A PREY-PREDATOR MODEL WITH ALTERNATIVE PREY: MATHEMATICAL MODEL AND ANALYSIS · 2011. 1. 13. · 2...

CANADIAN APPLIED

MATHEMATICS QUARTERLY

Volume 18, Number 2, Summer 2010

A PREY-PREDATOR MODEL WITH

ALTERNATIVE PREY: MATHEMATICAL

MODEL AND ANALYSIS

T. K. KAR, KUNAL CHAKRABORTY AND U. K. PAHARI

ABSTRACT. This paper describes a prey predator modelwith type II functional response where harvesting of each speciesis taken into consideration. Discrete type gestational delay ofpredators is incorporated and its effect on the dynamical be-haviour of the model system is analyzed. It is shown that har-vesting efforts may be used to control the prey predator systemconsidered. We have also studied the Hopf bifurcation of themodel system in the neighbourhood of the co-existing equilib-rium point considering delay as a variable bifurcation param-eter. Numerical simulations are given to verify the analyticalresults.

1 Introduction Mathematical modelling of exploitation of biolog-ical resources is still a very interesting field of research. Mathematicalmodelling is frequently an evolving process. Systematic mathematicalanalysis can often lead to better understanding of bioeconomic models.The exposed discrepancies in turn lead to the necessary modifications.The final model may or may not be free of any significant discrepanciesbut the analysis of the final model can thus be expected to reveal im-portant and nontrivial features of the system. In the last few decades,interest has been growing steadily in the designing and studying of math-ematical models of population interactions. There is a tendency amongresearchers to rush into the analysis of an existing model without know-ing the background or motivation of such model, or rush into the finalmodels and indulge in the analysis of certain mathematical propertiesof such models. There are several such literatures which mainly in-

Research of T. K. Kar is supported by the Council of Scientific and IndustrialResearch (CSIR), India (Grant no. 25(0160)/ 08 / EMR-II dated 17.01.08).

AMS subject classification: (2000): 34K20, 92D25.Keywords: Harvesting, global stability, time delay, Hopf bifurcation.

Copyright c©Applied Mathematics Institute, University of Alberta.

137

138 T. K. KAR, K. CHAKRABORTY AND U. K. PAHARI

clude prey predator interactions with the effects of time delay. Toahaet al. [15] studied analytically the necessary conditions of harvesting toensure the existence of the equilibrium points and the stabilities of amodel which consists of time delay and two populations subjected toconstant effort of harvesting. They proved that time delay can induceinstability and a Hopf bifurcation may appear. They also related thestable equilibrium point for the model with harvesting to profit func-tion problem and found that there exists a critical value of the effortthat maximizes the profit and the equilibrium point also remains sta-ble. The effect of constant rate harvesting on the dynamics of predator-prey systems has been investigated by Dai and Tang [4], Myerscough et

al. [13] and Xiao and Ruan [16], and they obtained very rich and inter-esting dynamical behaviours. Feng [6] considered a differential equationsystem with diffusion and time delays which models the dynamics ofpredator-prey interactions within three biological species. The resultsobtained by him explicitly present the effects of all the environmentaldata (growth rates and interaction rates) on the ultimate bounds of thethree biological species and numerical simulations of the model weregiven to demonstrate the pattern of dynamics (extinction, persistence,and permanence) in the ecological model. Nakaoka et al. [14] examineda Lotka-Volterra predator-prey system with two delays. They concludedthat a positive equilibrium of the system is globally asymptotically sta-ble for small delays and the system exhibits some chaotic behavior whentime delay becomes large. They analytically determined critical values oftime delay through which the system undergoes a Hopf bifurcation. Thedynamics of a prey-predator system, where predator population has twostages, juvenile and adult with harvesting were modelled using a systemof delay differential equation by Kar [9]. He concluded that both thedelay and harvesting effort may play a significant role on the stability ofthe system. Yafia [17] established an explicit algorithm for determiningthe direction of the Hopf bifurcation and the stability or instability ofthe bifurcating branch of periodic solutions through considering a modelwith one delay. A model in which the revenue is generated from fish-ing and the growth of fish depends upon the plankton which in turngrows logistically is developed by Dhar et al. [5]. They further formu-lated the model with delay in digestion of plankton by fish and foundthe threshold value of conversional parameter for Hopf-bifurcation. Karand Matsuda [10], and Kar and Pahari [11] discussed the predator preymodel with time delay and analyzed the effects of time delay on modeldynamics such as the time delay may change the stability of equilibriumpoints and even cause a switching of stabilities.

A PREY-PREDATOR MODEL 139

From the above literature survey it is very relevant to point out thatno attempt has been made to study the dynamics of a prey predatorsystem in the presence of an alternative resource by taking into accountfunctional response in the model. In the present paper we, therefore,model the dynamics of a prey-predator system where predator popula-tion partially dependent on a logistically growing resource with func-tional response. Also we have considered a gestation delay of predatorpopulation and harvesting of both the species. The main objective ofthis paper is to examine the effects of both fishing effort and delay on themodel dynamics. Conditions for the existence of nonnegative equilibria,criterion for their local and global stability behaviour are obtained. It isalso proved that the time delay can cause a stable equilibrium to becomeunstable and even a switching of stabilities may occur.

