Effects of Internet Sales Promotion on a Differential ...

Research ArticleEffects of Internet Sales Promotion on a DifferentialAdvertising Model

Hui Jiang and Junhai Ma

College of Management and Economics Tianjin University Tianjin 300072 China

Correspondence should be addressed to Junhai Ma mjhtjualiyuncom

Received 23 November 2017 Accepted 12 March 2018 Published 6 May 2018

Academic Editor Mustafa R S Kulenovic

Copyright copy 2018 Hui Jiang and Junhai Ma This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

Advertising and sales promotion are two important specific marketing communications tools In this paper Internet salespromotion is introduced into a differential advertising model and investigated quantitatively The conditions for the existence andstability of periodic solutions are obtained Flip bifurcation of periodic solution is investigated analyticallyThe results show that thesales promotion parameter can modify the stability of the differential advertising model and lead to chaos through flip bifurcationthe sales level will eventually be no less than a given value by adjusting the value of the sales promotion parameter and the optimalsales promotion strategy can lead tomaximumprofit Numerical results for periodic solutions bifurcation diagrams and the effectsof sales promotion strategies which are illustrated with an example are in good agreement with the theoretical analysis Theseresults have certain significant theoretical and practical value in related markets

1 Introduction

Advertising is an audio or visual form of marketing com-munication that employs an openly sponsored nonpersonalmessage to promote or sell a product service or ideaThe purpose of advertising is to convince customers thata companyrsquos services or products are the best point outand create a need for products or services announce newproducts and programs draw customers to the business andhold existing customers There has been a growing interestin the problem of determining profit maximizing advertisingstrategies by both practitioners and researchers [1ndash4]

Many discrete and differential game models are con-structed to investigate advertising in recent years A two-dimensional map is proposed to describe a market wherethere are two quantity-setting firms producing homogeneousgoods [5] A discrete market share attraction model whereagents are boundedly rational is introduced to investigatemultistability and path dependence [6] (in this model brandmanagers are assumed to be boundedly rational and overtime adapt their marketing efforts for their brand in corre-spondence to the marginal profits of the previous period)Compared to discrete models [7] (by the theory of nonlinear

dynamics the authors embed multiproduct structure inoutput duopolymixed dynamical game model with partialprivatization and cross-ownership in which a private firmand a public firm coexist) [8] (dynamic system conceptsare integrated into this game model for understandingthe evolution of the strategic decisions over multiple timeperiods) differential game models are more suitable for themathematical simulation of evolutionary processes in adver-tising Recent years have seen many results about differentialgame models (see eg [9]) Su et al [10] use the dynamicevolutionary game model and the multiagent modeling sim-ulation to discuss the APQSIS agents equilibrium strategiesand the effects of their interactive behaviors on the APQSISevolutionary stability with asymmetric information Feinberg[11] assumes that the market share evolves according to adifferential game model with an S-shaped function providesa rational as to when the nonconstant advertising policy isoptimal and uses the classic phase-plane analysis to deal withthe unspecified nonlinearity in the response function (castingthe dynamic between advertising and sales in a commonformat (an autonomous first-order relationship) extensionsare explored along three dimensions an S-shaped responsefunction the value of the discount rate and the possibility of

HindawiDiscrete Dynamics in Nature and SocietyVolume 2018 Article ID 8618146 11 pageshttpsdoiorg10115520188618146

2 Discrete Dynamics in Nature and Society

diffusion like response) A differential game model is used toobtain firmsmarketing communications decisions inmarketswith uncertainty and competition [12] Krishnamoorthy etal [13] consider a duopoly differential game model withjoint advertising and pricing decisions and obtain the closed-form Nash equilibria for both the game with symmetriccompetitors and that with asymmetric competitors by usingthe linear and isoelastic demand function forms (differen-tial game theory is used to analyze two different demandspecificationsmdashlinear demand and isoelastic demandmdashforsymmetric and asymmetric competitors) Despite the usefulresults gained from differential game models in advertisingthe complex dynamical behaviors such as periodic solutionand bifurcation are rarely investigated in advertising modelsSu et al [14] establish the trilateral game payoff matrix buildup the replicator dynamic equations and discuss possibleevolutionary stable states

Sales promotion [15] (this study proposes that theresponses of more and less deal-prone consumers to pricediscounts and premiums depend on the promotional benefitlevel low deal-prone consumers are concerned with obtain-ing price discounts) including things like contests and gamessweepstakes product giveaways samples coupons loyaltyprograms and discounts is another way to advertise Theultimate goal of sales promotions is to stimulate potentialcustomers to action and create an immediate boost in salesvolume Previous studies mainly use the methods of data-analysis and quality analysis to discuss the role of salespromotion For example the effects of sales promotions onbrand preference are discussed and it is found that salespromotions do not affect postpromotion brand preferenceby using meta-analysis [16] Promotion cost is consideredin a model which shows that keeping the fractions ofpromotion cost sharingwithin an appropriate range increasesprofits for all parties [17] (this paper utilizes a mathematicalmodel for solving retailers promotion and replenishmentdecisions under retailer competition and promotional effortwith the sales learning curve an appropriate range for thefraction of promotion cost sharing can achieve channelcoordination and determine the distribution of increasableprofit) Andrews [18] examines the interplay between twomarketing interventions that consumers encounter in retailmarketplaces diagnostic product information and multi-item sales promotions and discusses that the influence ofproduct information varies as a function of sales promotionformat The effect of sales promotion should be furtherinvestigated quantitatively

Internet sales promotion refers to various activities thatstimulate consumer demand arouse the desire to buy andfacilitate the purchase by using computers and networktechnology to deliver information about goods and servicesto the virtual market With the development of technologyInternet sales promotion has become a major promotionstrategy in business management [19] It plays a positiverole in promoting the development of firms and can dis-seminate promotion information to improve the sales levelInternet sales promotion has many advantages comparedto traditional sales promotion It can push right promotioninformation to right people break through the limitation of

time and space and spread around the world around theclock One successful case of Internet sales promotion isAlibabarsquos annual global online shopping carnival (OSC) heldevery November 11 (also known as Double 11 Day or SinglersquosDay) On Singlesrsquo Day this shopping carnival brought theentire Chinese electronic commerce industry an astonishingrecord volume of transactions worth 912 billion RMB in 2015and 1207 billion RMB in 2016 [20]

Advertising and sales promotion [21 22] which are twospecific marketing communications tools account for atleast 25 of UK marketing budgets [23] From the aboveliterature Internet sales promotion is rarely considered into adifferential gamemodel and studied quantitativelyMotivatedby these facts advertising and Internet sales promotion areintroduced into a differential advertising model in this studyThis research focuses on the complex dynamics of this modeland the effect of Internet sales promotion on sales level Wehope to find a more intuitive way to show how Internetsales promotion affects the market share and optimal salespromotion strategy

The paper is organized as follows In Section 2 we intro-duce the basic model and make this link with practice Theexistence and stability of periodic solutions and bifurcation ofperiodic solutions are investigated in Section 3 In Section 4different sales promotion strategies are discussed Numericalresults such as the time series of solutions bifurcation dia-gram and largest Lyapunov exponent are given in Section 5Finally some conclusions are drawn in Section 6

2 Model Description

The following notations will be used in the paper119878(119905) sales level at time 119905119878(119905) the derivative of 119878(119905) at time 119905119906(119905) the advertising expenditure at time 119905Δ119878(119905) the increment of 119878(119905) at time 119905119872 the size of the potential market or saturation level119879 the time between two consecutive Internet salespromotions119861 the advertising effectiveness of firm119867 sales promotion target120588 decay constant120582 the characteristic root119903 response rate to advertising119887 a sales promotion parameter reflecting the incre-ment of sales level119888 the maximum decay constant

Effective advertising strategy is a response to changes inmarket conditions and evolves with time As such differentialmodels have been generally accepted as a viable approachto the study of advertising decisions A differential modelmodels the instantaneous variation of market share as afunction of the advertising expenditure levels and market

Discrete Dynamics in Nature and Society 3

S

S(0)

S1

S+1

T 2T 3T 4T0

t

(a)

0 S

bc M

ΔS = (b minus cS)S

b(2c) BM(B + )

ΔS

(b)

Figure 1 (a) The solutions of (1) and (4) from the initial point (0 119878(0)) (b) sketch of Internet sales promotion strategy

positions To address the dynamics of advertising decisionsthe following advertising model is well studied

119878 (119905) = 119903119906 (119905) (1 minus 119878 (119905)119872 ) minus 120588119878 (119905) (1)

Now consider uniform advertising policy [24] in whichthe firm advertises at a constant level in this paper Hence119906(119905) = 119880 119905 ge 0 Set the advertising effectiveness of firm to be119861 = 119903119880119872 and the initial value 119878(0) gt 119861119872(119861 + 120588) One caneasily calculate that the solution to system (1) with the initialvalue 119878(0) is119878 (119905) = (119878 (0) minus 119861119872119861 + 120588) exp [minus (119861 + 120588) 119905] + 119861119872119861 + 120588119905 ge 0 (2)

