# Dynamic Behaviors of a Harvesting Leslie-Gower Predator-Prey Model

Authors: Na Zhang; Fengde Chen; Qianqian Su; Ting Wu
Journal: Discrete Dynamics in Nature and Society
 (2011)

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## Abstract

A Leslie-Gower predator-prey model incorporating harvesting is studied.
By constructing a suitable Lyapunov function, we show that the unique positive
equilibrium of the system is globally stable, which means that suitable harvesting
has no influence on the persistent property of the harvesting system. After that,
detailed analysis about the influence of harvesting is carried out, and an interesting finding
is that under some suitable restriction, harvesting has no influence on the final
density of the prey species, while the density of predator species is strictly decreasing
function of the harvesting efforts. For the practical significance, the economic profit
is considered, sufficient conditions for the presence of bionomic equilibrium are given,
and the optimal harvesting policy is obtained by using thePontryagin's maximal
principle. At last, an example is given to show that the optimal harvesting policy
is realizable.


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## Body

## 1. Introduction

Leslie [1, 2] introduced the following predator-prey model, where the “carrying capacity” of the predator's environment is proportional to the number of prey:
(1.1)dHdt=(r1-a1P-b1H)H,dPdt=(r2-a2PH)P,
where H and P are the density of prey species and the predator species at time t, respectively. The above system admits an unique coexisting fixed point
(1.2)H*=r1a2a1r2+a2b1,P*=r1r2a1r2+a2b1.
Recently, by constructing a suitable Lyapunov function, Korobeinikov [3] showed the positive equilibrium is globally stable; consequently, the system could not admits limit cycle. Such an finding is very interesting, since for predator-prey system incorporating Holling-type II or III functional response, limit cycle exists [4, 5]. After the work of Korobeinikov, many scholars have done works on Leslie-type predator prey ecosystem. Aguirre et al. [6] showed that Leslie-Gower predator-prey model with additive Allee effect is possible to admit two limit cycles; Aziz-Alaoui and Daher Okiye [7] argued that a suitable predator prey model should incorporate some kind of functional response, they proposed a predator-prey model with modified Leslie-Gower and Holling-type II schemes, they investigated the boundedness and global stability of the system; Nindjin et al. [8] further incorporated the time delay to the system considered in [7], and they showed that time delay plays important role on the dynamic behaviors of the system; Yafia et al. [9] studied the limit cycle bifurcated from time delay; Nindjin and Aziz-Alaoui [10], and Aziz-Alaoui [11] studied the dynamic behaviors of three Leslie-Gower-type species food chain system; Chen et al. [12] incorporated a prey refuge to system (1.1) and showed that the refuge has no influence on the persistent property of the system; Some scholars argued that nonautonomous case are more realistic if one consider the influence of seasonal effect of the environment. Huo and Li [13] studied the periodic solution of the nonautonomous case Leslie-Gower predator-prey system; Gakkhar and Singh [14] studied a Leslie-Gower predator-prey system with seasonally varying parameters; Song and Li [15] further considered the influence of impulsive effect. For more works on predator-prey ecosystem, one could refer to [4–27] and the references cited therein.As was pointed out by  Makinde [28]: “From the point of view of human needs, the exploitation of biological resources and harvesting of populations are commonly practiced in fishery, forestry, and wildlife management. There is a wide range of interest in the use of bioeconomic models to gain insight into the scientific management of renewable resources like fisheries and forestries.” Though there are numerous works on predator-prey system incorporating the harvesting, to this day, still no scholar studies the system (1.1) under the assumption of the harvesting on prey and predator species. In this paper, we assume that the predator and prey species in the model is both of commercial importance and they are subjected to constant effort harvesting with c1 and c2, two parameters that measures the effort being spent by a harvesting agency. Thus, we formulating the system as follows:
(1.3)dHdt=(r1-a1P-b1H)H-c1H,dPdt=(r2-a2PH)P-c2P,
where H and P are the density of prey species and the predator species at time t, respectively. To ensure the sustainable development, which means that we try to control the prey and predator species densities in a controllable range, but not to perish the species, it's natural to assume that 0<ci<ri, i=1,2.The rest of the paper is arranged as follows: we will study the stability property of positive equilibrium of system (1.3) in Section 2 and discuss the influence of the harvesting in Section 3. Bionomic equilibrium and optimal harvesting policy for system (1.3) are discussed in Section 4 and 5, respectively. An example of system (1.3) is given in Section 6 to show the feasibility of our results. We end this paper by a briefly discussion.