2 The model formulation We consider a prey-predator modeland it is assumed that the dynamics of both prey and predator pop-ulation follow logistic law of growth. Let us assume x and y are, re-spectively, the size of prey and predator population at time t. Thepredator population consumes the prey population with the functionalresponse, known as Holling type II functional response and contributesto its growth rate. It is considered that prey and predator populationare continuously harvested and hence the growth of prey and predatorpopulation are directly affected by harvesting. Keeping these aspectsin view, the dynamics of the system may be governed by the followingsystem of differential equations

dx

dt= r1x

(

1 − x

K1

)

− mxy

1 + ax− h1(t),

dy

dt= r2y

(

1 − y

K2

)

+αmxy

1 + ax− h2(t),

where r1 and r2 are, respectively, the intrinsic growth rate of prey andpredator population. K1 and K2 are their respective environmental car-rying capacities, m is the conversion of biomass constant, a is Michaelis-Menten constant, h1(t) and h2(t) are, respectively, the harvesting fromprey and predator population at time t, α is the maximum value ofpercapita reduction rate of x due to y.

Let us now consider this harvested prey predator system with timedelay due to gestation. It is inherently assumed that all the metabolic en-ergy a predator obtains through its food used for logistic growth which


ultimately enhance the predator population. Here the predator pop-ulation consumes the prey population at a constant rate α, but thereproduction of predators after predating the prey population is not in-stantaneous thus it will be incorporated by some time lag required forgestation of predators. Suppose the time interval between the momentswhen an individual prey is killed and the corresponding biomass is addedto the predator population is considered as the time delay τ . Under thisassumption the model becomes

dx

dt= r1x

(

1 − x

K1

)

− mxy

1 + ax− h1(t),

dy

dt= r2y

(

1 − y

K2

)

+αmx(t − τ)y(t − τ)

1 + ax(t − τ)− h2(t).

The functional form of harvest is generally considered using the phrasecatch-per-unit-effort (CPUE) hypothesis (Clark [3]) to describe an as-sumption that catch per unit effort is proportional to the stock level.Thus we consider h1 = q1E1x and h2 = q2E2y, where E1 and E2

are, respectively, the harvesting effort used to harvest from prey andpredator population, q1 and q2 are, respectively, the catchability co-efficient of prey and predator population. Here in this model we assumeq1 = q2 = 1.

Thus the final model becomes

(1)

dx

dt= r1x

(

1 − x

K1

)

− mxy

1 + ax− E1x,

dy

dt= r2y

(

1 − y

K2

)

+αmx(t − τ)y(t − τ)

1 + ax(t − τ)− E2y,

with initial conditions x(θ), y(θ) ≥ 0, θ ∈ [−τ, 0), and x(0) > 0, y(0) >0.

For a nonlinear delay system, there are two types of stabilities: ab-solute stability (independent of the delay) and conditional stability (de-pending on the delay). Here we consider two cases separately with andwithout time delay.

Case 1. τ = 0For τ = 0, the system (1) becomes

(2)

dx

dt= r1x

(

1 − x

K1

)

− mxy

1 + ax− E1x,

dy

dt= r2y

(

1 − y

K2

)

+αmxy

1 + ax− E2y,


Clearly, P0(0, 0) is the trivial equilibrium point of the system (2) andall other possible equilibrium points of the system (2), considered in thefirst quadrant of x-y plane, are as follows:

P1(x, 0) where x =K1

r1

(r1 − E1),

P2(0, y) where y =K2

r2

(r2 − E2),

P3(x∗, y∗) where y∗=

K2(r2 − E2)

r2

+K2αmx∗

r2(1 + ax∗),

and x∗ satisfying the following cubic equation

(3) Ax∗3

+ Bx∗2

+ Cx∗ + D = 0

where

A =a2r1

K1

> 0, B = a

[

2r1

K1

− a(r1 − E1)

]

,

C =r1

K1

+amK2(r2 − E2) + K2αm2

r2

− 2a(r1 − E1),

D =mK2(r2 − E2)

r2

− (r1 − E1).

For the existence of the equilibrium points P1(x, 0) and P2(0, y) it isassumed that

(4) E1 < r1 and E2 < r2.

Equation (3) has a unique positive solution if any one of the followinginequalities holds:

(i) B < 0, C < 0, D < 0,(ii) B > 0, C < 0, D < 0,(iii) B > 0, C > 0, D < 0.

Condition (i) holds if E1 < r1 − M1 where

M1 = max

[

2r1

aK1

,1

2a

(

r1

K1

+amK2(r2 − E2) + K2αm2

r2

)

,mK2(r2 − E2)

r2

]

.


Condition (ii) holds if

(

r1 −2r1

aK1

)

< E1 < (r1 − M2)

where

M2 = max

[

1

2a

(

r1

K1

+amK2(r2 − E2) + K2αm2

r2

)

,mK2(r2 − E2)

r2

]

.

Condition (iii) holds if

(r1 − M3) < E1 <

(

r1 −mK2(r2 − E2)

r2

)

where

M3 = max

[

2r1

aK1

,1

2a

(

r1

K1

+amK2(r2 − E2) + K2αm2

r2

)]

.

Once we get the unique positive solution of x∗ from equation (3) it iseasy to get the interior positive solution of y∗.

Thus, for a set of parameters of model system (2) different equilibriaexist for different levels of the harvesting efforts.

Before studying the stability of the model system, we show that thesolutions of the model system are bounded in a finite region initiatingat (x(0), y(0)).

Theorem 1. All the solutions of the system (2) which start in R2+ are

uniformly bounded.