It is observed from Figure 1(a) that the sales level 119878(119905)decreases and tends to 119861119872(119861 + 120588)

The five parts of the promotional mix are advertisingsales promotion personal selling direct marketing andpublic relations Advertising is the way a company informspotential customers about the products and services it offersin the hope that they will buy eventually while a salespromotion is an incentive provided to consumers to motivatethem to buy immediately Internet sales promotion [25]including things like online discount promotions online giftpromotions online lottery promotion bonus point promo-tion and online joint promotion can stimulate potentialcustomers to action and create an immediate boost in salesvolume

To improve sales Internet sales promotion strategy isapplied here Suppose that the act of Internet sales promotionis held periodically to create an immediate boost in salesvolume and the increment Δ119878(119905) depends on sales level 119878(119905)namelyΔ119878 (119905) = 119878 (119905+) minus 119878 (119905) = (119887 minus 119888119878 (119905)) 119878 (119905) 119905 = 119899119879 (3)

where 119887 gt 0 119888 gt 0 119899 = 1 2 119878(119905+) = lim120591rarr0+119878(119905 + 120591)1198872119888 le 119861119872(119861 + 120588) lt 119887119888 le 119872 Figure 1(b) provides

a schematic illustration of this Internet sales promotionstrategy The measures to promote sales level are supposed tobe taken atmoments 119905 = 119899119879 and the sales level 119878(119905) turns from119878(119899119879) to 119878(119899119879+) where 119878(119899119879+) = 119878(119899119879) + (119887 minus 119888119878(119899119879))119878(119899119879)and (119887 minus 119888119878(119899119879))119878(119899119879) is the increment of sales

Now build the following advertising model with Internetsales promotion119878 = minus120588119878 + 119861 (119872 minus 119878) 119905 = 119899119879

119878 (119905+) = 119878 (119905) + (119887 minus 119888119878 (119905)) 119878 (119905) 119905 = 119899119879 (4)

System (4) is an impulsive differential system [26] Figure 1(a)also shows the graph of one solution to (4) The red lineshows the decreasing sales level under the influence of decayconstant 120588 while the blue line shows the change of saleslevel under Internet sales promotion strategy The trajectoryoriginating from the initial point 1198600(0 119878(0)) reaches thepoint 1198601(119879 119878(119879)) at 119905 = 119879 and next jumps to the point119860+1(119879 119878(119879+)) due to the effect of sales promotion and so onHence

119878 (119905) = (119878 (0) minus 119861119872119861 + 120588) exp [minus (119861 + 120588) 119905] + 119861119872119861 + 1205880 le 119905 le 119879

119878 (119879+) = 119878 (119879) + (119887 minus 119888119878 (119879)) 119878 (119879) 119878 (119905) = (119878 (119879+) minus 119861119872119861 + 120588) exp [minus (119861 + 120588) (119905 minus 119879)]

+ 119861119872119861 + 120588 119879 lt 119905 le 2119879

(5)

In the above Internet sales promotion is introduced intoa differential advertising model In the following paper com-plex dynamic properties of system (4) and sales promotionstrategy will be investigated analytically


3 Periodic Solutions and Bifurcation

In order to facilitate management and supply most firmshope that themarket share remains fixed or changes regularlyTherefore we discuss the periodic change of companyrsquosmarket share under Internet sales promotion strategy in thissection

Suppose the solution of (4) arrives at the point (119896119879 119878119896)at moment 119905 = 119896119879 then the solution jumps to point (119896119879 119878+119896 )due to the effect of sales promotion and reaches the point ((119896+1)119879 119878119896+1) at moment 119905 = (119896 + 1)119879 where 119878119896 = 119878(119896119879) 119878+119896 =119878(119896119879+) = 119878119896 + (119887 minus 119888119878119896)119878119896 119878119896+1 = 119878((119896 + 1)119879) It follows from(4) that

119878 ((119896 + 1) 119879) = (119878+119896 minus 119861119872119861 + 120588) exp [minus (119861 + 120588) 119879]+ 119861119872119861 + 120588

(6)

and a discrete map

119878119896+1 = (119878119896 + (119887 minus 119888119878119896) 119878119896 minus 119861119872119861 + 120588) exp [minus (119861 + 120588) 119879]+ 119861119872119861 + 120588

(7)

namely 119878119896+1 = 11986611198782119896 + 1198662 (119887) 119878119896 + 1198663 fl 119865 (119878119896) (8)where1198661 = minus119888 exp[minus(119861+120588)119879]1198662(119887) = (1+119887) exp[minus(119861+120588)119879]1198663 = (119861119872(119861 + 120588))(1 minus exp[minus(119861 + 120588)119879])

If 119878(119896119879) = 119878((119896+1)119879) then the evolution of the dynamicsrepeats itself For this to hold there must be a fixed point 1198780in discrete map (8) that isminus119866111987820 + (1 minus 1198662 (119887)) 1198780 minus 1198663 = 0 (9)Note that minus1198661 gt 0 and 1198663 gt 0 Equation (9) has one positiveroot and one negative root and the positive root is

11987801 = 1 minus 1198662 (119887) minus radic(1198662 (119887) minus 1)2 minus 41198661119866321198661 gt 0 (10)

The fixed point 11987801 of map (8) corresponds to thefollowing period-119879 solution of system (4)

119878 (119905) = (11987801 + (119887 minus 11988811987801) 11987801 minus 119861119872119861 + 120588)sdot exp [minus (119861 + 120588) (119905 minus 119899119879)] + 119861119872119861 + 120588

119899119879 lt 119905 le (119899 + 1) 119879(11)

where 11987801 is shown in (10)Furthermore it follows from (8) and (10) that the eigen-

value of the fixed point 1198780 is120582 = 1198651015840 (1198780) = 211986611198780 + 1198662 (119887)= 1 minus radic(1198662 (119887) minus 1)2 minus 411986611198663 (12)

When |120582| lt 1 namely

minus 1 lt 1 minus radic(1198662 (119887) minus 1)2 minus 411986611198663 lt 1(1198662 (119887) minus 1)2 minus 411986611198663 lt 4 (13)

the period-119879 solution (11) of system (4) is stable Thus thediscussion above gives the following result

Proposition 1 Suppose that (13) holds Then system (4) has aunique stable period-119879 solution which is shown in (11)

With the implementation of Internet sales promotionstrategy great changes have taken place in sales level 119878(119905)Proposition 1 shows that sales level 119878(119905) will change peri-odically with period-119879 as the sales promotion parameter 119887meets condition (13) In Proposition 1 the period-119879 solutionis stable under condition (1198662(119887) minus 1)2 minus 411986611198663 lt 4 In thecase of (1198662(119887) minus 1)2 minus 411986611198663 gt 4 are there other kinds ofperiodic solutions in system (4) To answer this question itneeds to investigate bifurcation of periodic solutions by usingthe following lemma [27]

Lemma 2 Let 119891120583 Rrarr R be a one-parameter family of mapsuch that 1198911205830 has a fixed point 1199090 with eigenvalue minus1 Assumethe following conditions

(1198621) ((120597119891120597120583)(12059721198911205971199092) + 2(1205972119891120597119909120597120583)) = 0 at (1199090 1205830)(1198622) 119886 = (12)(12059721198911205971199092)2+(13)(12059731198911205971199093) = 0 at (1199090 1205830)

Then there is a smooth curve of fixed points of 119891120583 passingthrough (1199090 1205830) the stability of which changes at (1199090 1205830)There is also a smooth curve 120574 passing through (1199090 1205830) so that120574 (1199090 1205830) is a union of hyperbolic period-2 orbits

In (1198622) the sign of 119886 determines the stability and thedirection of bifurcation of the orbits of period-2 If 119886 ispositive the orbits are stable if 119886 is negative they are unstable

Suppose the eigenvalue of the fixed point 11987801 is 120582 = minus1 for119887 = 1198870 then120582 = 1 minus radic(1198662 (1198870) minus 1)2 minus 411986611198663 = minus1 (14)

One can calculate that

1198870 = (1 + radic4 + 411986611198663) exp [(119861 + 120588) 119879] minus 111987801 (1198870) = minus1 minus radic1 + 119866111986631198661 (15)

An eigenvaluewithminus1 is associatedwith a flip bifurcationSo (11987801(1198870) 1198870) is a candidate for flip bifurcation point

By using (8) it follows that

120597119865120597119887 12059721198651205971198782119896

+ 2 1205972119865120597119878119896120597119887 = 21198661 exp [minus (119861 + 120588) 119879] 11987801+ 2 exp [minus (119861 + 120588) 119879]


