A mathematical model to study the simultaneous effect of ...€¦ · and non-wood industries has...

ISSN 1 746-7233, England, UKWorld Journal of Modelling and Simulation

Vol. 12 (2016) No. 1, pp. 34-47

A mathematical model to study the simultaneous effect of pollutants emittedfrom wood and non-wood based industries on forest resources

Abhinav Tandon∗ , Kumari Jyotsna

Department of Mathematics, Birla Institute of Technology, Mesra, Ranchi-835215 Jharkhand, India

(Received June 23 2015, Accepted September 05 2015)

Abstract. In this paper, we develop a model to study the effects of industrialization and associated pollutionon forest resources. Here, we assume that wood and non-wood based industries coexist in the forest habitatand both of these industries also generate pollution and affect the forest resources. Wood-based industriesdirectly depend on forest resources, whereas a constant rate of resources (which does not depend on theforest) is provided to non-wood industries. The proposed model is analyzed using the stability theory ofdifferential equations and numerical simulation. From the obtained results, it is inferred that the density offorest resources decreases, not only with an increase in the density of wood- based industries, but also withincrease in the density of non-wood industries. Although non-wood based industries do not depend on forestresources directly. It is also concluded that forest resources may become extinct if the pollution from woodand non-wood industries is not held in check.

Keywords: wood-based industries, non-wood industries, forest resources, stability, pollutants, simulation

1 Introduction

Jharkhand is one of the richest states in India for all kind of mineral deposits and forests. It has hugereserves of iron ores, coal, mica, bauxite, limestone, copper, uranium, zinc, manganese and other resources.Because of Jharkhand’s numerous natural resources, it is a suitable place for the establishment of all types ofwood and non-wood based industries. Non-wood based industries mean all types of industries such as miningindustries which are based on metallurgy of different types of mineral deposits or, in general, we can saythe industries that do not depend on forest directly. It is well known that forest resources play an importantrole for the survival of human beings and other organisms, whereas industrialization is driven by the needs ofgrowing human population of various countries[8]. The rapid growth of industrialization in the form of woodand non-wood industries has diminished the forest habitat. Damodar Valley, Nowamundi and Saranda aretypical examples where the forest habitat has been devastated significantly[19]. In particular, Damodar Valley,which once had forest cover of 65%, now stands at only 0.05%[20]. In recent years, scientists and ecologistshave observed that the global temperature of the environment is slowly increasing due to the emission oftoxicants (or pollutants) from industries leading to adverse effects on human being and the environment[10, 18].The pollution associated with these industries is absorbed by the plants and damages them and thus, the growthrate of forestry biomass is affected by the pollutants emitted by different sources[3, 5, 23, 24].

Therefore, in the above scenario, it is appropriate to develop and study the behavior of systems thatconsider the impact of industrialization on the forest resources. For this purpose, mathematical models areone of the efficient tools available. The use of mathematical models of environmental systems is not a newendeavor. Previously, many investigators have studied the effects of industrialization and pollution on forestresources (or forestry biomass) through mathematical models. Freedman and Shukla[11] studied the effects of

∗ Corresponding author. E-mail address: [email protected]

Published by World Academic Press, World Academic Union

World Journal of Modelling and Simulation, Vol. 12 (2016) No. 1, pp. 34-47 35

toxicant on the growth rate of the single species and its carrying capacity in a predator-prey system. Naresh etal.[17] proposed a mathematical model on the effects of toxicant on plant biomass. In their model, they gavean intimate relationship between plant biomass and toxicant and show that as the emission rate of toxicantincreases, the density of plant biomass decreases. More recently, Dubey and Hussain[9] analyzed a model inwhich emission rate of pollutant is assumed to be constant, zero or periodic and dependent upon industrial-ization. In this model, it has been shown that the resource biomass may tend to extinction due to the emissionof pollutants into the environment. Dubey and Narayanan[7] discussed the effects of industrialization, pop-ulation and pollution on the depletion of a renewable resource through mathematical model and found thatthe resource biomass decreases as the density of industrialization, population and pollution increases. Shuklaet al.[21] studied the effects of primary and secondary toxicants on the natural resources and showed that theresources may tend to extinct if the emission rate of primary toxicants and formation rate of secondary tox-icants are very high. Dubey et al.[8] found that the forestry resources lead to extinction on the account ofpopulation and industrialization driven by population pressure. They also discussed the control measures formaintaining the ecological stability. Agarwal et al.[2] investigated a ratio-dependent model to study the inter-action among forestry biomass, wildlife population and pressure of industrialization. They concluded that thedensity of forestry biomass and wildlife species decreases or it may be extinct if the industrialization continueswithout any delay. Misra and his coauthors[15, 16, 22] have also considered the depletion and conservation offorestry resources through mathematical models. They show that advanced technological efforts can conservethe forestry resources. Chaudhary et al.[4] studied a mathematical model of forestry resources conservationconsidering wood based industries and synthetic industries. In their paper, they have shown that forestry re-sources are being depleting through wood based industries and also discussed the conservation of forestryresources through the replacement of wood industries by synthetic industries. But, they did not consider thenegative effects of synthetic industries on the forestry resources.