Proof. Let (x(t), y(t)) be any solution of the system with positive initialconditions. Now we define the function W = x + y/α. Therefore, timederivative gives

dW

dt=

dx

dt+

1

α

dy

dt

= r1x

(

1 − x

K1

)

− mxy

1 + ax− E1x +

r2y

α

(

1 − y

K2

)

+mxy

1 + ax− E2y

α

= r1x − r1

K1

x2 − E1x +r2

αy − r2

αK2

y2 − E2

αy.


Now for each v > 0, we have

dW

dt+ vW = r1x − r1

K1

x2 − E1x +r2

αy − r2

αK2

y2 − E2

αy + vx +

v

αy.

Thus,dW

dt+ vW ≤ u,

where

u =r1

4K1

(

E1K1

r1

− K1 −vK1

r1

)2

+r2

4αK2

(

E2K2

r2

− K2 −vK2

r2

)2

.

Applying the theory of differential inequality (Birkoff and Rota [1]), weobtain

0 ≤ W (x, y) ≤ u

v+

W (x(0), y(0))

evtand for t → ∞, 0 ≤ W ≤ u

v.

Thus, all the solutions of the system (4) enter into the region

B =

{

(x, y) : 0 ≤ W ≤ u

v+ ε, for any ε > 0

}

.

This completes the proof.

3 Stability analysis In this section we discuss the stability of dif-ferent equilibrium points of the system (2), by computing the variationalmatrix at various equilibrium points and using the Routh-Hurwitz cri-terion [12].

At P0(0, 0), we find the roots of the characteristic equation are (r1 −E1) and (r2 − E2). Thus, the trivial equilibrium P0(0, 0) is stable ifE1 > r1 and E2 > r2.For the boundary equilibrium P1(x, 0), the eigenvalues of the character-istic equation are λ1

1 and λ21 where

λ11 = −(r1 − E1), λ2

1 = (r2 − E2) +αmK1(r1 − E1)

r1 + aK1(r1 − E1).

It is noted that P1(x, 0) is stable if

E1 < r1 and E2 > r2 +αmK1(r1 − E1)

r1 + aK1(r1 − E1).


Again, for the boundary equilibrium P2(0, y), the eigenvalues of the char-acteristic equation are λ1

2 and λ22 where

λ12 = −(r2 − E2), λ2

2 = (r1 − E1) −mK2

r2

(r2 − E2).

Therefore, P2(0, y), is stable if

E2 < r2 and E1 > r1 −mK2

r2

(r2 − E2).

It is observed that even in the absence of prey the predator may exists inits equilibrium level and this is happened due to the alternative sourceof prey.

The characteristic equation for the interior equilibrium P3(x∗, y∗) is

given by

(5) µ2 + f(x∗(E1, E2), y∗(E1, E2))µ + g(x∗(E1, E2), y

∗(E1, E2)) = 0,

where

f(x∗(E1, E2), y∗(E1, E2)) =

r1x∗

K1

+r2y

∗

K2

− may∗x∗

(1 + ax∗)2

and

g(x∗(E1, E2), y∗(E1, E2)) =

r1r2x∗y∗

K1K2

+mx∗y∗

(1 + ax∗)2

(

αm

1 + ax∗− ar2y

∗

K2

)

.

Therefore, stability of the interior equilibrium P3(x∗, y∗) depends on the

sign of f(x∗(E1, E2), y∗(E1, E2)) and g(x∗(E1, E2), y

∗(E1, E2)). If

(H1) f(x∗(E1, E2), y∗(E1, E2)) > 0 and g(x∗(E1, E2), y

∗(E1, E2)) > 0,then P3(x

∗, y∗) is a stable node or a stable spiral.

(H2) f(x∗(E1, E2), y∗(E1, E2)) < 0 and g(x∗(E1, E2), y

∗(E1, E2)) > 0,then P3(x

∗, y∗) is an unstable node or an unstable spiral.

(H3) g(x∗(E1, E2), y∗(E1, E2)) < 0, then P3(x

∗, y∗) is a saddle point.

(H4) f(x∗(E1, E2), y∗(E1, E2)) = 0, then there are some limit cycles.

Now we want to study the global behaviour of the system (2). Westate the following theorem.


Theorem 2. Let us define ρ(E1, E2) = r1

K1− ay∗ and assume that the

positive equilibrium is locally stable. If ρ(E1, E2) > 0, then it is globally

asymptotically stable.

Proof. To show the global stability of system (2), we define a Lyapunovfunction as follows:

V (x, y) = L1

(

x − x∗ − x∗ lnx

x∗

)

+ L2

(

y − y∗ − y∗ lny

y∗

)

where L1 and L2 are positive constants to be determined in the subse-quent steps. It can be easily verified that the function V is zero at theequilibrium P3(x

∗, y∗) and is positive for all other positive values of xand y.

The time derivative of along the trajectories of (2) is

dV

dt= L1

x − x∗

x

dx

dt+ L2

y − y∗

y

dy

dt

= L1

x − x∗

x

[

r1x

(

1 − x

K1

)

− mxy

1 + ax− E1x

]

+ L2

y − y∗

y

[

r2y

(

1 − y

K2

)

+αmxy

1 + ax− E2y

]

= −[

r1L1

K1

(x − x∗)2 +r2L2

K2

(y − y∗)2]

− m[(L1 − L2α)(x − x∗)(y − y∗) + L1a(x − x∗)(x∗y − xy∗)]

= −[(

r1L1

K1

− L1ay∗

)

(x − x∗)2 +r2L2

K2

(y − y∗)2

+ m(L1 − L2α + L1ax∗)(x − x∗)(y − y∗)

]

.