= minus2 exp [minus (119861 + 120588) 119879]radic1 + 11986611198663= 0

119886 = 12 (12059721198651205971198782119896

)2 + 13 (12059731198651205971198783119896

) = 211986621gt 0

(16)

at (119878119896 119887) = (11987801 1198870)Conditions (1198621) and (1198622) hold So a flip bifurcation

occurs in view of Lemma 2 A positive period-2119879 solutionbifurcates from the positive period-119879 solution at 119887 =1198870 Since 119886 gt 0 in (1198622) then the positive period-2119879solution is stable It also means that for some 120598 gt 0system (4) has a positive stable period-2119879 solution for 119887 isin(1198870 1198870 + 120598) At the same time the positive period-119879 solutionloses its stability The analysis above leads to the followingresults

Proposition 3 Suppose that 119887 = 1198870 where 1198870 is shown in (15)Then a flip bifurcation occurs at 119887 = 1198870 For some 120598 gt 0 system(4) has a positive stable period-2119879 solution for 119887 isin (1198870 1198870 + 120598)

The sales promotion parameter 119887 plays an important rolein advertising model For some 119887 (119887 lt 1198870) sales level 119878(119905)changes periodically with period-119879 Proposition 3 shows thata flip bifurcation occurs at 119887 = 1198870 and sales level 119878(119905) changesperiodically with period-2119879 for 119887 isin (1198870 1198870 + 120598) Hence system(4) changes from a steady state (period-119879) to another one(period-2119879)

To study the period-2119879 solution in Proposition 3 con-sider the following iterative map

119878119896+1 = 1198652 (119878119896)= 119866311198784119896 + 2119866211198662 (119887) 1198783119896 + 1198661 (119887) 1198782119896 + 1198662 (119887) 119878119896+ 1198663 (119887) (17)

where 1198661(119887) = 11986611198662(119887) + 119866111986622(119887) + 2119866211198663 1198662(119887) = 11986622(119887) +2119866111986631198662(119887) 1198663(119887) = 119866111986623 + 1198662(119887)1198663 + 1198663Suppose that 1198780 is a fixed point of map (17) Then 1198780 =1198652(1198780) namely

1198663111987840 + 2119866211198662 (119887) 11987830 + 1198661 (119887) 11987820 + (1198662 (119887) minus 1) 1198780+ 1198663 (119887) = 0 (18)

It follows from the facts 11987801 = 119865(11987801) and 11987802 = 119865(11987802) that11987801 = 1198652(11987801) and 11987802 = 1198652(11987802) where 11987801 and 11987802 are shownin (10) Hence 11987801 and 11987802 are also the fixed points of map(17) Furthermore

1198663111987840 + 2119866211198662 (119887) 11987830 + 1198661 (119887) 11987820 + (1198662 (119887) minus 1) 1198780+ 1198663 (119887) = (1198780 minus 11987801) (1198780 minus 11987802) [1198662111987820+ (11986611198662 (119887) + 1198661) 1198780 + 11986611198663 + 1198662 (119887) + 1]

(19)

It follows from (18) that

1198662111987820 + (11986611198662 (119887) + 1198661) 1198780 + 11986611198663 + 1198662 (119887) + 1 = 0 (20)

Suppose that

(1198662 (119887) minus 1)2 minus 411986611198663 gt 4 (21)

holds A straightforward calculation leads to the two positiveroots to (20)

11987803 = minus11986611198662 (119887) minus 1198661 + radic[11986611198662 (119887) + 1198661]2 minus 411986621 (11986611198663 + 1198662 (119887) + 1)211986621 11987804 = minus11986611198662 (119887) minus 1198661 minus radic[11986611198662 (119887) + 1198661]2 minus 411986621 (11986611198663 + 1198662 (119887) + 1)211986621

(22)

Thus 11987803 and 11987804 are fixed points of map (17) 11987803 = 119865(11987804)and 11987804 = 119865(11987803)

The positive fixed points 11987803 and 11987804 of map (17) corre-spond to a period-2119879 solution of system (4) which startsfrom the point (0 11987803(1+119887minus11988811987803)) reaches the point (119879 11987804)jumps to the point (119879 11987804(1 + 119887 minus 11988811987804)) due to the effect ofsales promotion and reaches the point (2119879 11987803) The period-2119879 solution of system (4) is

119878 (119905) = 1198651 (11987803) exp [minus (119861 + 120588) (119905 minus 119899119879)] + 119861119872119861 + 120588119899119879 lt 119905 le (119899 + 1) 119879119878 (119905) = 1198651 (11987804) exp [minus (119861 + 120588) (119905 minus (119899 + 1) 119879)]+ 119861119872119861 + 120588 (119899 + 1) 119879 lt 119905 le (119899 + 2) 119879(23)

where 1198651(119909) = (1 + 119887 minus 119888119909)119909 minus 119861119872(119861 + 120588)


Proposition 4 Suppose that (21) holds Then system (4) has aunique period-2119879 solution which is shown in (23)

4 Sales Promotion Strategy

Propositions 1 and 3 show that system (4) has stable periodicsolutionsThe solutions from different initial points will tendto these stable periodic solutions

Suppose the sales target is that the sales level 119878(119905) is noless than119867 Now we discuss the sales promotion strategy toachieve this goal In the case of stable period-119879 solution thesales level 119878(119905) changes according to119878 (119905) = (11987801 + (119887 minus 11988811987801) 11987801 minus 119861119872119861 + 120588)

sdot exp [minus (119861 + 120588) (119905 minus 119899119879)] + 119861119872119861 + 120588119899119879 lt 119905 le (119899 + 1) 119879(24)

and 119878(119905) isin [11987801 11987801 + (119887 minus 11988811987801)11987801)To achieve the above sales target it requires that 11987801 ge 119867

Suppose that minus21198661119867 minus (1198662(119887) minus 1) gt 0 namely

119887 le (1 minus 21198661119867) exp [(119861 + 120588) 119879] minus 1 fl 1198870 (25)

It follows from 11987801 ge 119867 that

1 minus 1198662 (119887) minus radic(1198662 (119887) minus 1)2 minus 41198661119866321198661 ge 119867

radic(1198662 (119887) minus 1)2 minus 411986611198663 ge minus21198661119867 minus (1198662 (119887) minus 1) 119887 ge (1 minus 1198663 + 11986611198672119867 ) exp [(119861 + 120588) 119879] minus 1 fl 1198871

(26)

If 1198871 lt min1198870 1198870 the periodic solution (11) is stable for1198871 le 119887 le min1198870 1198870 in view of Proposition 1 The analysisabove leads to the following results

Proposition 5 Suppose that 1198871 le 119887 le min1198870 1198870 where 11988701198870 and 1198871 are shown in (15) (25) and (26) respectively Thenthe sales level 119878(119905) will eventually be no less than119867 and 119878(119905) isin[11987801 11987801 + (119887 minus 11988811987801)11987801)

In the case of the period-2119879 solution the sales level 119878(119905)changes according to (23)

From (22) it follows that 11987803 gt 11987804 gt 0 and 119878(119905) isin[11987804 11987804 + (119887 minus 11988811987804)11987804) To make sure the sales level 119878(119905)will eventually be no less than119867 in the case of stable period-2119879 solution it requires that 11987804 ge 119867 Suppose that minus211986621119867 minus11986611198662(119887) minus 1198661 gt 0 namely

119887 ge (minus1 minus 21198661119867) exp [(119861 + 120588) 119879] minus 1 fl 1198870 (27)


minus11986611198662 (119887) minus 1198661 minus radic[11986611198662 (119887) + 1198661]2 minus 411986621 (11986611198663 + 1198662 (119887) + 1)211986621 ge 119867(1 + 1198661119867) (1198662 (119887) + 1) ge minus11986611198663 minus 119866211198672

(28)

Under condition 1 + 1198661119867 lt 0 one can calculate that

119887 le (minus1 minus 11986611198663 + 1198662111986721 + 1198661119867 ) exp [(119861 + 120588) 119879] minus 1 fl 1198872 (29)

If the period-2119879 solution (23) is stable for max1198870 1198870 lt 119887 lt1198872 119878(119905) will eventually be no less than the threshold119867

Proposition 6 Suppose that 1 + 1198661119867 lt 0 and the period-2119879 solution (23) is stable for max1198870 1198870 lt 119887 lt 1198872 where 11988701198870 and 1198872 are shown in (15) (27) and (29) respectively Thenthe sales level 119878(119905) will eventually be no less than119867 and 119878(119905) isin[11987804 11987804 + (119887 minus 11988811987804)11987804)

Each firm cares about not only sales but also profit Forsimplicity stable period-119879 solution (119887 lt 1198870) is used to discussfirmrsquos profit Set the profit function as