Here, we propose and analyze a non-linear mathematical model to study the depletion of forest resourcesfrom industrialization-augmented pollution. It is assumed that wood-based as well as non-wood based in-dustries coexist in the forest habitat and the pollutants are also emitted from both types of industries. Here,wood-based industries directly depend on forest resources, whereas non-wood is growing at a constant rate.The aim is to investigate the simultaneous impacts of wood and non-wood industries and the pollution causedby them on the forest resources.

2 Mathematical model

Consider a forest resources that grow logistically and simultaneously decreases due to wood-based indus-tries as well as pollutants emitted through wood and non-wood based industries. It is assumed that the growthrate of wood-based industries is directly proportional to the density of forest resources whereas, a constantrate of resources (which does not depend on the forest resources) is provided to non-wood industries. Thedynamics of the problem are governed by the following system of nonlinear ordinary differential equation.

dF

dt= rF

(1− F

L

)− βFW − κ1FPW − κ2FPI ,

dW

dt= λF + β1FW − π1WI − ϕ1W,

dI

dt= Q− π2WI − ϕ2I, (1)

dPW

dt= θ1W − δ1FPW − λW PW ,

dPI

dt= θ2I − δ2FPI − λIPI ,

where F (0) ≥ 0, W (0) ≥ 0, I(0) ≥ 0, PW (0) ≥ 0, PI(0) ≥ 0.In model system (1), F is the density of forest resources, W is the density of wood-based industries,

I is the density of non-wood based industries. PW and PI are the concentrations of pollutants caused by

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36 A. Tandon & K. Jyotsna: A mathematical model to study the simultaneous effect of pollutants

wood and non-wood industries respectively. The constants r and L are intrinsic growth rate and carryingcapacity of the forest resources respectively. β represents the rate of depletion of forest resources due towood-based industries, whereas β1 signifies the growth rate of wood-based industries due to forest resources.The coefficients κ1 and κ2 are the depletion rates of forest resources caused by the pollutant generated throughwood and non- wood industries respectively. Here, we consider the migration of wood-based industries to theforest region and it is represented through the growth rate λ, which directly depends on the density of forestresources[22]. π1 and π2 are the competition rates between wood and non-wood industries, where is the rateof competition effect of I on W and π2 is the rate of competition effect of W on I . The constants ϕ1 andϕ2 are control rate coefficients applied by government authorities to wood and non-wood based industriesrespectively. The proportionality constants θ1 and θ2 are the growth rates of pollutant generated by woodand non-wood based industries respectively. δ1 and δ1 are the constants representing the loss of pollutantgenerated by wood and non-wood industries due to forest resources respectively. Q is the constant rate ofresources (which does not depend on forest resources) provided to non-wood based industries. The constantsλW and λI are the natural depletion rate coefficients of pollutants emitted from wood and non-wood basedindustries, respectively.

3 Qualitative analysis

Here, we analyze the model (1) by using stability theory of differential equations. For this in the followinglemma, first we show that all solutions of model (1) are nonnegative and bounded.

3.1 Boundedness of solutions

Lemma 1. The set

Ω =

(F, W, I, PW , PI) : 0 ≤ F ≤ L, 0 ≤ W ≤ Wm, 0 ≤ I ≤ Q

ϕ2, 0 ≤ PW ≤ PWm , 0 ≤ PI ≤ PIm

,

where Wm =λL

(ϕ1 − β1L), PWm =

θ1λL

λW (ϕ1 − β1L), PIm =

θ2Q

λIϕ2attracts all solutions initiating in the

interior of the positive octant, where ϕ1 − β1L > 0.