Choosing L1 = 1 and L2 = 1+ax∗

α, a little algebric manipulation yeilds

dv

dt= −

[(

r1

K1

− ay∗

)

(x − x∗)2 +r2(1 + αx∗)

K2α(y − y∗)2

]

.

Thus, if ρ(E1, E2) > 0, then dVdt

< 0 and dVdt

= 0 if and only if x = x∗

and y = y∗. This completes the proof.


FIGURE 1: The condition ρ(E1, E2) > 0 is plotted in the parametricspace (E1, E2) corresponding to the numerical values: r1 = 1.8, r2 =2.5, K1 = 30, K2 = 20, m = 0.5, a = 3, α = 1.5, E2 = 2.

The plot of ρ(E1, E2) indicates that the assumption ρ(E1, E2) > 0makes some sense.

In the next section we will study the uniform persistence of the sys-tem (2) using the average Lyapunov function. By the permanence orpersistence of a system, we mean that all the species are present andnone of them will go to extinction. The persistence of a system havebeen studied by Freedman [7] and Butler et al. [2] and some other au-thors.

Theorem 3. The system (2) is uniformly persistent if E1 < r1, E2 < r2

and (r1 − E1) > mK2

r2(r2 − E2).

Proof. To show the uniform persistence of system (2). We define anaverage Lyapunov function as follows:

φ(X) = xcyd

where c and d are positive constants, φ(x) is non-negative function de-fined in R

2+ and X is a function of x and y. After differentiating, we

have

θ(X) =φ(x)

φ(x)= c

x

x+ d

y

y

= c

[

r1

(

1 − x

K1

)

− my

1 + ax− E1

]


+ d

[

r2

(

1 − y

K2

)

+αmx

1 + ax− E2

]

.

Hence, the uniform persistence exists, if there exists c and d, such thatθ(X) is positive at P0(0, 0), P1(x, 0) and P2(0, y). Now

θ(P0(0, 0)) = c(r1 − E1) + d(r2 − E2) > 0(i)

if E1 < r1 and E2 < r2,

θ(P1(x, 0)) = d

[

(r2 − E2) +αmK1(r1 − E1)

r1 + aK1(r1 − E1)

]

> 0(ii)

if E1 < r1 and E2 < r2,

θ(P2(0, y)) = c

[

(r1 − E1) −mK2

r2

(r2 − E2)

]

> 0(iii)

if (r1 − E1) >mK2

r2

(r2 − E2).

Thus, if E1 < r1, E2 < r2 and (r1 −E1) > mK2

r2(r2 −E2), then θ(X) > 0

for all equilibrium points. This completes the proof.

4 Bifurcation in the parametric space (E1, E2) In this sec-tion, we investigate the existence and stability of interior equilibriumin the parametric space (E1, E2). For this purpose the conditions forexistence of interior equilibrium, local stability and global stability areplotted in the parametric space (E1, E2). After plotting the aforesaidconditions in the parametric space (E1, E2), it is observed that the para-metric space (E1, E2) can be divided into three regions which consist ofeleven subregions. Let us now check in how many regions the inte-rior equilibrium is locally asymptotically stable. To analyze the sys-tem behaviour we assign numerical value of the parameters as follows:r1 = 1.8, r2 = 2.5, K1 = 30, K2 = 20, m = 0.5, a = 3, α = 1.5. Forthe purpose of computations we mainly use the software MATLAB 7.0.

Region-A:{(E1, E2) ∈ R2+ = I ∪ IX ∪ X ∪ XI} where

I :f(x∗(E1, E2), y∗(E1, E2)) > 0, g(x∗(E1, E2), y

∗(E1, E2)) > 0,

B < 0, C > 0, D > 0.

IX :f(x∗(E1, E2), y∗(E1, E2)) > 0, g(x∗(E1, E2), y

∗(E1, E2)) > 0,


FIGURE 2: Existence and stability regions of the equilibrium points inthe parametric space (E1, E2) corresponding to the numerical values asmentioned earlier.

B < 0, C < 0, D < 0.

X :f(x∗(E1, E2), y∗(E1, E2)) > 0, g(x∗(E1, E2), y

∗(E1, E2)) > 0,

B < 0, C < 0, D > 0.

XI :f(x∗(E1, E2), y∗(E1, E2)) > 0, g(x∗(E1, E2), y

∗(E1, E2)) > 0,

B < 0, C > 0, D < 0.

In each subregion of the region A we find that TraceJ < 0 and det J > 0,therefore, the interior equilibrium is a stable node or a stable spiral.The following figures ensure the stability of interior equilibrium for anarbitrarily taken point (E1, E2) in each subregion of the region A.

Phase plane trajectories of Figures 3, 4, 5 and 6 are drawn with dif-ferent initial levels. Trajectories clearly indicate that the interior equi-librium is asymptotically stable.

Region-B:{(E1, E2) ∈ R2+ = II ∪ III} where

II :f(x∗(E1, E2), y∗(E1, E2)) > 0, g(x∗(E1, E2), y

∗(E1, E2)) < 0,

B < 0, C > 0, D > 0.