119875119878 (119887) = int1198790(120590119878 (119905) minus 1199062) 119889119905

minus 119901 [(119887 minus 119888119878 (119879)) 119878 (119879)]2 (30)

where 120590 is firmrsquos unit profit margin 119906 is constant advertisingexpenditure 119901 is unit promotion cost 119878(119905) is shown in (11)119887 lt 1198870 and 119878(119879) = 11987801

It follows from (11) that(119887 minus 11988811987801) 11987801= (11987801 minus 119861119872119861 + 120588) (exp ((119861 + 120588) 119879) minus 1) (31)

and then

119875119878 (119887) = int1198790[120590119878 (119905) minus 1199062] 119889119905 minus 119901 [(119887 minus 11988811987801) 11987801]2

= minus119901[(11987801 minus 119861119872119861 + 120588) (exp ((119861 + 120588) 119879) minus 1)]2+ exp ((119861 + 120588) 119879) minus 1119861 + 120588 12059011987801 + 120590119861119872119879119861 + 120588 minus 1198802119879minus (exp ((119861 + 120588) 119879) minus 1)2 120590119861119872(119861 + 120588)2 exp ((119861 + 120588) 119879)

(32)


Hence

1198751198781015840 (119887)= (minus2119901(11987801 minus 119861119872119861 + 120588) (exp ((119861 + 120588) 119879) minus 1)2+ exp ((119861 + 120588) 119879) minus 1119861 + 120588 120590) 119878101584001 (119887)

(33)

By using (10) it is easy to see that

119878101584001 (119887) = exp (minus (119861 + 120588) 119879)21198661 (1

minus 1198662 (119887) minus 1radic(1198662 (119887) minus 1)2 minus 411986611198663) = 01198781015840101584001 (119887) = 21198663 exp (minus2 (119861 + 120588) 119879)[(1198662 (119887) minus 1)2 minus 411986611198663]32 gt 0

(34)

Let 1198751198781015840(1198873) = 0 Then 119887 = 1198873 is the solution to equation

minus 2119901(11987801 minus 119861119872119861 + 120588) ((exp ((119861 + 120588) 119879) minus 1))2+ exp ((119861 + 120588) 119879) minus 1119861 + 120588 120590 = 0 (35)

Similar to the discussion to (26) one can calculate that

1198873 = (1 minus 1198663 + 1198661119867211198671 ) exp [(119861 + 120588) 119879] minus 1 (36)

where1198671 = 119861119872(119861 + 120588) + 1205902119901(119861 + 120588)(exp((119861 + 120588)119879) minus 1)1198870 = 119887 le (1 minus 211986611198671) exp[(119861 + 120588)119879] minus 1 and 1198873 lt min1198870 1198870Furthermore

11987511987810158401015840 (1198873)= [minus2119901(11987801 minus 119861119872119861 + 120588) (exp ((119861 + 120588) 119879) minus 1)2+ exp ((119861 + 120588) 119879) minus 1119861 + 120588 120590] 1198781015840101584001 (1198873)minus 2119901 [exp ((119861 + 120588) 119879) minus 1]2 (119878101584001 (119887))2 = 0minus 2119901 [exp ((119861 + 120588) 119879) minus 1]2 (119878101584001 (119887))2 lt 0

(37)

In view of the facts1198751198781015840(1198873) = 0 and11987511987810158401015840(1198873) lt 0 the profitfunction119875119878(119887) reaches amaximal value at 119887 = 1198873 which givesthe following optimal sales promotion strategy

Proposition 7 Given system (4) and profit function (30) Theoptimal sales promotion strategy is

Δ119878 (119905) = (1198873 minus 119888119878 (119905)) 119878 (119905) 119905 = 119899119879 (38)

where 1198873 is shown in (36)

5 Numerical Results

To show the effect of Internet sales promotion strategy andcomplex dynamics of system (4) numerical results such asthe time series of period-119879 solution and period-2119879 solutionbifurcation diagram and largest Lyapunov exponents (LLE)are given in this section

Consider 119878 = minus09119878 + 01 (100 minus 119878) 119905 = 119899119878 (119905+) = 119878 (119905) + (119887 minus 02119878 (119905)) 119878 (119905) 119905 = 119899 (39)

where 120588 = 09 119861 = 01119872 = 100 119888 = 02 and 119879 = 1 Onecan easily calculate that 119861119872(119861 + 120588) = 10 1198661 asymp minus007361198662(119887) = exp(minus1)(1 + 119887) and 1198663 asymp 63212

Set 119887 = 48 in system (39) The positive fixed point ofmap (15) is 11987801 asymp 197571 and hence 119878+01 asymp 3652 Figure 2(a)shows that system (39) has a period-119879 solution which startsfrom the initial point (0 3652) and reaches (1 197571) at119905 = 1 Since (1198662(119887) minus 1)2 minus 411986611198663 asymp 31456 lt 4 condition(13) holds Figure 2(b) shows that the solution starting fromthe point (0 50) tends to the period-119879 solution in Figure 2(a)Illustrative simulations shown in Figure 2 are in agreementwith Proposition 1

A straightforward calculation gives 1198870 = (1 +radic4 + 411986611198663) exp[(119861 + 120588)119879] minus 1 asymp 56945 It follows fromProposition 3 that flip period-doubling bifurcation occursat 119887 = 56945 One-dimensional bifurcation diagram andlargest Lyapunov exponents (LLE) in Figure 3(a) can wellshow the dynamic characteristics of system (39) with the salespromotion parameter 119887 changes Along with the increase of 119887from 5 to 56945 system (39) has a stable period-119879 solutionand hence is in stable state System (39) is in stable statewith period-2119879 when 119887 changes from 56945 to 725 whilesystem is in stable state with period-4119879 when 119887 changesfrom 725 to 755 When the sales promotion parameter 119887 gt785 system (39) will go into chaotic state The LLE inFigure 3(a) corresponding to the bifurcation diagramcan alsoshow the system stability features The LLE equals zero for119887 = 56945 and flip bifurcation occurs When the first timeLLE gt 0 the system goes into chaotic state Simulationsshown in Figure 3(a) verify Proposition 3 In a word the salespromotion parameter 119887 can cause the advertising model todemonstrate complicated character

Set 119887 = 59 in (39)Then (1198662(119887) minus 1)2 minus411986611198663 asymp 43993 gt4 and condition (21) holds One can calculate that 11987803 asymp272830 11987804 asymp 208084 119878+04 = 11987804 + (59 minus 0211987804)11987804 asymp 5698119878+03 = 11987803 + (59 minus 0211987803)11987803 asymp 3938 Figure 3(b) showsthe unique period-2119879 solution which starts from (0 5698)reaches the point (1 27283) jumps to the point (1 3938)due to the effect of sales promotion arrives at the point(2 208084) jumps to the point (2 5698) due to the effect of


S(0) = 3652

10

15

20

25

30

35

40

45

50

55S

1 2 3 4 5 6 7 8 9 100

t

(a)

S(0) = 3652S(0) = 50

10

15

20

25

30

35

40

45

50

55

S

1 2 3 4 5 6 7 8 9 100

t

(b)

Figure 2 (a)The period-119879 solution to (39) with 119887 = 48 and the initial point (0 3652) (b) two solutions to system (39) with the initial points(0 3652) and (0 50)

0

10

20

30

40

LLE

andS k

55 6 65 7 75 85b

(a)

S(0) = 5698

152025303540455055606570

S

2 4 6 8 100t

(b)

Figure 3 (a) The bifurcation diagram and largest Lyapunov exponent of system (39) for 119887 isin (5 8) (b) the period-2119879 solution of (39) with119887 = 59 and the initial point (0 5698)sales promotion and so on Simulation shown in Figure 3(b)is in agreement with Proposition 4

Figure 3(a) also shows that the path to chaos is period-double bifurcation The period-4119879 solution to system (39)with 119887 = 725 is shown in Figure 4(a) while the solution tosystem (39) with 119887 = 785 is shown in Figure 4(b)

The value of 1198871 (25) for 119867 isin (10 235) is shown inFigure 5(a) Choose 119867 = 20 it follows that 1198871 asymp 4859 Nowset 119887 = 5 in system (39) It is observed from Figure 5(b) thatthe value of 119878(119905) will be eventually larger than119867 = 20 whichis in agreement with Proposition 5

Figure 6(a) shows the value of 1198872 (27) for 119867 isin (18 23)Choose 119867 = 20 it follows that 1198872 asymp 6084 Now set 119887 = 6 insystem (39) Conditions 1 + 1198661119867 asymp minus0472 lt 0 and 1198870 lt 119887 lt1198872 hold It is observed from Figure 6(b) that the value of 119878(119905)

will be eventually larger than119867 = 20 which is in agreementwith Proposition 6

Set the profit function as

119875119878 (119887) = int1198790(20119878 (119905) minus 200) 119889119905

minus 119901 [(119887 minus 021198782 (119879)) 1198782 (119879)]2 (40)