## 2. Stability Property of Positive Equilibrium

By simple computation, under the assumption0<ci<ri, i=1,2, system (1.3) admits an unique positive equilibrium
(2.1)H1*=(r1-c1)a2a1(r2-c2)+a2b1,P1*=(r1-c1)(r2-c2)a1(r2-c2)+a2b1.
Obviously, (H1*,P1*) satisfies the equalities
(2.2)r1-c1-b1H1*-a1P1*=0,r2-c2=a2P1*H1*.Our result about the local stability property of this equilibrium is stated as follows.Theorem 2.1.
The positive equilibrium(H1*,P1*) of system (1.3) is locally asymptotically stable.Proof.
The variational matrixJ*(H,P) of the system (1.3) is given by
(2.3)(r1-c1-a1P-2b1H-a1Ha2P2H2r2-c2-2a2PH).
So, the characteristic equation for J*(H*,P*) is given by λ2+aλ+b=0, where
(2.4)a=a2b1(r1-c1)a1(r2-c2+a2b1)+r2-c2(>0),b=(r1-c1)(r2-c2)(>0).
It is clear that the roots of the characteristic equation are negative or have negative real parts. Hence, the unique positive equilibrium of system (H1*,P1*) is stable. This completes the proof of the Theorem 2.1.
Concerned with the global stability property of the positive equilibrium, we have the following.Theorem 2.2.
The positive equilibrium(H1*,P1*) of system (1.3) is globally stable.Definition 2.3.
System (1.3) is called permanent if for any positive solution (H(t),P(t))T of system (1.3) there exist positive constants mi, Mi, i=1,2, which are independent of the solution of the system, such that
(2.5)m1≤liminf⁡t→+∞H(t)≤limsup⁡t→+∞H(t)≤M1,m2≤liminf⁡t→+∞P(t)≤limsup⁡t→+∞P(t)≤M2.
Theorem2.2 shows that
(2.6)lim⁡t→+∞H(t)=H1*>0,lim⁡t→+∞P(t)=P1*>0.
Noticing that H1* and P1* are only dependent on the coefficients of the system (1.3) and independent of the solution of system (1.3). Thus, (2.6) clearly shows that suitable harvesting (more precisely, with restriction 0<ci<ri, i=1,2) has no influence on the persistent property of the system.Proof of Theorem2.2.
We will adapt the idea of Korobeinikov [3] to prove Theorem 2.2. More precisely, we construct the following Lyapunov function:
(2.7)V(H,P)=ln⁡HH1*+H1*H-1+a1H1*a2(ln⁡PP1*+P1*P-1).
Obviously, V(H,P) is well defined and continuous for all H,P>0. By simple computation, we have
(2.8)∂V∂H=1H(1-H1*H),∂V∂P=a1H1*a2P(1-P1*P).
Equation (2.8) shows that the positive equilibrium (H1*,P1*) is the only extremum of the function V(H,P) in the positive quadrant. Noting that
(2.9)∂2V∂H2=1H2(-1+2H1*H),∂2V∂P∂H=0,∂2V∂P2=a1H1*a2P2(-1+2P1*P).
Therefore,
(2.10)d2V|(H1*,P1*)=1H1*dH2+a1H1*a2P1*dP2>0.
Above analysis shows that (H1*,P1*) is the only minimum extremum of the function V(H,P) in the positive quadrant. One could easily verify that
(2.11)lim⁡H→0V(H,P)=lim⁡P→0V(H,P)=lim⁡H→+∞V(H,P)=lim⁡P→+∞V(H,P)=+∞.
From (2.8) and (2.11), we can see that the positive equilibrium (H1*,P1*) is the global minimum, that is,
(2.12)V(H,P)>V(H1*,P1*)=0,
for all H,P>0.
Calculating the derivative ofV along the solution of the system (1.3), by using equalities (2.2), we have
(2.13)dVdt=1H(1-H1*H)(r1-c1-b1H-a1P)H+a1H1*a2P(1-P1*P)⋅(r2-c2-a2PH)P=H-H1*H(b1H1*+a1P1*-b1H-a1P)+a1H1*a2⋅(1-P1*P)⋅(a2P1*H1*-a2PH)=-b1H(H-H1*)2-a1P(P-P1*)2.
Obviously, (dV/dt)<0 strictly for all H,P>0 except the positive equilibrium (H1*,P1*), where (dV/dt)=0. Thus, V(H,P) satisfies Lyapunov's asymptotic stability theorem, and the positive equilibrium (H1*,P1*) of system (1.3) is globally stable. This ends the proof of Theorem 2.2.Remark 2.4.
With the restriction0<ci<ri, i=1,2, system (1.3) always admits an unique positive equilibrium and from Theorems 2.1 and 2.2 we can see that this equilibrium is globally attractive, since it's stability property is not changed with the variation of parameter ci, the system could not undergoes Hopf's bifurcation and there is no limit cycle of system (1.3) in R+2. In fact, we can also prove this declare by using Bendixson-Dulac theorem.Let(2.14)F(H,P)≐(r1-a1P-b1H)H-c1H,G(H,P)≐(r2-a2PH)P-c2P,B(H,P)≐1HP.
Obviously, F(H,P), G(H,P), and B(H,P)∈C1(R+2).Calculating from above equations we get(2.15)∂BF∂H+∂BG∂P=-b1P-a2H2<0,(H,P)∈R+2.According toBendixson-Dulac theorem, we know that there is no limit cycle of system (1.3) in R+2.