Proof. By using comparison theorem found in [12], From the first equation of the model (1), we have

dF

dt≤ rF

(1− F

L

).

This gives 0 ≤ F ≤ L. From the second equation of the model, we get

dW

dt≤ λF + β1FW − ϕ1L.

This gives, 0 ≤ W ≤ λL

(ϕ1 − β1L), ϕ1 − β1L > 0, or L <

ϕ1

β1.

This condition implies that the carrying capacity of forest resources is dependent on control rate ϕ1 ofwood industries and the growth rate β1 of wood industries through forest resources. From the third equationof the model, we have

dI

dt≤ Q− ϕ2I.

That gives 0 ≤ I ≤ Qϕ2

.Similarly from the fourth and last equation of the model, we have

dPW

dt≤ θ1W − λW PW .

This gives

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0 ≤ PW ≤ θ1λL

λW (ϕ1 − β1L),

dPI

dt≤ θ2I − λIPI .

This gives

0 ≤ PI ≤θ2Q

λIϕ2.

Hence the lemma follows.

3.2 Equilibrium points and their existence

There are three equilibrium points.

(1) E1

(0, 0, Q

ϕ20, θ2Q

λIϕ2

)exists without any condition.

(2) E2(F ∗2 , 0, I∗2 , 0, P ∗

I2) exists provided the following condition is satisfied:

(rϕ2λI − κ2θ2Q) > 0.

(3) E3 (F ∗, W ∗, I∗, P ∗W , P ∗

I ) exists provided the following conditions are satisfied:

(rϕ2λI − κ2θ2Q) > 0,

(ϕ1 − β1F∗) > 0.

Proof. The equilibrium points of the model (1) may be obtained by solving the following algebraic equations:

rF

(1− F

L

)− βFW − κ1FPW − κ2FPI = 0, (2)

λF + β1FW − π1WI − ϕ1W = 0, (3)

Q− π2WI − ϕ2I = 0, (4)

θ1W − δ1FPW − λW PW = 0, (5)

θ2I − δ2FPI − λIPI = 0. (6)

(1) E1

(0, 0, Q

ϕ2, 0,

θ2Q

λIϕ2

)is obvious.

(2) E2(F ∗2 , 0, I∗2 , 0, P ∗

I2)

From first, third and fifth equation of the model (1), we have

rF

(1− F

L

)− κ2FPI = 0, (7)

Q− ϕ2I = 0, (8)

θ2I − δ2FPI − λIPI = 0. (9)

From Eqs. (8) and (9), we have

I =Q

ϕ2, (10)

PI =θ2I

δ2F + λI=

θ2Q

ϕ2 (δ2F + λI). (11)

From the Eqs. (7) and (11), we get an equation in F ,

R1 (F ) = r

(1− F

L

)− κ2θ2Q

ϕ2 (δ2F + λI). (12)

From the Eq. (12), we may easily conclude the following results.



(a) R1 (0) = r − κ2θ2Q

ϕ2λI=

(rϕ2λI − κ2θ2Q)ϕ2λI

, which is positive if

(rϕ2λI − κ2θ2Q) > 0. (13)

(b) R1 (L) = − κ2θ2Q

ϕ2 (δ2L + λI)which is negative.

The above points (a) and (b) together imply that there exists a positive root (F = F ∗2 ) of Eq. (12) in the

interval (0, L). This root will be unique provided R′(F ) is negative in (0, L). With the help of this uniquepositive value of F = F ∗

2 in Eqs. (8) and (9), we get a unique positive value of PI = P ∗I2

and I = I∗2 .(3) E3(F ∗, W ∗, I∗, P ∗

W , P ∗I )

From the third equation of the model (1), we have

I =Q

π2W + ϕ2. (14)

From the second equation of the model (1) and Eq. (14), we have

π2 (ϕ1 − β1F ) W 2 − (λπ2F − ϕ2 (ϕ1 − β1F )− π1Q) W − λϕ2F = 0, (15)

or

W =(λπ2F − ϕ2 (ϕ1 − β1F )− π1Q) +

√(λπ2F − ϕ2 (ϕ1 − β1F )− π1Q)2 + 4 (ϕ1 − β1F ) λϕ2π2F

2π2 (ϕ1 − β1F )= f(F ) (Say). (16)

W is positive if

ϕ1 − β1F > 0. (17)