Take 119901 = 06 A straightforward calculation gives that 1198874 asymp4756 and the optional sales promotion strategy isΔ119878 (119905) = (4756 minus 02119878 (119905)) 119878 (119905) 119905 = 119899 (41)

Figure 7(a) shows the value of the profit 119875119878 (40) for 119901 =06 and 119887 isin (2 57) while Figure 7(b) shows the value of


20

30

40

50

60

70

80

90S

4 8 12 160

t

(a)

10

40

70

100

110

S

305 310 315 320 325 330 335 340300

t

(b)

Figure 4 (a) The period-4119879 solution of (39) with 119887 = 725 (b) the solution of (39) with 119887 = 785

2

25

3

35

4

45

5

55

6

b 1

12 14 16 18 20 22 2410

H

(a)

H = 20

10

15

20

25

30

35

40

45

50S

5 10 15 200

t

(b)

Figure 5 (a) The value of 1198871 for119867 isin (10 235) (b) the solution of (39) with 119887 = 5 and the initial point (0 15)

b 2

56

58

6

62

64

66

68

7

19 20 21 22 2318

H

(a)

H = 20

10

15

20

25

30

35

40

45

50

55

60

65

S

5 10 15 20 25 300

t

(b)



25 3 35 4 45 5 55 62

b

50

100

150

200

250

300

350

400PS

(a)

2 3 4 5 604

0608

1

bp

p = 06

50

100

150

200

250

300

350

400

PS

(b)

Figure 7 (a) The value of 119875119878(119887) (40) for 119901 = 06 and 119887 isin (2 57) (b) the value of 119875119878(119887) (40) for 119901 isin (04 1) and 119887 isin (2 57)the profit 119875119878 (40) for 119901 isin (04 1) and 119887 isin (2 57) Thesenumerical results are in agreement with Proposition 7

6 Conclusions

In this paper Internet sales promotion is introduced into adifferential advertising model Results show that the systempossesses complex dynamic behavior Advertising model hasa unique stable period-119879 solution or period-2119879 solution forparameter 119887 satisfies some conditions In these cases of peri-odic solutions sales level 119878(119905) tends to a periodic solution andeventually changes periodically which provides regulation todirect manufacture The sales promotion parameter 119887 whichcanmodify the system stability and lead to chaos through flipbifurcation plays an important role in the sales promotionstrategy

Internet sales promotion strategy is proposed to improvesales level andmaximize firmrsquos profit Suppose the sales targetis that the sales level 119878(119905) is no less than 119867 The firm shouldchoose the sales promotion parameter 119887 such that conditionsin Propositions 5 and 6 hold Profit is a key element infirmrsquos decision to employ what sales promotion strategy Theoptimal sales promotion strategy is obtained and used tomaximize firmrsquos profit in Proposition 7 These results havesome significant theoretical and practical values in relatedmarkets

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

Thiswork is jointly supported by theNatural Science Founda-tion of China under no 11662001 and the Science Foundationof Guangxi Province under no 2016GXNSFDA380031

References

[1] N Fukawa and Y Zhang ldquoProfit-sharing between an open-source firm and application developers - maximizing profits

from applications and in-application advertisementsrdquo Indus-trial Marketing Management vol 48 pp 111ndash120 2015

[2] C Zhang and H Zheng ldquoD3-Equivariant coupled advertisingoscillators modelrdquo Procedia Eng vol 192 pp 342ndash347 2011

[3] C F Nicholson and H M Kaiser ldquoDynamic market impacts ofgeneric dairy advertisingrdquo Journal of Business Research vol 61no 11 pp 1125ndash1135 2008

[4] A Buratto L Grosset and B Viscolani ldquoAdvertising channelselection in a segmented marketrdquoAutomatica vol 42 no 8 pp1343ndash1347 2006

[5] G-I Bischi L Stefanini and L Gardini ldquoSynchronizationintermittency and critical curves in a duopoly gamerdquo Mathe-matics and Computers in Simulation vol 44 no 6 pp 559ndash5851998

[6] G I Bischi and M Kopel ldquoMultistability and path dependencein a dynamic brand competition modelrdquo Chaos Solitons ampFractals vol 18 no 3 pp 561ndash576 2003

[7] F Wu and J Ma ldquoThe complex dynamics of a multi-productmixed duopoly model with partial privatization and cross-ownershiprdquo Nonlinear Dynamics vol 80 no 3 pp 1391ndash14012015

[8] J Ma and Z Guo ldquoThe parameter basin and complex ofdynamic game with estimation and two-stage considerationrdquoApplied Mathematics and Computation vol 248 pp 131ndash1422014

[9] J Huang M Leng and L Liang ldquoRecent developments indynamic advertising researchrdquo European Journal of OperationalResearch vol 220 no 3 pp 591ndash609 2012

[10] X Su H Liu and S Hou ldquoThe Trilateral Evolutionary Game ofAgri-Food Quality in FARmer-Supermarket Direct PurchaseA SIMulation APProachrdquo Complexity Article ID 5185497 11pages 2018

[11] FM Feinberg ldquoOn continuous-time optimal advertising underS-shaped responserdquo Management Science vol 47 no 11 pp1476ndash1487 2001

[12] A Prasad and S P Sethi ldquoIntegrated marketing communica-tions in markets with uncertainty and competitionrdquo Automat-ica vol 45 no 3 pp 601ndash610 2009

[13] A Krishnamoorthy A Prasad and S P Sethi ldquoOptimal pricingand advertising in a durable-good duopolyrdquo European Journalof Operational Research vol 200 no 2 pp 486ndash497 2010


[14] X Su S Duan S Guo and H Liu ldquoEvolutionary Games in theAgricultural Product Quality and Safety Information SYStemA MUltiagent SIMulation APProachrdquo Complexity Article ID7684185 13 pages 2018

[15] M Palazon and E Delgado-Ballester ldquoThe expected benefitas determinant of deal-prone consumers response to salespromotionsrdquo Journal of Retailing and Consumer Services vol 18no 6 pp 542ndash547 2011

[16] D DelVecchio D H Henard and T H Freling ldquoThe effect ofsales promotion on post-promotion brand preference A meta-analysisrdquo Journal of Retailing vol 82 no 3 pp 203ndash213 2006

[17] Y Tsao and G Sheen ldquoEffects of promotion cost sharing policywith the sales learning curve on supply chain coordinationrdquoComputers amp Operations Research vol 39 no 8 pp 1872ndash18782012

[18] D Andrews ldquoProduct information and consumer choice confi-dence in multi-item sales promotionsrdquo Journal of Retailing andConsumer Services vol 28 pp 45ndash53 2016

[19] H Martin K Eva and M Radovan The importance of e-mailmarketing in e-commerce Marketing Communication ModernPromotional Strategy Ronald Press Company New York NYUSA 2017

[20] X Xu Q Li L Peng et al ldquoThe impact of informationalincentives and social influence on consumer behavior duringAlibabas online shopping carnivalrdquo Comput Hum Behav vol76 pp 245ndash254 2017

[21] C W Park M S Roth and P F Jacques ldquoEvaluating the effectsof advertising and sales promotion campaignsrdquo Industrial Mar-keting Management vol 17 no 2 pp 129ndash140 1988

[22] I Buil L de Chernatony and E Martınez ldquoExamining the roleof advertising and sales promotions in brand equity creationrdquoJournal of Business Research vol 66 no 1 pp 115ndash122 2013

[23] J Byun and S Jang ldquoEffective promotions for membershipsubscriptions and renewals totourist attractions Discount vsbonusrdquo Tourism Management vol 50 pp 194ndash203 2015

[24] H I Mesak A Bari and Q Lian ldquoPulsation in a competitivemodel of advertising-firmrsquos cost interactionrdquo European Journalof Operational Research vol 246 no 3 pp 916ndash926 2015

[25] H K Cheng and K Dogan ldquoCustomer-centric marketing withInternet couponsrdquo Decision Support Systems vol 44 no 3 pp606ndash620 2008

[26] D D Bainov and P S Simeonov ldquoImpulsive DifferentialEquations Periodic Solutions and Applicationsrdquo Tech RepLongman Scientific amp Technical New York NY USA 1993

[27] J Guckenheimer andPHolmesNonlinearOscillationsDynam-ical Systems and Bifurcations of Vector Fields Springer-VerlagNew York NY USA 1983

Hindawiwwwhindawicom Volume 2018

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Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

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Dierential EquationsInternational Journal of