III :f(x∗(E1, E2), y∗(E1, E2)) < 0, g(x∗(E1, E2), y

∗(E1, E2)) > 0,

B > 0, C > 0, D > 0.


10 15 20 25 305

10

15

20

25

30

35

x

y

FIGURE 3: For (E1, E2) = (0.72289, 1.5261) taken from subregion-I.

10 15 20 25 300

5

10

15

20

25

30

35

x

y

FIGURE 4: For (E1, E2) = (0.56225, 2.49) taken from subregion-II.


10 15 20 25 305

10

15

20

25

30

35

x

y

FIGURE 5: For (E1, E2) = (0.16064, 2.008) taken from subregion-X.

0 5 10 15 20 25 300

5

10

15

20

25

30

35

x

y

FIGURE 6: For (E1, E2) = (1.4458, 2.49) taken from subregion-XI.


Interior equilibrium does not exist in the region-B.

Region-C:{(E1, E2) ∈ R2+ = IV ∪ V ∪ V I ∪ V II ∪ V III} where

IV :f(x∗(E1, E2), y∗(E1, E2)) < 0, g(x∗(E1, E2), y

∗(E1, E2)) < 0,

B > 0, C > 0, D < 0.

V :f(x∗(E1, E2), y∗(E1, E2)) > 0, g(x∗(E1, E2), y

∗(E1, E2)) < 0,

B > 0, C > 0, D < 0.

V I :f(x∗(E1, E2), y∗(E1, E2)) < 0, g(x∗(E1, E2), y

∗(E1, E2)) < 0,

B > 0, C < 0, D < 0.

V II :f(x∗(E1, E2), y∗(E1, E2)) < 0, g(x∗(E1, E2), y

∗(E1, E2)) < 0,

B < 0, C < 0, D < 0.

V III :f(x∗(E1, E2), y∗(E1, E2)) > 0, g(x∗(E1, E2), y

∗(E1, E2)) < 0,

B < 0, C < 0, D < 0.

It is found in each subregion of the region C that for an arbitrary valueof (E1, E2) the predator population is reached to its extinction thoughthe prey population exists. Following figures ensure the analysis.

0 10 20 30 40 50−5

−4

−3

−2

−1

0

1

2

t

x an

d y

Prey biomassPredator biomass

FIGURE 7: For (E1, E2) = (2.8514, 3.1727) taken from subregion-V.


0 10 20 30 40 50−7

−6

−5

−4

−3

−2

−1

0

1

2

3

t

x an

d y


FIGURE 8: For (E1, E2) = (2.0482, 3.494) taken from subregion-VI.

0 10 20 30 40 50−8

−6

−4

−2

0

2

4

6

8

t

x an

d y


FIGURE 9: For (E1, E2) = (1.4859, 3.6145) taken from subregion-VII.

Variation of prey and predator biomass in Figures 7, 8, 9 and 10 withthe increasing time clearly indicate that the predator population goes toextinction.


0 10 20 30 40 50−5

0

5

10

15

20

t

x an

d y


FIGURE 10: For (E1, E2) = (0.60241, 3.2932) taken from sub region-VIII.

5 Numerical study of the system behaviour In order to studythe numerical behaviour of the model system, the parameters of the sys-tem (2) are assigned by the following numerical values: r1 = 1.8, r2 =2.5, K1 = 30, K2 = 20, m = 0.5, a = 3, α = 1.5, E1 = 1.2, E2 = 2.Figure 11 depicts the isoclines of prey and predator population whenthe fishing effort used to harvest predator population is considered tobe constant E2 = 2. The isoclines are obtained for different fishing ef-forts used to harvest prey population, i.e., E1 = 0.6, E1 = 0.8, E1 =1, E1 = 1.2, E1 = 1.4. When fishing effort associated to the predatorpopulation is assumed to be constant, the exploitation of the prey pop-ulation is simultaneously done by fishing agency and a constant rate bythe predator population. As a result the equilibrium point for the preypopulation gradually decreases when the respective fishing effort usedto harvest prey population is simultaneously increased. Consequentlythe prey population collapses for a small time interval compare to thesituation where both the population simultaneously harvested.

Figure 12 depicts the isoclines of prey and predator population whenthe fishing effort used to harvest prey population is considered to beconstant E1 = 1.2. The isoclines are obtained for different fishing effortsused to harvest predator population, i.e., E2 = 1.5, E2 = 1.75, E2 =2, E2 = 2.25, E2 = 2.5. It is observed that the predator isoclines aregradually decreases with the increasing fishing effort used to harvestpredator population though effort on prey population is constant. This


is due to the fact that the reproduction of predators after predating theprey population is not instantaneous. Thus the predator population isultimately reached to its extinction limit for a particular fishing effort.

FIGURE 11: Isoclines of prey and predator population when E2 = 2 isconstant.

FIGURE 12: Isoclines of prey and predator population when E1 = 1.2is constant.


020

4060

80

0

20

40

60

80−200

−100

0

100

200

300

E1

E2

Pre

y eq

uilib

rium

yie

ld

FIGURE 13: Variation of prey equilibrium yield curves of the system (2)with the increasing prey and predator fishing effort.

020

4060

80

0

20

40

60

80−8

−6

−4

−2

0

2

4

x 104

E1

E2

Pre

dato

r equ

ilibr

ium

yie

ld

FIGURE 14: Variation of predator equilibrium yield curves of the sys-tem (2) with the increasing prey and predator fishing effort.