From the third, fourth and fifth equations of the model (1), we have

I =Q

π2W + ϕ2=

Q

π2f(F ) + ϕ2= u(F ), (18)

PW =θ1W

δ1F + λW=

θ1f(F )δ1F + λW

= g(F ), (19)

PI =θ2I

δ2F + λI=

θ2u(F )δ2F + λI

= h(F ). (20)

From Eqs. (16), (19), (20) and first equation of the model (1), we get an equation in F ,

R2 (F ) = r

(1− F

L

)− βf(F )− κ1g(F )− κ2h(F ). (21)

(a) R2 (0) = r − κ2θ2Q

ϕ2λI=

(rϕ2λI − κ2θ2Q)ϕ2λI

, which is positive if

(rϕ2λI − κ2θ2Q) > 0, (22)

orQ <

rϕ2λI

κ2θ2,

this gives a threshold value of Q.(b) R2 (L) = −βf(L) − κ1g(L) − κ2h(L) < 0, which is negative as f(L), g(L) and h(L) all are

positive. The above points (a) and (b) together imply that there exists a positive root (F = F ∗) of Eq.(21) in the interval (0, L). This root will be unique provided R′

2(F ) is negative in (0, L). With the helpof this unique positive value of F = F ∗ in Eqs. (16), (18), (19) and (20), we get a unique positivevalue of W = W ∗, I = I∗, PW = P ∗

W and PI = P ∗I .



Remark 1. From Eqs. (16), (18), (19) and (20), we get the following results.

(1)dW

dπ1< 0, this implies that if the value of π1 increases, then the density of non-wood industries increases

and the density of wood-based industries decreases.

(2)dI

dπ2< 0, this implies that if the value of π2 increases, then the density of wood-based industries increases

and the density of non-wood industries decreases.

(3)dPW

dF> 0, this implies that pollutants of wood-based industries increases as the density of forest resources

increases, because growth of wood industries is dependent on the density of forest resources and theconcentration of pollutants emitted from them is directly proportional to the density of wood industries.

(4)dPI

dF< 0, this implies that pollutants of non-wood industries decrease as the density of forest resources

increases.

3.3 Stability analysis

The local stability behavior of equilibrium points may be determined by eigenvalues of the correspondingJacobian matrix M . The general Jacobian matrix M for the model (1) is given as,

M =

a11 −βF 0 −κ1F −κ2Fλ + β1W a22 −π1W 0 00 −π2I −π2W − ϕ2 0 0−δ1PW θ1 0 −δ1F − λW 0−δ2PI 0 θ2 0 −δ2F − λI

,

where a11 = r − 2rL F − βW − κ1PW − κ2PI , a22 = β1F − π1I − ϕ1. Let Mi be the matrix M evaluated

at the equilibrium Ei(i = 1, 2, 3) then

M1 =

rϕ2λI−κ2θ2Q

ϕ2λI0 0 0 0

λ −π1Qϕ2

− ϕ1 0 0 00 −π2Q

ϕ2−ϕ2 0 0

0 θ1 0 −λW 0− δ2θ2Q

ϕ2λI0 θ2 0 −λI

.

As we know that at equilibrium E2, F ∗2 has a unique positive value. For this, we have a condition from equation

(13), rϕ2λI − κ2θ2Q > 0. This gives positive eigenvalue of the matrix Thus, E1 is unstable in F -directionwhenever E2 or E3 exists.

Now, we study the local stability behavior of the equilibriums E2 and E3 by using Routh-Hurwitz crite-rion. Then the matrix M2 at equilibrium point E2 as follows.

M2 =

− r

LF ∗2 −β F ∗

2 0 −κ1F∗2 −κ2F

∗2

λ β1F∗2 − π1I

∗2 − ϕ1 0 0 0

0 −π2I∗2 −ϕ2 0 0

0 θ1 0 −δ1F∗2 − λW 0

−δ2P∗I2

0 θ2 0 −δ2F∗2 − λI

.