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Submit your manuscripts atwwwhindawicom


diffusion like response) A differential game model is used toobtain firmsmarketing communications decisions inmarketswith uncertainty and competition [12] Krishnamoorthy etal [13] consider a duopoly differential game model withjoint advertising and pricing decisions and obtain the closed-form Nash equilibria for both the game with symmetriccompetitors and that with asymmetric competitors by usingthe linear and isoelastic demand function forms (differen-tial game theory is used to analyze two different demandspecificationsmdashlinear demand and isoelastic demandmdashforsymmetric and asymmetric competitors) Despite the usefulresults gained from differential game models in advertisingthe complex dynamical behaviors such as periodic solutionand bifurcation are rarely investigated in advertising modelsSu et al [14] establish the trilateral game payoff matrix buildup the replicator dynamic equations and discuss possibleevolutionary stable states

Sales promotion [15] (this study proposes that theresponses of more and less deal-prone consumers to pricediscounts and premiums depend on the promotional benefitlevel low deal-prone consumers are concerned with obtain-ing price discounts) including things like contests and gamessweepstakes product giveaways samples coupons loyaltyprograms and discounts is another way to advertise Theultimate goal of sales promotions is to stimulate potentialcustomers to action and create an immediate boost in salesvolume Previous studies mainly use the methods of data-analysis and quality analysis to discuss the role of salespromotion For example the effects of sales promotions onbrand preference are discussed and it is found that salespromotions do not affect postpromotion brand preferenceby using meta-analysis [16] Promotion cost is consideredin a model which shows that keeping the fractions ofpromotion cost sharingwithin an appropriate range increasesprofits for all parties [17] (this paper utilizes a mathematicalmodel for solving retailers promotion and replenishmentdecisions under retailer competition and promotional effortwith the sales learning curve an appropriate range for thefraction of promotion cost sharing can achieve channelcoordination and determine the distribution of increasableprofit) Andrews [18] examines the interplay between twomarketing interventions that consumers encounter in retailmarketplaces diagnostic product information and multi-item sales promotions and discusses that the influence ofproduct information varies as a function of sales promotionformat The effect of sales promotion should be furtherinvestigated quantitatively

Internet sales promotion refers to various activities thatstimulate consumer demand arouse the desire to buy andfacilitate the purchase by using computers and networktechnology to deliver information about goods and servicesto the virtual market With the development of technologyInternet sales promotion has become a major promotionstrategy in business management [19] It plays a positiverole in promoting the development of firms and can dis-seminate promotion information to improve the sales levelInternet sales promotion has many advantages comparedto traditional sales promotion It can push right promotioninformation to right people break through the limitation of

time and space and spread around the world around theclock One successful case of Internet sales promotion isAlibabarsquos annual global online shopping carnival (OSC) heldevery November 11 (also known as Double 11 Day or SinglersquosDay) On Singlesrsquo Day this shopping carnival brought theentire Chinese electronic commerce industry an astonishingrecord volume of transactions worth 912 billion RMB in 2015and 1207 billion RMB in 2016 [20]

Advertising and sales promotion [21 22] which are twospecific marketing communications tools account for atleast 25 of UK marketing budgets [23] From the aboveliterature Internet sales promotion is rarely considered into adifferential gamemodel and studied quantitativelyMotivatedby these facts advertising and Internet sales promotion areintroduced into a differential advertising model in this studyThis research focuses on the complex dynamics of this modeland the effect of Internet sales promotion on sales level Wehope to find a more intuitive way to show how Internetsales promotion affects the market share and optimal salespromotion strategy

The paper is organized as follows In Section 2 we intro-duce the basic model and make this link with practice Theexistence and stability of periodic solutions and bifurcation ofperiodic solutions are investigated in Section 3 In Section 4different sales promotion strategies are discussed Numericalresults such as the time series of solutions bifurcation dia-gram and largest Lyapunov exponent are given in Section 5Finally some conclusions are drawn in Section 6

2 Model Description

The following notations will be used in the paper119878(119905) sales level at time 119905119878(119905) the derivative of 119878(119905) at time 119905119906(119905) the advertising expenditure at time 119905Δ119878(119905) the increment of 119878(119905) at time 119905119872 the size of the potential market or saturation level119879 the time between two consecutive Internet salespromotions119861 the advertising effectiveness of firm119867 sales promotion target120588 decay constant120582 the characteristic root119903 response rate to advertising119887 a sales promotion parameter reflecting the incre-ment of sales level119888 the maximum decay constant

Effective advertising strategy is a response to changes inmarket conditions and evolves with time As such differentialmodels have been generally accepted as a viable approachto the study of advertising decisions A differential modelmodels the instantaneous variation of market share as afunction of the advertising expenditure levels and market


S

S(0)

S1

S+1

T 2T 3T 4T0

t

(a)

0 S

bc M

ΔS = (b minus cS)S

b(2c) BM(B + )

ΔS

(b)



119878 (119905) = 119903119906 (119905) (1 minus 119878 (119905)119872 ) minus 120588119878 (119905) (1)








119878 (119905+) = 119878 (119905) + (119887 minus 119888119878 (119905)) 119878 (119905) 119905 = 119899119879 (4)




+ 119861119872119861 + 120588 119879 lt 119905 le 2119879

(5)






119878 ((119896 + 1) 119879) = (119878+119896 minus 119861119872119861 + 120588) exp [minus (119861 + 120588) 119879]+ 119861119872119861 + 120588

(6)

and a discrete map


(7)






119899119879 lt 119905 le (119899 + 1) 119879(11)









(1198621) ((120597119891120597120583)(12059721198911205971199092) + 2(1205972119891120597119909120597120583)) = 0 at (1199090 1205830)(1198622) 119886 = (12)(12059721198911205971199092)2+(13)(12059731198911205971199093) = 0 at (1199090 1205830)








120597119865120597119887 12059721198651205971198782119896




119886 = 12 (12059721198651205971198782119896

)2 + 13 (12059731198651205971198783119896

) = 211986621gt 0

(16)






119878119896+1 = 1198652 (119878119896)= 119866311198784119896 + 2119866211198662 (119887) 1198783119896 + 1198661 (119887) 1198782119896 + 1198662 (119887) 119878119896+ 1198663 (119887) (17)


1198663111987840 + 2119866211198662 (119887) 11987830 + 1198661 (119887) 11987820 + (1198662 (119887) minus 1) 1198780+ 1198663 (119887) = 0 (18)



(19)


1198662111987820 + (11986611198662 (119887) + 1198661) 1198780 + 11986611198663 + 1198662 (119887) + 1 = 0 (20)

Suppose that

(1198662 (119887) minus 1)2 minus 411986611198663 gt 4 (21)



(22)

















(26)








(28)






119875119878 (119887) = int1198790(120590119878 (119905) minus 1199062) 119889119905

minus 119901 [(119887 minus 119888119878 (119879)) 119878 (119879)]2 (30)



and then



(32)


Hence


(33)


119878101584001 (119887) = exp (minus (119861 + 120588) 119879)21198661 (1


(34)




1198873 = (1 minus 1198663 + 1198661119867211198671 ) exp [(119861 + 120588) 119879] minus 1 (36)



(37)



Δ119878 (119905) = (1198873 minus 119888119878 (119905)) 119878 (119905) 119905 = 119899119879 (38)


5 Numerical Results








S(0) = 3652

10

15

20

25

30

35

40

45

50

55S

1 2 3 4 5 6 7 8 9 100

t

(a)

S(0) = 3652S(0) = 50

10

15

20

25

30

35

40

45

50

55

S

1 2 3 4 5 6 7 8 9 100

t

(b)


0

10

20

30

40

LLE

andS k

55 6 65 7 75 85b

(a)

S(0) = 5698

152025303540455055606570

S

2 4 6 8 100t

(b)







119875119878 (119887) = int1198790(20119878 (119905) minus 200) 119889119905

minus 119901 [(119887 minus 021198782 (119879)) 1198782 (119879)]2 (40)




20

30

40

50

60

70

80

90S

4 8 12 160

t

(a)

10

40

70

100

110

S

305 310 315 320 325 330 335 340300

t

(b)


2

25

3

35

4

45

5

55

6

b 1

12 14 16 18 20 22 2410

H

(a)

H = 20

10

15

20

25

30

35

40

45

50S

5 10 15 200

t

(b)


b 2

56

58

6

62

64

66

68

7

19 20 21 22 2318

H

(a)

H = 20

10

15

20

25

30

35

40

45

50

55

60

65

S

5 10 15 20 25 300

t

(b)



25 3 35 4 45 5 55 62

b

50

100

150

200

250

300

350

400PS

(a)

2 3 4 5 604

0608

1

bp

p = 06

50

100

150

200

250

300

350

400

PS

(b)


6 Conclusions





Acknowledgments


References





































Journal of












Journal of







Volume 2018






Volume 2018









S

S(0)

S1

S+1

T 2T 3T 4T0

t

(a)

0 S

bc M

ΔS = (b minus cS)S

b(2c) BM(B + )

ΔS

(b)