## 3. The Influence of Harvesting

We will discuss this topic on three aspects.(1) The case of only harvesting prey species
In this case,(3.1)H1*=(r1-c1)a2a1r2+a2b1,P1*=(r1-c1)r2a1r2+a2b1.
Obviously, H1*, P1* are all continuous differentiable function of parameter c1 and
(3.2)dH1*dc1=-a2a1r2+a2b1<0,dP1*dc1=-r2a1r2+a2b1<0.
The above inequalities show that H1* and P1* are both the strictly decreasing function of c1, that is, increasing the capture rate of prey species leads to the decreasing of the density of both prey and predator species.(2) The case of only harvesting predator species
In this case,(3.3)H1*=r1a2a1(r2-c2)+a2b1,P1*=r1(r2-c2)a1(r2-c2)+a2b1.
that is, H1*, P1* are all continuous differentiable function of parameter c2. Noticing that
(3.4)dH1*dc2=r1a2a1(a1(r2-c2)+a2b1)2>0,dP1*dc2=-r1a2b1(a1(r2-c2)+a2b1)2<0.
It is easy to see that H1* is the strictly increasing function of parameter c2, while P1* is the strictly decreasing function of c2, that is, increasing the capture rate of predator species leads to the increasing the density of prey species and the decreasing of predator species.(3) The case of harvesting predator and prey species together
In this case, it follows from (2.1) that H1* and P1* are all continuous differential functions of parameters ci, i=1,2. Though we had made the assumption 0<ci<ri, it still not an easy thing to give an detailed analysis of all of the cases. Here, we only investigate the following problem, which seems very interesting.Problem 3.1.
Is it possible to choose some suitable parametersci such that after the harvesting of predator and prey, the densities of prey species as t→+∞ still has no change? That is,  H1*=H*. If this is possible, what about the dynamic behaviors of predator species in this case?The first part of the question is equivalent to say that in what caseH1*=H*, that is,
(3.5)(r1-c1)a2a1(r2-c2)+a2b1=r1a2a1r2+a2b1.
Solving the above equality, we obtain
(3.6)c2=a1r2+a2b1r1a1⋅c1.
It means that with the suitable capture efforts (ci which satisfy the equality (3.6)), prey species will converge to H* as t→+∞.Now substituting (3.6) into the second equality of (2.1), we have
(3.7)P1*=r12a1r2-r1c1(a1r2+a2b1)a1(r1-c1)(a1r2+a2b1).
Obviously, P1* is the continuous differential function of parameter c1 and
(3.8)dP1*dc1=-b1a2r12a1(r1-c1)2(a1r2+a2b1)<0,
that is, if the capture rate of predator and prey species satisfies (3.6), then increasing the harvesting of prey (and of predator) will lead to the finally decreasing of predator densities (P1*).