6 Analysis of delay model We shall now investigate the dynam-ics of delay system (1).

Let (x∗, y∗) be the only interior equilibrium of system (1) and letX = x− x∗ and Y = y − y∗ be the perturbed variables. After removing


the nonlinear terms and by using equilibrium conditions, we obtain thelinear variational system as

(6)

dX

dt=

[

r1

(

1 − 2x∗

K1

)

− my∗

(1 + ax∗)2− E1

]

X − mx∗

1 + ax∗Y,

dY

dt=

αmy∗

(1 + ax∗)2X(t − τ) +

[

r2

(

1 − 2y∗

K2

)

− E2

]

Y

+αmx∗

1 + ax∗Y (t − τ).

The characteristic equation of the linearized system (6) is a transcen-dental equation of the following form

(7) ∆(λ, τ) = λ2 + Pλ + Q + (Rλ + S)e−λt = 0

where

P =my∗

(1 + ax∗)2+ E1 + E2 − r1

(

1 − 2x∗

K1

)

− r2

(

1 − 2y∗

K2

)

,

Q =

(

r2

(

1 − 2y∗

K2

)

− E2

)(

r1

(

1 − 2x∗

K1

)

− my∗

(1 + ax∗)2− E1

)

,

R = − αmx∗

1 + ax∗,

S =αmx∗

1 + ax∗

(

r1

(

1 − 2x∗

K1

)

− E1

)

.

It is known that the steady state is asymptotically stable if all roots ofthe characteristic equation (7) have negative real parts. Now we canfind the condition for nonexistence of delay induced instability by usingthe following theorem given by Gopalsamy [8].

Theorem 4. Necessary and sufficient conditions for P3(x∗, y∗) to be

locally asymptotically stable in the presence of a time delay τ are

(i) the real parts of all the roots of ∆(λ, τ) = 0 are negative;

(ii) for all real ω and for τ > 0, ∆(ıω, τ) 6= 0, where ı =√−1.

If ı ω is a root of equation (7), then

−ω2 + Q + S cosωτ + Rω sinωτ + ı(Pω + Rωcosωτ − Ssinωτ) = 0.


Separating real and imaginary parts we get

(8)S cosωτ + Rω sinωτ = ω2 − Q,

Rω cosωτ − S sinωτ = −Pω.

Squaring and adding both of equation (8) we finally have

(9) ω4 − (R2 − P 2 + 2Q)ω2 + (Q2 − S2) = 0.

Now by Descartes’ rule of sign, for an equation with real coefficient, itis evident from (9) that if

(H5) (P 2 − R2 − 2Q) > 0 and (Q2 − S2) > 0,

then equation (9) does not have positive roots. Therefore, characteristicequation (7) does not have purely imaginary roots. Since conditionsin ensure that all roots of equation (5) have negative real parts, byTheorem 4 stated above, it follows that the roots of equation (7) havenegative real parts as well.

Again, it is observed that if

(H6) (Q2 − S2) < 0,

then equation (9) has a unique positive root ω20 . Substituting ω2

0 in (8)and solving for τ , we get

(10) τn =1

ω0

cos−1

{

S(ω20 − Q) − PRω2

0

R2ω20 + S2

}

+2nπ

ω0

, n = 0, 1, 2, . . . .

Again, if

(H7)(R2 − P 2 + 2Q) > 0, (Q2 − S2) > 0 and

(R2 − P 2 + 2Q)2 > 4(Q2 − S2),

then equation (9) has two positive roots

ω2± =

(R2 − P 2 + 2Q) ±√

(R2 − P 2 + 2Q)2 − 4(Q2 − S2)

2.

Substituting ω2± in (9) we get

(11) τ±

k =1

ω±

cos−1

{

S(ω2± − Q) − PRω2

±

R2ω2± + S2

}

+2kπ

ω±

, k = 0, 1, 2, . . . .


From (7) we obtain

dλ

dt=

(Rλ2 + Sλ)e−λτ

2λ + P + (R − Rλτ − Sτ)e−λτ.

Thus, we have

sign

{

d

dτRe (λ)

}

λ=ı ω

= sign {ω2[2ω2 − (R2 − P 2 + 2Q)]}

= (R2 − P 2 + 2Q)2ω2 − 2(Q2 − S2).

Let us now compute the following transversality conditions:

[

d

dτRe (λ)

]

τ=τ0,ω=ω0

= [ω4 − (Q2 − S2)] > 0,

[

d

dτRe (λ)

]

τ=τ+

k,ω=ω+

=1

2

{

[(R2 − P 2 + 2Q)2 − 4(Q2 − S2)]

+√

(R2 − P 2 + 2Q)2 − 4(Q2 − S2)}

> 0,

[

d

dτRe (λ)

]

τ=τ−

k,ω=ω−

=1

2

√

(R2 − P 2 + 2Q)2 − 4(Q2 − S2)

×[

√

(R2 − P 2 + 2Q)2 − 4(Q2 − S2)

− (R2 − P 2 + 2Q)]

< 0.

Hence, the transversality conditions are verified. We summarize theabove analysis by the following theorem.

Theorem 5. (i) If (H1) and (H5) hold, then all roots of equation (7)have negative real parts for all τ ≥ 0.

(ii) If (H1) and (H6) hold, then equilibrium (x∗, y∗) is asymptotically

stable for τ < τ0 and unstable for τ > τ0. Further, as τ increases through

τ0, (x∗, y∗) bifurcates into small amplitude periodic solutions, where

τ0 =1

ω0

cos−1

{

S(ω20 − Q) − PRω2

0

R2ω20 + S2

}

.