Let C22 = ϕ1−β1F∗2 +π1I

∗2 , C44 = δ1F

∗2 +λW and C55 = δ2F

∗2 +λI . Now the corresponding characteristic

equation of the Jacobian matrix M2 is given as,

B5 + q1B4 + q2B

3 + q3B2 + q4B + q5 = 0, (23)

where



q1 =r

LF ∗

2 + C22 + ϕ2 + C44 + C55,

q2 =C55

(C22 +

r

LF ∗

2 + ϕ2 + C44

)+

((ϕ2 + C44)

(C22 +

r

LF ∗

2

)+ C44ϕ2 +

r

LF ∗

2 + λβF ∗2

)− κ2δ2P

∗I2F

∗2 ,

q3 =C55

((ϕ2 + C44)

(C22 +

r

LF ∗

2

)+ C44ϕ2 +

r

LF ∗

2 C22 + λβF ∗2

)+

(κ1θ1λF ∗

2 + (ϕ2 + C44)( r

LF ∗

2 C22 + λβF ∗2

)+ C44ϕ2

(C22 +

r

LF ∗

2

))− κ2δ2PIF

∗2 (ϕ2 + C44)− κ2F

∗2 C44,

q4 =C55

(κ1θ1λF ∗

2 + (ϕ2 + C44)( r

LF ∗

2 C22 + λβF ∗2

)+ C44ϕ2

(C22 +

r

LF ∗

2

))+

(κ1θ1λϕ2F

∗2 + C44ϕ2

( r

LF ∗

2 C22 + λβF ∗2

))− κ2C44 (ϕ2 + C44) F ∗

2

− κ2C22ϕ2F∗2 − κ2θ2π2λF ∗

2 I∗2 ,

q5 =C55

(κ1θ1λϕ2F

∗2 + C44ϕ2

( r

LF ∗

2 C22 + λβF ∗2

))− κ2C44C22ϕ2F

∗2 − κ2θ2π2λI∗2C44F

∗2 .

Here it is easy to see that q1 > 0 if ϕ1−β1F∗2 +π1I

∗2 > 0. Thus for the local stability of the equilibrium

E2, we have the following result.

Theorem 1. The equilibrium E2 is locally asymptotically stable if the following conditions are satisfied:q5 >, q1q2 − q3 > 0, q3(q1q2 − q3) − q1(q1q4 − q5) > 0 and q4 (q3 (q1q2 − q3)− q1 (q1q4 − q5)) −q5 (q2 (q1q2 − q3)− (q1q4 − q5)) > 0, where q1, q2, q3, q4 and q5 are defined as above.

Now the matrix at the equilibrium E3 point is given as,

M3 =

− r

LF ∗ −βF ∗ 0 −κ1δ1F∗ −κ2δ2F

∗

λ + β1W∗ β1F

∗ − π1I∗ − ϕ1 −π1 W ∗ 0 0

0 −π2I∗ −π2W

∗ − ϕ2 0 0−δ1P

∗W θ1ϕ1 0 −δ1F

∗ − λW 0−δ2F

∗I 0 θ2ϕ2 0 −δ2F

∗ − λI

.

Let a∗21 = λ+β1W∗, a∗22 = ϕ1−β1F

∗+π1I∗, a∗33 = π2W

∗+ϕ2, a∗44 = δ1F∗+λW and a∗55 = δ2F

∗+λI .Now the corresponding characteristic equation of the Jacobian matrix M3 as follows,

A5 + p1A4 + p2A

3 + p3A2 + p4A + p5 = 0, (24)

where

p1 =r

LF ∗ + a∗22 + a∗33 + a∗44 + a∗55,

p2 =a∗55

(a∗22 + a∗33 +

r

LF ∗ + a∗44

)+

(a∗44

(a∗22 + a∗33 +

r

LF ∗

)+

(a∗22a

∗33 +

r

LF ∗ (a∗22 + a∗33)− π1π2W

∗I∗ + βF ∗a∗21

)− κ1δ1F

∗P ∗W

)− κ2δ1F

∗P ∗I ,

p3 =a∗44

(a∗22a

∗33 +

r

LF ∗ (a∗22 + a∗33)− π1π2W

∗I∗ + βF ∗a∗21

)+

( r

LF ∗a∗22a

∗33 + βF ∗a∗21a

∗33 −

r

Lπ1π2W

∗I∗F ∗)