## 4. Bionomic Equilibrium

This section is devoted to study the bionomic equilibrium of system (1.3) since it has the practical significance.The term bionomic equilibrium is an amalgamation concepts of biological equilibrium and economic equilibrium. As we know, a biological equilibrium is given by(dH/dt)=(dP/dt)=0. And the economic equilibrium is said to be achieved when the TR (total revenue obtained by selling the harvested predators H and P) equals TC (the total cost for the effort devoted to harvesting).Some symbols should be given at first.Letp1 is the price per unit biomass of the prey H,p2 is the price per unit biomass of the predator P,q1 is the fishing cost per unit effort of the prey H,q2 is the fishing cost per unit effort of the predator P.Then, the economic rent (revenue at any time) is given by(4.1)N=TR-TC=(p1H-q1)c1+(p2P-q2)c2=defN1+N2,
where N1=def(p1H-q1)c1, N2=def(p2P-q2)c2, that is, N1 and N2 represent the net revenues for the population H and P, respectively.For convenience, we take the price per unit biomass of the predators and the fishing cost per unit effort of the predators to be constant. So, the bionomic equilibrium is given by the following simultaneous equations(4.2)dHdt=(r1-a1P-b1H)H-c1H,(4.3)dPdt=(r2-a2PH)P-c2P,(4.4)N=(p1H-q1)c1+(p2P-q2)c2=0.Since the price and the cost of the predators are not sure, we will consider the following cases in order to determine the bionomic equilibrium.Case 4.1.
If(4.5)q1p1>H,
that is,
(4.6)p1H-q1<0
holds, that is to say the total cost exceed the total revenue for the harvesting of prey, obviously, the prey harvesting will be stopped (i.e., c1=0) and the predator harvesting remains operational if  p2P-q2>0.
Then, from (4.4), we have
(4.7)P1∞=q2p2.
Substituting it into (4.3), it follows that
(4.8)H1∞=r1p2-a1q2b1p2.
Again, substituting (4.7) and (4.8) into (4.2) leads to
(4.9)c2∞=r2-a2p1∞H1∞=r2-a2b1q2r1p2-a1q2.
So, ifr1>a2(q2/p2) and r2>(a2b1q2/r1p2-a1q2) hold together, we have the bionomic equilibrium [H1∞,P1∞,0,c2∞].Case 4.2.
If(4.10)q2p2>P,
that is,
(4.11)p2P-q2<0
holds, that is to say the total cost exceeds the total revenue for the harvesting of predator obviously, the prey harvesting will be stopped (i.e., c2=0) and the predator harvesting remains operational if p1H-q1>0.
Then, it follows from (4.4) that
(4.12)H1∞=q1p1.
Substituting (4.12) into (4.3), we obtain
(4.13)P1∞=r2q1a2p1.
Again, substituting (4.12) and (4.13) into (4.2) leads to
(4.14)c1∞=r1-a1P1∞-b1H1∞=r1-(a1r2-a2b1)q1a2p1.
So, if r1>((a1r2-a2b1)q1/a2p1) hold, we have the bionomic equilibrium [H1∞,P1∞,c1∞,0].Case 4.3.
If(4.15)q2p2>P,q1p1>H,
that is,
(4.16)p2P-q2<0,p1H-q1<0
hold, then it is equivalent to say that the total cost exceeds the total revenue for two populations.
Obviously, the harvesting will be stopped, that is,c1=0, c2=0. In this case, there is no bionomic equilibrium.Case 4.4.
If(4.17)q2p2<P,q1p1<H,
that is,
(4.18)p2P-q2>0,p1H-q1>0
hold. In this case, the total revenue exceeds the total cost for two populations and the harvesting is in operational because it can bring profit for fishery.
From (4.4), we have
(4.19)H1∞=q1p1,P1∞=q2p2.
Substituting above equalities into (4.2) and (4.3), it is easy to obtain
(4.20)c1∞=r1-(a1p2q1-b1p1q2)p1p2,c2∞=r2-a2p1q2p2q1.
So, if
(4.21)r1>(a1p2q1-b1p1q2)p1p2,r2>a2p1q2p2q1
hold together, we have the bionomic equilibrium [H1∞,P1∞,c1∞,c2∞].It is obviously that the bionomic equilibrium may exist if the intrinsic growth rates of two species exceed some value.