(iii) If (H1) and (H7) hold, then there is a positive integer m such that

there are m switches from stability to instability and to stability. In


other words, when τ ∈ [0, τ+

0 ), (τ−

0 , τ+

1 ), . . . , (τ−

m−1, τ+m), the equilibrium

is stable and when τ ∈ [τ+

0 , τ−

0 ), (τ+

1 , τ−

1 ), . . . , (τ+

m−1, τ−m), the equilib-

rium is unstable. Therefore, there are bifurcations at (x∗, y∗) when

τ = τ±

k , k = 0, 1, 2, 3, . . . . The positive equilibrium is unstable for

τ > τ+m where m is the minimum nonnegative integer at which, for the

first time

τ+m+1 =

θ+ + (m + 1)2π

ω+

≤ τ−

m =θ− + m2π

ω−

,

i.e., m is the minimum nonnegative integer satisfying

m ≥ (θ+ + 2π)ω− − θ−ω+

2π(ω+ − ω−).

It is clear from the above discussion that the delay has a great impacton the dynamics of the system.

As an example, we consider the system

dx

dt= 1.5x

(

1 − x

80

)

− 0.5xy

1 + 2x− 1.2x,

dy

dt= 1.2y

(

1 − y

60

)

+3.9 × 0.5x(t − τ)y(t − τ)

1 + 2x(t − τ)− 2y.

When τ = 0, we see from Figure 15, that the equilibrium (7.792, 5.92)is a stable node. Now we wish to study the effect of the delay τ onthe dynamics of the model. By using Theorem 5, it is found that thereis a critical value τ0 = 0.27. At τ0 = 0.27 the interior equilibrium(x∗, y∗) = (5.594, 4.746). The equilibrium (x∗, y∗) = (5.594, 4.746) isasymptotically stable for τ < 0.27 becomes unstable for τ > 0.27, andthere is a small amplitude periodic solution.

Therefore, if we consider τ = 0.17, then it is observed from Figures 16and 17 that P3(x

∗, y∗) is locally asymptotically stable and the popula-tions x, y converge to their steady states in finite time. Now, if wegradually increase the value of τ keeping other parameters fixed, thenwe have got a critical value τ = τ0 = 0.27 such that P3(x

∗, y∗) loses itsstability as τ passes through τ0. Figures 18 and 19 clearly show the re-sult. It is also noted that if we consider τ = 0.37, then it is evident fromFigures 20 and 21 that the positive equilibrium P3(x

∗, y∗) is unstableand there is a periodic orbit near P3(x

∗, y∗).The effects of stability of equilibrium points of the model system

with the increasing time and particular harvesting efforts are shown in


FIGURE 15: Phase plane trajectories of prey and predator biomassbeginning with different initial levels. Trajectories clearly indicate thatthe interior equilibrium (x∗, y∗) = (7.792, 5.92) is asymptotically stable.

0 500 1000 1500 20000

2

4

6

8

10

12

14

16

18

20τ =0.17

Time

Pop

ulat

ions

FIGURE 16: Variation of prey and predator biomass with the increasingtime when τ = 0.17.


0 5 10 15 20 25 300

5

10

15

20

25

30

35

x

y

τ=0.17

FIGURE 17: Phase plane trajectories of prey and predator biomassbeginning with different initial levels when τ = 0.17.

0 500 1000 1500 20000

2

4

6

8

10

12

14

16

18

20τ =0.27

Time

Pop

ulat

ions



0 5 10 15 20 25 300

5

10

15

20

25

30

35

x

y

τ=0.27


0 500 1000 1500 20000

2

4

6

8

10

12

14

16

18

20τ =0.37

Time

Pop

ulat

ions



0 5 10 15 20 25 300

5

10

15

20

25

30

35

x

y

τ=0.37


Figures 22–27 for a fixed value of τ . It is clear from Figures 22–27 thatfor a given value of harvesting effort has great influence on stability ofequilibrium points of the model system. Figure 22 depicts that preyand predator population remains stable if we consider the harvestingefforts E1 = 1.4 and E2 = 2.2 for τ = 0.17 but in this figure it is notedthat the predator population tends to its extinction. It is evident fromFigure 23 that for the harvesting efforts E1 = 2 and E2 = 3 both thepopulations reach to their extinction though τ = 0.17 remains same.Again, it is observed that for the harvesting efforts E1 = 0.5, E2 = 1.5and τ = 0.17, the populations are not only stable and but also exist witha greater steady state value.

We have already found that the interior equilibrium becomes unstablefor τ = 0.37 but from Figure 25 it is observed that interior equilibriumbecomes stable for the changed harvesting efforts E1 = 1.5 and E2 = 2.5though τ = 0.37 remains same. Here in this figure it is also observed thatthe predator population has already reached to its extinction and withthe increasing time prey population also reaches to its extinction whereas in Figure 26 it is clearly shown that though both the populationsare collapsed for E1 = 2, E2 = 3 and τ = 0.37, in finite time. It isnoted from Figure 27 that for E1 = 0.5, E2 = 1.5 and τ = 0.37, thepopulations exist for finite time and clearly stable, though it is unstablefor E1 = 1.2 and E2 = 2 (see Figures 20 and 21).