− κ1δ1F∗P ∗

W (a∗22 + a∗33)− κ1θ1F∗a∗21

+ a∗55

(a∗44

(a∗22 + a∗33 +

r

LF ∗

)+

(a∗22a

∗33+

r

LF ∗ (a∗22 + a∗33)−π1π2W

∗I∗ + βF ∗a∗21

)−κ1δ1F

∗P ∗W

)− κ2F

∗ (δ2P∗I (a∗22 + a∗33) + a∗44δ2P

∗I ) ,



p4 =a∗55

(a∗44

(a∗22a

∗33 + r

LF ∗ (a∗22 + a∗33)− π1π2W∗I∗ + βF ∗a∗21

)+

(rLF ∗a∗22a

∗33F

∗ + βF ∗a∗21a∗33 − r

Lπ1π2W∗I∗F ∗)− κ1δ1F

∗P ∗W (a∗22 + a∗33)− κ1θ1F

∗a∗22

)+ a∗44

( r

LF ∗a∗22a

∗33 + βF ∗a∗21a

∗33 −

r

Lπ1π2W

∗I∗F ∗)− κ1F

∗ (a∗22a∗33δ1P

∗W − π1π2δ1W

∗P ∗W )

− κ1θ1F∗a∗21a

∗33 − κ2F

∗ (θ2π2I∗a∗21 + a∗22a

∗33δ2P

∗I − π1π2δ2W

∗P ∗I + δ2P

∗I a∗44 (a∗22 + a∗33)) ,

p5 =a∗55

(a∗44

(rLF ∗a∗22a

∗33 + βF ∗a∗21a

∗33 − r

Lπ1π2W∗I∗F ∗)− κ1F

∗ (a∗22a∗33δ1P

∗W − π1π2δ1W

∗P ∗W )

−κ1θ1F∗a∗21a

∗33

)− a∗44 (π2θ2I

∗a∗21 + a∗22a∗33δ2P

∗I − π1π2δ2W

∗P ∗I ) .

Here it is easy to see that p1 > 0 if ϕ1 − β1F∗ + π1I

∗ > 0. Thus for the local stability of the equilibrium E3,we have the following result.

Theorem 2. The equilibrium is locally asymptotically stable if the following conditions are satisfied: p5 >0, p1p2 − p3 > 0, p3(p1p2 − p3) − p1(p1p4 − p5) > 0 and p4 (p3 (p1p2 − p3)− p1 (p1p4 − p5)) −p5 (p2 (p1p2 − p3)− (p1p4 − p5)) > 0, where p1, p2, p3, p4 and p5 are defined as above. The result onglobal stability is given in the following theorem.

Theorem 3. If the following inequalities hold in Ω,

(1) 4κ1δ1P∗W < r

LλW ,(2) 4κ2δ2P

∗I < r

LλI ,(3) 4π1π2W

∗I∗ < ϕ2 (ϕ1 − β1F∗),

(4) θ21 <

(m1m3

)(ϕ1 − β1F

∗) λW ,

(5) θ22 <

(m2m4

)ϕ2λI .

Then equilibrium E3 is globally asymptotically stable in the region Ω.

Proof. To prove this, we take a positive definite function and under some conditions, we show that its deriva-tive is negative. Proof of this theorem can be seen in Appendix A.

4 Numerical simulation

In order to check the feasibility of the results obtained in the above section for the existence of E3 weconduct some numerical computations of the model (1) using MATLAB. The following parameter values havebeen chosen:

L = 100, Q = 50, λ = 0.8, r = 4, β = 0.04, β1 = 0.003, λI = 1, λW = 1, ϕ1 = 3, ϕ2 = 5,

π1 = 0.5, π2 = 0.3, δ1 = 0.02, δ2 = 0.01, κ1 = 0.5, κ2 = 0.5, θ1 = 0.1, θ2 = 0.7

It is found that under the above mentioned numerical values, the equilibrium values of E3(F ∗, W ∗, I∗,P ∗

W , P ∗I ), are obtained as:

F ∗ = 45.8055, W ∗ = 5.5399, I∗ = 7.506, P ∗W = 0.2891, P ∗

I = 3.6033.

The eigenvalues of the Jacobian matrix M3 corresponding to the equilibrium E3 (F ∗, W ∗, I∗, P ∗W , P ∗

I ) forthe model system (1) are:

−9.1568,−3.4231− 1.1845i,−3.4231 + 1.1845i,−1.9972,−0.4837.

Here, it may be noted that three eigenvalues of matrix are negative and the other two eigenvalues are withnegative real parts. Therefore, it implies that the interior equilibrium E3 (F ∗, W ∗, I∗, P ∗

W , P ∗I ) is locally

asymptotically stable.