## 5. Optimal Harvesting Policy

In order to determine the optimal harvesting policy, we consider the present valueJ of a continuous time-stream of revenue
(5.1)J=∫0∞e-δt{(p1H-q1)c1(t)+(p2P-q2)c2(t)}dt,
where δ denotes the instantaneous annual rate of discount and ci(t)(i=1,2) are the control variables, which are subject to the assumption 0<ci(t)<ri, i=1,2.Now, our objective is to maximizeJ subject to the state equations (1.3) by invoking Pontryagin's maximal principle.The Hamiltonian for the problem is given at first(5.2)H¯=e-δt{(p1H-q1)c1(t)+(p2P-q2)c2(t)}+λ1[(r1-a1P-b1H)H-c1H]+λ2[(r2-a2PH)P-c2P],
where λ1(t) and λ2(t) are the adjoint variables.Obviously, the control variablesc1 and c2 appear linearly in the Hamiltonian function H¯. So, the conditions
(5.3)∂H¯∂ci=0,i=1,2
are necessary for the singular control to be optimal.Then, we have(5.4)λ1=e-δt(p1-q1H),λ2=e-δt(p2-q2P),
that is
(5.5)λ1eδt=p1-q1H,λ2eδt=p2-q2P.
Therefore, the shadow prices eδtλi(t) do not vary with time in the optimal equilibrium. Hence they remain bounded as t→∞.By the maximal principle, the adjoint variables satisfydλ1/dt=-∂H¯/∂H and dλ2/dt=-(∂H¯/∂P), for all t≥0, that is.
(5.6)dλ1dt=-{e-δtp1c1+λ1(r1-c1a1p1-2b1H)-λ2a2PH2},dλ2dt=-{e-δtp2c2+λ2(r2-c2-2a2PH)-λ1a1H}.
Substituting (5.4) and (5.5) into (5.6), we get
(5.7)2a2p2P2+a1p1H2P+(p2δ-r2p2-a1q1)HP-a2q2P-q2δH=0,2b1p1H3+a1p1PH2+(p1δ-r1p1-b1q1)H2-q1δH-a2p2P+a2q2=0.
From (5.7), we may find the positive values of (Hδ,Pδ).Substituting(Hδ,Pδ) into (1.3), we get the equations as follows:
(5.8)c1δ=r1-a1Pδ-b1Hδ,c2δ=r2-a2PδHδ.
Then, we may have the optimal equilibrium effort levels c1δ and c2δ, if
(5.9)r1>a1Pδ+b1Hδ,r2>a2PδHδ.It means that, under the harvesting for both prey and predator species, the optimal harvesting policy can be obtained only under the assumption that the intrinsic growth rates of two species exceed some values.The optimal harvesting policy means the economic rent for harvesting will be the maximal in the future time, with the populations of the system are permanent.


## 6. Numerical Example

Let(6.1)r1=1.6,a1=0.7,b1=0.3,p1=0.4,q1=0.2,r2=1,a2=0.25,δ=1.25,p2=0.5,q2=0.2.From Section6, we know that the optimal harvesting policy can be obtained by substituting the result of (5.7) into (1.3). So, the corresponding equations of (5.7) are
(6.2)0.25P2+0.28H2P-0.015HP-0.05P-0.25H=0,0.24H3+0.28PH2-0.2H2-0.25H-0.125P+0.05=0.Solving (6.2) by using Maple, we get the results as follow:
(6.3){H=0.03875658339,P=0.3212762700},{H=0.3442951974,P=-0.5444632570},{H=1.121042647,P=0.6323992784},{H=0.8269704200+1.125716729I,P=0.1137766719-1.326783285I},{H=0.8269704200-1.125716729I,P=0.1137766719+1.326783285I},{H=-0.7529121920+0.04647885793I,P=-0.2550446252-0.7947461329I}.From the above results, one could easily see that there is only one result (Hδ=1.121042647, Pδ=0.6323992784) meeting the condition(6.4)Hδ>q1p1(=0.5),Pδ>q2p2(=0.4).
In this case, the corresponding system of (1.3) is as follows:
(6.5)dHdt=(1.6-0.7P-0.3H)H-c1H,dPdt=(1-0.25PH)P-c2P.Substituting (Hδ=1.121042647, Pδ=0.6323992784) into (6.5), we obtain the optimal harvesting efforts
(6.6)c1δ=0.821008<r1(=1.6),c2δ=0.858971<r2(=1).


## 7. Conclusion

A Leslie-Gower predator-prey model incorporating harvesting is studied in this paper. We first show that suitable harvesting has no influence on the persistent property of the harvesting system. After that, we try to give the detail analysis of harvesting on the dynamic behaviors of the system. Our study shows that for the system having both harvesting on predator and prey species, it admits some interesting phenomenon; maybe such a finding could be applied to help human improving the scientific management of renewable resources such as fisheries and forest trees. Then, for the practical significance, we consider the economic profit of the harvesting. The bionomic equilibrium and optimal harvesting policy are studied. The results show that the optimal harvesting policy may exist. Finally, an example is given to show that the optimal harvesting policy of system (1.3) is realizable.



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*Source: 473949-2011-04-04.xml*