119878 (119905) = 119903119906 (119905) (1 minus 119878 (119905)119872 ) minus 120588119878 (119905) (1)








119878 (119905+) = 119878 (119905) + (119887 minus 119888119878 (119905)) 119878 (119905) 119905 = 119899119879 (4)




+ 119861119872119861 + 120588 119879 lt 119905 le 2119879

(5)






119878 ((119896 + 1) 119879) = (119878+119896 minus 119861119872119861 + 120588) exp [minus (119861 + 120588) 119879]+ 119861119872119861 + 120588

(6)

and a discrete map


(7)






119899119879 lt 119905 le (119899 + 1) 119879(11)









(1198621) ((120597119891120597120583)(12059721198911205971199092) + 2(1205972119891120597119909120597120583)) = 0 at (1199090 1205830)(1198622) 119886 = (12)(12059721198911205971199092)2+(13)(12059731198911205971199093) = 0 at (1199090 1205830)








120597119865120597119887 12059721198651205971198782119896




119886 = 12 (12059721198651205971198782119896

)2 + 13 (12059731198651205971198783119896

) = 211986621gt 0

(16)






119878119896+1 = 1198652 (119878119896)= 119866311198784119896 + 2119866211198662 (119887) 1198783119896 + 1198661 (119887) 1198782119896 + 1198662 (119887) 119878119896+ 1198663 (119887) (17)


1198663111987840 + 2119866211198662 (119887) 11987830 + 1198661 (119887) 11987820 + (1198662 (119887) minus 1) 1198780+ 1198663 (119887) = 0 (18)



(19)


1198662111987820 + (11986611198662 (119887) + 1198661) 1198780 + 11986611198663 + 1198662 (119887) + 1 = 0 (20)

Suppose that

(1198662 (119887) minus 1)2 minus 411986611198663 gt 4 (21)



(22)

















(26)








(28)






119875119878 (119887) = int1198790(120590119878 (119905) minus 1199062) 119889119905

minus 119901 [(119887 minus 119888119878 (119879)) 119878 (119879)]2 (30)



and then



(32)


Hence


(33)


119878101584001 (119887) = exp (minus (119861 + 120588) 119879)21198661 (1


(34)




1198873 = (1 minus 1198663 + 1198661119867211198671 ) exp [(119861 + 120588) 119879] minus 1 (36)



(37)



Δ119878 (119905) = (1198873 minus 119888119878 (119905)) 119878 (119905) 119905 = 119899119879 (38)


5 Numerical Results








S(0) = 3652

10

15

20

25

30

35

40

45

50

55S

1 2 3 4 5 6 7 8 9 100

t

(a)

S(0) = 3652S(0) = 50

10

15

20

25

30

35

40

45

50

55

S

1 2 3 4 5 6 7 8 9 100

t

(b)


0

10

20

30

40

LLE

andS k

55 6 65 7 75 85b

(a)

S(0) = 5698

152025303540455055606570

S

2 4 6 8 100t

(b)







119875119878 (119887) = int1198790(20119878 (119905) minus 200) 119889119905

minus 119901 [(119887 minus 021198782 (119879)) 1198782 (119879)]2 (40)




20

30

40

50

60

70

80

90S

4 8 12 160

t

(a)

10

40

70

100

110

S

305 310 315 320 325 330 335 340300

t

(b)


2

25

3

35

4

45

5

55

6

b 1

12 14 16 18 20 22 2410

H

(a)

H = 20

10

15

20

25

30

35

40

45

50S

5 10 15 200

t

(b)


b 2

56

58

6

62

64

66

68

7

19 20 21 22 2318

H

(a)

H = 20

10

15

20

25

30

35

40

45

50

55

60

65

S

5 10 15 20 25 300

t

(b)



25 3 35 4 45 5 55 62

b

50

100

150

200

250

300

350

400PS

(a)

2 3 4 5 604

0608

1

bp

p = 06

50

100

150

200

250

300

350

400

PS

(b)


6 Conclusions





Acknowledgments


References





































Journal of












Journal of







Volume 2018






Volume 2018












119878 ((119896 + 1) 119879) = (119878+119896 minus 119861119872119861 + 120588) exp [minus (119861 + 120588) 119879]+ 119861119872119861 + 120588

(6)

and a discrete map


(7)






119899119879 lt 119905 le (119899 + 1) 119879(11)









(1198621) ((120597119891120597120583)(12059721198911205971199092) + 2(1205972119891120597119909120597120583)) = 0 at (1199090 1205830)(1198622) 119886 = (12)(12059721198911205971199092)2+(13)(12059731198911205971199093) = 0 at (1199090 1205830)








120597119865120597119887 12059721198651205971198782119896




119886 = 12 (12059721198651205971198782119896

)2 + 13 (12059731198651205971198783119896

) = 211986621gt 0

(16)






119878119896+1 = 1198652 (119878119896)= 119866311198784119896 + 2119866211198662 (119887) 1198783119896 + 1198661 (119887) 1198782119896 + 1198662 (119887) 119878119896+ 1198663 (119887) (17)


1198663111987840 + 2119866211198662 (119887) 11987830 + 1198661 (119887) 11987820 + (1198662 (119887) minus 1) 1198780+ 1198663 (119887) = 0 (18)



(19)


1198662111987820 + (11986611198662 (119887) + 1198661) 1198780 + 11986611198663 + 1198662 (119887) + 1 = 0 (20)

Suppose that

(1198662 (119887) minus 1)2 minus 411986611198663 gt 4 (21)



(22)

















(26)








(28)






119875119878 (119887) = int1198790(120590119878 (119905) minus 1199062) 119889119905

minus 119901 [(119887 minus 119888119878 (119879)) 119878 (119879)]2 (30)



and then



(32)


Hence


(33)


119878101584001 (119887) = exp (minus (119861 + 120588) 119879)21198661 (1


(34)




1198873 = (1 minus 1198663 + 1198661119867211198671 ) exp [(119861 + 120588) 119879] minus 1 (36)



(37)



Δ119878 (119905) = (1198873 minus 119888119878 (119905)) 119878 (119905) 119905 = 119899119879 (38)


5 Numerical Results








S(0) = 3652

10

15

20

25

30

35

40

45

50

55S

1 2 3 4 5 6 7 8 9 100

t

(a)

S(0) = 3652S(0) = 50

10

15

20

25

30

35

40

45

50

55

S

1 2 3 4 5 6 7 8 9 100

t

(b)


0

10

20

30

40

LLE

andS k

55 6 65 7 75 85b

(a)

S(0) = 5698

152025303540455055606570

S

2 4 6 8 100t

(b)







119875119878 (119887) = int1198790(20119878 (119905) minus 200) 119889119905

minus 119901 [(119887 minus 021198782 (119879)) 1198782 (119879)]2 (40)




20

30

40

50

60

70

80

90S

4 8 12 160

t

(a)

10

40

70

100

110

S

305 310 315 320 325 330 335 340300

t

(b)


2

25

3

35

4

45

5

55

6

b 1

12 14 16 18 20 22 2410

H

(a)

H = 20

10

15

20

25

30

35

40

45

50S

5 10 15 200

t

(b)


b 2

56

58

6

62

64

66

68

7

19 20 21 22 2318

H

(a)

H = 20

10

15

20

25

30

35

40

45

50

55

60

65

S

5 10 15 20 25 300

t

(b)



25 3 35 4 45 5 55 62

b

50

100

150

200

250

300

350

400PS

(a)

2 3 4 5 604

0608

1

bp

p = 06

50

100

150

200

250

300

350

400

PS

(b)


6 Conclusions





Acknowledgments


References





































Journal of












Journal of







Volume 2018






Volume 2018










119886 = 12 (12059721198651205971198782119896

)2 + 13 (12059731198651205971198783119896

) = 211986621gt 0

(16)






119878119896+1 = 1198652 (119878119896)= 119866311198784119896 + 2119866211198662 (119887) 1198783119896 + 1198661 (119887) 1198782119896 + 1198662 (119887) 119878119896+ 1198663 (119887) (17)


1198663111987840 + 2119866211198662 (119887) 11987830 + 1198661 (119887) 11987820 + (1198662 (119887) minus 1) 1198780+ 1198663 (119887) = 0 (18)



(19)


1198662111987820 + (11986611198662 (119887) + 1198661) 1198780 + 11986611198663 + 1198662 (119887) + 1 = 0 (20)

Suppose that

(1198662 (119887) minus 1)2 minus 411986611198663 gt 4 (21)



(22)

















(26)








(28)






119875119878 (119887) = int1198790(120590119878 (119905) minus 1199062) 119889119905

minus 119901 [(119887 minus 119888119878 (119879)) 119878 (119879)]2 (30)



and then



(32)


Hence


(33)


119878101584001 (119887) = exp (minus (119861 + 120588) 119879)21198661 (1


(34)