For the above set of parameters, the computer generated graphs are plotted in Figs. 1-7 to predict thesensitivity of the model system under the change of different parameter values. Fig. 1 shows the change in thedensity of forest resources (F ) with respect to time (t) for different values of β. Here, it may be noted thatF decreases as β increases and finally attains its equilibrium. Figs. 2 and 3 respectively depict the effect ofpollutants emitted from wood and non-wood industries on the density of forest resources by taking differentvalues of δ1 and δ2. These figures also show that density of forest resources decreases as δ1 and δ2 decreases.The same effects (as in Fig. 1) can also be seen (Fig. 4) on the density of forest resources with change in thegrowth rate of wood industries due to forest resources β1. But, β1 as increases beyond certain limit, the modelsystem has a destabilizing effect in the direction of density of forest resources. This is because of equilibriumcondition (ϕ1 − β1F

∗) > 0, which may be violated for larger values of β1. Thus, β1 is an important parameterand it should remain under control in order to have the stable system i.e. the use of forest resources for wood-based industries should be held in check.

Fig. 1: Effect of β on forest resource

Fig. 2: Effect of δ1 on forest resource

The variation of density of forest resources (F ) and wood-based industries (W ) with time for differentvalues of Q are plotted in Figs. 5 and 6 respectively. The graphs show that both F and W decrease as the sizeof Q increases. This implies that though F and W based industries are not directly dependent on the growth ofnon-wood industries (Q), but indirectly it affects both. Fig. 5 also reveals that as Q increases beyond certainlevel (i.e. for Q = 60), the density of forest resources (F ) leads to extinction. Similar effect can also bevisualized on density of wood-based industries W for the same value of Q. This also describes that there is acompetition effect between wood and non-wood industries, so if we increase the size of non-wood industries,the size of wood-based industries decreases. From this discussion, it can be said that Q has an important rolein the above model system and it should be controlled, otherwise forest resources can become extinct.



Fig. 3: Effect of δ2 on forest resource

Fig. 4: Effect of β1 on forest resource

Fig. 5: Effect of Q on wood industry

Fig. 6: Effect of Q on wood industry



Fig. 7:

In order to illustrate the global stability of E3 in F −W −I plane, numerical simulation is performed fordifferent initial starts and displayed in Fig. 7. From this figure, we can see that all trajectories initiating insidethe region of attraction approach towards the equilibrium value.

5 Conclusions

In this paper, we have developed a mathematical model to study the effects of wood, non-wood basedindustries and the pollution emitted from them on the depletion of forest resources. In the modeling process,logistic growth of forest resources and migration of wood based industries to the forest region have been con-sidered. Using the stability theory of differential equations, model has been analyzed qualitatively to establishthe following:

(1) boundedness and nonnegativeness of solutions(2) identification of equilibrium points along with their existence and uniqueness(3) the stability of each equilibrium point.

The qualitative analysis shows that that the equilibrium point E1 is always unstable, whenever the otherequilibrium points E2 or E3 exists, while the equilibrium points E2 and E3 are stable under certain conditions.Numerical simulation has also been performed to illustrate the feasibility of the obtained results.

The results of the model show that wood-based industries deplete the forest resources both directly(through harvesting) and indirectly (through pollutants), but that non-wood industries only deplete the for-est resources indirectly (through pollution). On the other side, it has also been derived that the pollutants ofnon-wood industries also get reduced with increase in the density of forest resources, because forest resourcesalso serve as a sink for these pollutants. It has also been observed that in order to have a stable model system,the utilization of forest resources for wood- based industries should be limited. From this, it can be said thatnon-wood industries can be the alternative for wood-based industries to fulfill with the increasing demandsof population. But, because pollution generated from non-wood industries also affects the forest resourcesindirectly, non-wood industries can only be enhanced up to certain extent.

Therefore, it can be concluded that with more and more industrialization either in the form of woodor non-wood, the forest resources are affected and may become extinct. So in order to avoid this, effortsshould be made to limit the wood and non-wood industries in a forest habitat. However, it appears difficult toachieve because of increasing population and their demands. In view of this, the better way out is to regulatethe pollution emitted from these industries. Polluting industries should be held in check, while non-pollutingindustries should be motivated in a forest habitat. This aspect of regulating pollution through different controlstrategies through mathematical modeling in a forest habitat has been left for future research.