0 10 20 30 40 500

2

4

6

8

10

12

14

16

18

20τ =0.17

Time

Pop

ulat

ions

Prey populationPredator population

FIGURE 22: Variation of prey and predator biomass with the increasingtime when τ = 0.17 and E1 = 1.4, E2 = 2.2.

0 5 10 150

2

4

6

8

10

12

14

16

18

20τ =0.17

Time

Pop

ulat

ions


FIGURE 23: Variation of prey and predator biomass with the increasingtime when τ = 0.17 and E1 = 2, E2 = 3.


0 10 20 30 40 500

5

10

15

20

25

30

35

40

45

50τ =0.17

Time

Pop

ulat

ions



0 5 10 15 20 250

2

4

6

8

10

12

14

16

18

20τ =0.37

Time

Pop

ulat

ions




0 5 10 150

2

4

6

8

10

12

14

16

18

20τ =0.37

Time

Pop

ulat

ions


FIGURE 26: Variation of prey and predator biomass with the increasingtime when τ = 0.37 and E1 = 2, E2 = 3.

0 10 20 30 40 5015

20

25

30

35

40

45τ =0.37

Time

Pop

ulat

ions




7 Conclusion In the present paper we have considered a deter-ministic model of a prey-predator system having alternative prey andharvesting of each species. We have stated and proved several resultsgiving criterion for the existence of several equilibrium points, stabilityand bifurcations for the model system. We have obtained important re-sults for stability of the differential equations and the delay differentialequation model system in order to study the effect of gestation delay onvarious dynamic behaviours with different levels of harvesting efforts.It is observed that harvesting efforts have the ability to derive a stableequilibrium point to an unstable one and an unstable equilibrium to astable one. Here, it is noted that time delay also plays an important roleto the dynamics of the system.

REFERENCES

1. G. Birkoff and G. C. Rota, Ordinary Differential Equations, Ginn, 1982.2. G. Butler, H. I. Freedman and P. Waltman, Uniformly persistent system, Proc.

Amer. Math. Soc. 96 (1986), 420–430.3. C. W. Clark, Mathematical Bioeconomics: The Optimal Management of Re-

newable Resources, 2nd ed., John Wiley and Sons, New York, 1990.4. G. Dai and M. Tang, Coexistence region and global dynamics of a harvested

predator-prey system, SIAM J. Appl. Math. 58 (1998), 193–210.5. J. Dhar, A. K. Sharma and S. Tegar, The role of delay in digestion of plankton

by fish population: a fishery model, J. Nonlinear Sci. Appl. 1(1) (2008), 13–19.6. W. Feng, Dynamics in 3-species predator-prey models with time delays, Dis-

crete Contin. Dyn. Syst. Supplement (2007), 364–372.7. H. I. Freedman and P. Waltman, Persistence in models of three interacting

predator-prey populations, Math. Biosci. 68 (1984), 213–231.8. K. Gopalswamy, Stability and Oscillation in Delay Differential Equations of

Population Dynamics, Kluwer Academic Publisher, The Netherlands, 1992.9. T. K. Kar, Dynamics of a ratio-dependent prey-predator system with selective

harvesting of predator species, J. Appl. Math. Comput. 23(1, 2) (2007), 385–395.

10. T. K. Kar and H. Matsuda, Controllability of a harvested prey-predator systemwith time delay, J. Biol. Syst. 14(2) (2006), 243–254.

11. T. K. Kar and U. K. Pahari, Modelling and analysis of a prey-predator sys-tem with stage-structure and harvesting, Nonlinear Anal. Real World Appl. 8

(2007), 601–609.12. R. M. May, Stability and Complexity in Model Ecosystems, Princeton Univer-

sity Press, New Jersey, 2001.13. M. R. Myerscough, B. F. Gray, W. L. Hogarth and J. Norbury, An analysis of

an ordinary differential equation model for a two-species predator-prey systemwith harvesting and stocking, J. Math. Biol. 30 (1992), 389–411.

14. S. Nakaoka, Y. Saito and Y. Takeuchi, Stability, delay, and chaotic behaviorin a Lotka-volterra predator-prey system, Math. Biosci. Engrg. 3(1) (2006),173–187.


15. S. Toaha, M. H. A. Hassan, F. Ismail and L. W. June, Stability analysis andmaximum profit of predator - prey population model with time delay and con-stant effort of harvesting, Malaysian J. Math. Sci. 2(2) (2008), 147–159.

16. D. Xiao and S. Ruan, Bogdanov-Takens bifurcations in predator-prey systemswith constant rate harvesting, Fields Inst. Commun. 21 (1999), 493–506.

17. R. Yafia, F. E. Adnani and H. T. Alaoui, Stability of limit cycle in a predator-prey model with modified Leslie-Gower and Holling-type II schemes with timedelay, Appl. Math. Sci. 1(3) (2007), 119–131.

Corresponding author: T. K. Kar

Department of Mathematics, Bengal Engineering and Science University

Shibpur, Howrah-711103, West Bengal, India

E-mail address: [email protected]

Department of Mathematics, MCKV Institute of Engineering

243 G.T.Road (N), Liluah, Howrah-711204, West Bengal, India

E-mail address: kc [email protected]

Department of Mathematics, Sree Chaityanya College

Habra, 24 PGs (North), West Bengal, India

A PREY-PREDATOR MODEL WITH ALTERNATIVE PREY: MATHEMATICAL MODEL AND ANALYSIS · 2011. 1. 13. · 2...

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