1198873 = (1 minus 1198663 + 1198661119867211198671 ) exp [(119861 + 120588) 119879] minus 1 (36)



(37)



Δ119878 (119905) = (1198873 minus 119888119878 (119905)) 119878 (119905) 119905 = 119899119879 (38)


5 Numerical Results








S(0) = 3652

10

15

20

25

30

35

40

45

50

55S

1 2 3 4 5 6 7 8 9 100

t

(a)

S(0) = 3652S(0) = 50

10

15

20

25

30

35

40

45

50

55

S

1 2 3 4 5 6 7 8 9 100

t

(b)


0

10

20

30

40

LLE

andS k

55 6 65 7 75 85b

(a)

S(0) = 5698

152025303540455055606570

S

2 4 6 8 100t

(b)







119875119878 (119887) = int1198790(20119878 (119905) minus 200) 119889119905

minus 119901 [(119887 minus 021198782 (119879)) 1198782 (119879)]2 (40)




20

30

40

50

60

70

80

90S

4 8 12 160

t

(a)

10

40

70

100

110

S

305 310 315 320 325 330 335 340300

t

(b)


2

25

3

35

4

45

5

55

6

b 1

12 14 16 18 20 22 2410

H

(a)

H = 20

10

15

20

25

30

35

40

45

50S

5 10 15 200

t

(b)


b 2

56

58

6

62

64

66

68

7

19 20 21 22 2318

H

(a)

H = 20

10

15

20

25

30

35

40

45

50

55

60

65

S

5 10 15 20 25 300

t

(b)



25 3 35 4 45 5 55 62

b

50

100

150

200

250

300

350

400PS

(a)

2 3 4 5 604

0608

1

bp

p = 06

50

100

150

200

250

300

350

400

PS

(b)


6 Conclusions





Acknowledgments


References





































Journal of












Journal of







Volume 2018






Volume 2018




















(26)








(28)






119875119878 (119887) = int1198790(120590119878 (119905) minus 1199062) 119889119905

minus 119901 [(119887 minus 119888119878 (119879)) 119878 (119879)]2 (30)



and then



(32)


Hence


(33)


119878101584001 (119887) = exp (minus (119861 + 120588) 119879)21198661 (1


(34)




1198873 = (1 minus 1198663 + 1198661119867211198671 ) exp [(119861 + 120588) 119879] minus 1 (36)



(37)



Δ119878 (119905) = (1198873 minus 119888119878 (119905)) 119878 (119905) 119905 = 119899119879 (38)


5 Numerical Results








S(0) = 3652

10

15

20

25

30

35

40

45

50

55S

1 2 3 4 5 6 7 8 9 100

t

(a)

S(0) = 3652S(0) = 50

10

15

20

25

30

35

40

45

50

55

S

1 2 3 4 5 6 7 8 9 100

t

(b)


0

10

20

30

40

LLE

andS k

55 6 65 7 75 85b

(a)

S(0) = 5698

152025303540455055606570

S

2 4 6 8 100t

(b)







119875119878 (119887) = int1198790(20119878 (119905) minus 200) 119889119905

minus 119901 [(119887 minus 021198782 (119879)) 1198782 (119879)]2 (40)




20

30

40

50

60

70

80

90S

4 8 12 160

t

(a)

10

40

70

100

110

S

305 310 315 320 325 330 335 340300

t

(b)


2

25

3

35

4

45

5

55

6

b 1

12 14 16 18 20 22 2410

H

(a)

H = 20

10

15

20

25

30

35

40

45

50S

5 10 15 200

t

(b)


b 2

56

58

6

62

64

66

68

7

19 20 21 22 2318

H

(a)

H = 20

10

15

20

25

30

35

40

45

50

55

60

65

S

5 10 15 20 25 300

t

(b)



25 3 35 4 45 5 55 62

b

50

100

150

200

250

300

350

400PS

(a)

2 3 4 5 604

0608

1

bp

p = 06

50

100

150

200

250

300

350

400

PS

(b)


6 Conclusions





Acknowledgments


References





































Journal of












Journal of







Volume 2018






Volume 2018









Hence


(33)


119878101584001 (119887) = exp (minus (119861 + 120588) 119879)21198661 (1


(34)




1198873 = (1 minus 1198663 + 1198661119867211198671 ) exp [(119861 + 120588) 119879] minus 1 (36)



(37)



Δ119878 (119905) = (1198873 minus 119888119878 (119905)) 119878 (119905) 119905 = 119899119879 (38)


5 Numerical Results








S(0) = 3652

10

15

20

25

30

35

40

45

50

55S

1 2 3 4 5 6 7 8 9 100

t

(a)

S(0) = 3652S(0) = 50

10

15

20

25

30

35

40

45

50

55

S

1 2 3 4 5 6 7 8 9 100

t

(b)


0

10

20

30

40

LLE

andS k

55 6 65 7 75 85b

(a)

S(0) = 5698

152025303540455055606570

S

2 4 6 8 100t

(b)







119875119878 (119887) = int1198790(20119878 (119905) minus 200) 119889119905

minus 119901 [(119887 minus 021198782 (119879)) 1198782 (119879)]2 (40)




20

30

40

50

60

70

80

90S

4 8 12 160

t

(a)

10

40

70

100

110

S

305 310 315 320 325 330 335 340300

t

(b)


2

25

3

35

4

45

5

55

6

b 1

12 14 16 18 20 22 2410

H

(a)

H = 20

10

15

20

25

30

35

40

45

50S

5 10 15 200

t

(b)


b 2

56

58

6

62

64

66

68

7

19 20 21 22 2318

H

(a)

H = 20

10

15

20

25

30

35

40

45

50

55

60

65

S

5 10 15 20 25 300

t

(b)



25 3 35 4 45 5 55 62

b

50

100

150

200

250

300

350

400PS

(a)

2 3 4 5 604

0608

1

bp

p = 06

50

100

150

200

250

300

350

400

PS

(b)


6 Conclusions





Acknowledgments


References





































Journal of












Journal of







Volume 2018






Volume 2018









S(0) = 3652

10

15

20

25

30

35

40

45

50

55S

1 2 3 4 5 6 7 8 9 100

t

(a)

S(0) = 3652S(0) = 50

10

15

20

25

30

35

40

45

50

55

S

1 2 3 4 5 6 7 8 9 100

t

(b)


0

10

20

30

40

LLE

andS k

55 6 65 7 75 85b

(a)

S(0) = 5698

152025303540455055606570

S

2 4 6 8 100t

(b)







119875119878 (119887) = int1198790(20119878 (119905) minus 200) 119889119905

minus 119901 [(119887 minus 021198782 (119879)) 1198782 (119879)]2 (40)




20

30

40

50

60

70

80

90S

4 8 12 160

t

(a)

10

40

70

100

110

S

305 310 315 320 325 330 335 340300

t

(b)


2

25

3

35

4

45

5

55

6

b 1

12 14 16 18 20 22 2410

H

(a)

H = 20

10

15

20

25

30

35

40

45

50S

5 10 15 200

t

(b)


b 2

56

58

6

62

64

66

68

7

19 20 21 22 2318

H

(a)

H = 20

10

15

20

25

30

35

40

45

50

55

60

65

S

5 10 15 20 25 300

t

(b)



25 3 35 4 45 5 55 62

b

50

100

150

200

250

300

350

400PS

(a)

2 3 4 5 604

0608

1

bp

p = 06

50

100

150

200

250

300

350

400

PS

(b)


6 Conclusions





Acknowledgments


References





































Journal of












Journal of







Volume 2018






Volume 2018









20

30

40

50

60

70

80

90S

4 8 12 160

t

(a)

10

40

70

100

110

S

305 310 315 320 325 330 335 340300

t

(b)


2

25

3

35

4

45

5

55

6

b 1

12 14 16 18 20 22 2410

H

(a)

H = 20

10

15

20

25

30

35

40

45

50S

5 10 15 200

t

(b)


b 2

56

58

6

62

64

66

68

7

19 20 21 22 2318

H

(a)

H = 20

10

15

20

25

30

35

40

45

50

55

60

65

S

5 10 15 20 25 300

t

(b)



25 3 35 4 45 5 55 62

b

50

100

150

200

250

300

350

400PS

(a)

2 3 4 5 604

0608

1

bp

p = 06

50

100

150

200

250

300

350

400

PS

(b)


6 Conclusions





Acknowledgments


References





































Journal of












Journal of







Volume 2018






Volume 2018









25 3 35 4 45 5 55 62

b

50

100

150

200

250

300

350

400PS

(a)

2 3 4 5 604

0608

1

bp

p = 06

50

100

150

200

250

300

350

400

PS

(b)


6 Conclusions





Acknowledgments


References





































Journal of












Journal of







Volume 2018






Volume 2018






























Journal of












Journal of







Volume 2018






Volume 2018








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