Appendix. proof of theorem 3

To prove this theorem we consider a positive definite function as follows,



U =(

F − F ∗ − F ∗ lnF

F ∗

)+

12m1(W −W ∗)2 +

12m2(I − I∗)2 +

12m3(PW − P ∗

W )2

+12m4(PI − P ∗

I )2, (25)

where m1, m2, m3 and m4 are positive constants. On differentiating Eq. (25) with respect to t′ we get,

dU

dt= (F − F ∗)

F

F+ m1 (W −W ∗) W + m2 (I − I∗) I + m3 (PW − P ∗

W ) PW + m4 (PI − P ∗I ) PI .

Substituting the values of the derivative from the model (1), we have

dU

dt=− r

L(F − F ∗)2 −m1π1I(W −W ∗)2 −m3δ1F (PW − P ∗

W )2 −m4δ2F (PI − P ∗I )2

−m1 (ϕ1 − β1F∗) (W −W ∗)2 −m2ϕ2(I − I∗)2 −m2π2W (I − I∗)2 −m3λW (PW − P ∗

W )2

−m4λI(PI − P ∗I )2 + (m1β1W

∗ + m1λ− β) (W −W ∗) (F − F ∗)− (m1π1W

∗ + m2π2I∗) (W −W ∗) (I − I∗)− (κ2 + m4δ2P

∗I ) (PI − P ∗

I ) (F − F ∗)− (κ1 + m2δ1P

∗W ) (PW−P ∗

W ) (F−F ∗) + m3θ1 (F−F ∗) (PW−P ∗W ) + m4θ2 (I−I∗) (PI−P ∗

I ) .

Choosing m1(β1W∗ + λ)− β = 0 which gives m1 = β

β1W ∗+λ is positive constant. Now U can be written as,

dU

dt=−m1π1I(W −W ∗)2 −m2π2W (I − I∗)2 −m3δ1F (PW − P ∗

W )2 −m4δ2F (PI − P ∗I )2

− 12Z11(F − F ∗)2 − Z12 (F − F ∗) (PW − P ∗

W )− 12Z22(PW − P ∗

W )2

− 12Z11(F − F ∗)2 − Z13 (F − F ∗) (PI − P ∗

I )− 12Z33(PI − P ∗

I )2

− 12Z44(W −W ∗)2 + Z42 (W −W ∗) (PW − P ∗

W )− 12Z22 (PW − P ∗

W )

− 12Z44(W −W ∗)2 − Z45 (W −W ∗) (I − I∗)− 1

2Z55(I − I∗)2

− 12Z55(I − I∗)2 + Z53 (I − I∗) (PI − P ∗

I )− 12Z33(PI − P ∗

I )2,

where Z11 = rL , Z12 = κ1 + m3δ1P

∗W , Z22 = m3λW , Z13 = κ2 + m4δ2P

∗I , Z33 = m4λI , Z44 =

m1 (ϕ1 − β1F∗), Z42 = m3θ1, Z45 = m1π1W

∗ + m2π2I∗, Z55 = m2ϕ2 and Z53 = m4θ2.

Now according to Z2ij < ZiiZjj we have,

(κ1 + m3δ1P∗W )2 <

r

Lm3λW , (26)

(κ2 + m4δ2P∗I )2 <

r

Lm4λI , (27)

(m1π1W∗ + m2π2I

∗)2 < m1m2 (ϕ1 − β1F∗) ϕ2, (28)

m23θ

21 < m1m3 (ϕ1 − β1F

∗) λW , (29)

m24θ

22 < m2m4ϕ2λI . (30)

Eqs. (26), (27) and (28) can be written as,

(κ1 −m3δ1P∗W )2 + 4κ1δ1m3P

∗W <

r

Lm3λW .

Choosing m3 = κ1δ1P ∗

Wthen we have

4κ1δ1P∗W <

r

LλW . (31)



Again(κ2 −m4δ2P

∗I )2 + 4κ2δ2m4P

∗I <

r

Lm4λI .

Choosing m4 = κ2δ2P ∗

I, then we have

4κ2δ2P∗I <

r

LλI . (32)

Now from Eq. (28),

(m1π1W∗ −m2π2I

∗)2 + 4m1m2π1π2W∗I∗ < m1m2 (ϕ1 − β1F

∗) ϕ2.

Choosing m2 = π1W ∗

π2I∗ m1, then we have

4π1π2W∗I∗ < ϕ2 (ϕ1 − β1F

∗) . (33)

Hence all conditions of Theorem 3 are satisfied for dUdt to be negative in the region Ω. Therefore equilibrium

E3 is globally asymptotically stable in the region Ω.

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