119899)119870

1198992(119901 minus 119901)

2

+ (119886119870 + 119890119886119871)1003816100381610038161003816119901 minus 119901

1003816100381610038161003816 |119911 minus 119911|

+ (119890119886119870 minus 120583) (119911 minus 119911)2

)119889119909

le intΩ

(((1198702

+ 119898119870 minus 119898119899)119870

1198992+119886119870 + 119890119886119871

2120576) (119901 minus 119901)

2

+(119890119886119871 + 119886119870

2120576+ 119890119886119870 minus 120583) (119911 minus 119911)

2

)119889119909

(53)


intΩ

(11988911205821

1003816100381610038161003816119901 minus 11990110038161003816100381610038162

+ 11988921205821|119911 minus 119911|

2

) 119889119909

le intΩ

(((1198702

+ 119898119870 minus 119898119899)119870

1198992+119886119870 + 119890119886119871

2120576) (119901 minus 119901)

2

+ (119890119886119871 + 119886119870

2120576+ 119890119886119870 minus 120583) (119911 minus 119911)

2

)119889119909

(54)



2

+ 119898119870 minus 119898119899)1198701198992 a


+ 119898119870 minus

119898119899)1198701198992

+((119886119870+119890119886119871)2)120576 Let1198632= ((119890119886119871+119886119870)2120576+119890119886119870minus


proof


119894(119909)


1198972 and

120601119894(119909) = cos(119894119909119897) Let

119879119894(119906) = minus120582

119894(1198891+ 1198892) + 119860 (119906)

119863119894(119906) = 119889

111988921205822

119894minus 119860 (119906) 119889

2120582119894minus 119861 (119906) 119862 (119906)

(55)



119863119894(119906) = 0 119879

119894(119906) = 0 119863

119895(119906) = 0 119879

119895(119906) = 0

for 119895 = 119894

119889119863119894(119906)

119889119906= 0

(56)


119894(1 le 119894 le 119896

1015840




most three 119906119894for 119863

119894(119906119894) = 119863(119906

119894 120582119894) = 0 In particular if


119894isin (119898 119906

119904) such that

119863119894(119906119894) = 119863(119906



2120582119894minus 119861(119906)119862

1015840


21198601015840

(119906)120582plusmn(119906) lt



119898

4 Discussion




79998

79999

8

80001

80002

80003

80004

80005

80006

80007

80008

0 20 40 60 80 100 120

(a)

23076

23077

23077

23078

23078

23079

23079

2308

2308

0 20 40 60 80 100 120

(b)


1= 0001 119889

2= 0004 119890 = 05 119886 = 01 119899 = 5






120582

u

D(u 120582) = 0

T(u 120582) = 0

900

800

700

600

500

400

300

200

100

0

0 1 2 3 4 5 6 7


0001 1198892= 0004 119890 = 05 119886 = 1 119899 = 5 119898 = 5 120583 = 04 119903 = 2 and


120582

u

D(u 120582) = 0

T(u 120582) = 0

0 1 2 3 4

300

200

100

0


0001 1198892= 0004 119890 = 05 119886 = 1 119899 = 5 119898 = 1 120583 = 04 119903 = 2 and





Acknowledgments


References















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











2

+ 119898119870 minus 119898119899)1198701198992 a


+ 119898119870 minus

119898119899)1198701198992

+((119886119870+119890119886119871)2)120576 Let1198632= ((119890119886119871+119886119870)2120576+119890119886119870minus


proof


119894(119909)


1198972 and

120601119894(119909) = cos(119894119909119897) Let

119879119894(119906) = minus120582

119894(1198891+ 1198892) + 119860 (119906)

119863119894(119906) = 119889

111988921205822

119894minus 119860 (119906) 119889

2120582119894minus 119861 (119906) 119862 (119906)

(55)



119863119894(119906) = 0 119879

119894(119906) = 0 119863

119895(119906) = 0 119879

119895(119906) = 0

for 119895 = 119894

119889119863119894(119906)

119889119906= 0

(56)


119894(1 le 119894 le 119896

1015840




most three 119906119894for 119863

119894(119906119894) = 119863(119906

119894 120582119894) = 0 In particular if


119894isin (119898 119906

119904) such that

119863119894(119906119894) = 119863(119906



2120582119894minus 119861(119906)119862

1015840


21198601015840

(119906)120582plusmn(119906) lt



119898

4 Discussion




79998

79999

8

80001

80002

80003

80004

80005

80006

80007

80008

0 20 40 60 80 100 120

(a)

23076

23077

23077

23078

23078

23079

23079

2308

2308

0 20 40 60 80 100 120

(b)


1= 0001 119889

2= 0004 119890 = 05 119886 = 01 119899 = 5






120582

u

D(u 120582) = 0

T(u 120582) = 0

900

800

700

600

500

400

300

200

100

0

0 1 2 3 4 5 6 7


0001 1198892= 0004 119890 = 05 119886 = 1 119899 = 5 119898 = 5 120583 = 04 119903 = 2 and


120582

u

D(u 120582) = 0

T(u 120582) = 0

0 1 2 3 4

300

200

100

0


0001 1198892= 0004 119890 = 05 119886 = 1 119899 = 5 119898 = 1 120583 = 04 119903 = 2 and





Acknowledgments


References















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










120582

u

D(u 120582) = 0

T(u 120582) = 0

900

800

700

600

500

400

300

200

100

0

0 1 2 3 4 5 6 7


0001 1198892= 0004 119890 = 05 119886 = 1 119899 = 5 119898 = 5 120583 = 04 119903 = 2 and


120582

u

D(u 120582) = 0

T(u 120582) = 0

0 1 2 3 4

300

200

100

0


0001 1198892= 0004 119890 = 05 119886 = 1 119899 = 5 119898 = 1 120583 = 04 119903 = 2 and





Acknowledgments


References















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Research Article Stability and Dynamical Analysis of a...

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