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

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 the Pontryagin’s maximal principle. At last, an example is given to show that the optimal harvesting policy is realizable.


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: 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 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 Hollingtype 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 predatorprey 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 c 1 and c 2 , two parameters that measures the effort being spent by a harvesting agency.Thus, we formulating the system as follows: 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 < c i < r i , 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.

Stability Property of Positive Equilibrium
By simple computation, under the assumption 0 < c i < r i , i 1, 2, system 1.3 admits an unique positive equilibrium Obviously, H 1 * , P 1 * satisfies the equalities Our result about the local stability property of this equilibrium is stated as follows.
Proof.The variational matrix J * H, P of the system 1.3 is given by So, the characteristic equation for J * H * , P * is given by λ 2 aλ b 0, where It is clear that the roots of the characteristic equation are negative or have negative real parts.Hence, the unique positive equilibrium of system H 1 * , P 1 * 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 H 1 * , P 1 * 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 m i , M i , i 1, 2, which are independent of the solution of the system, such that Noticing that H 1 * and P 1 * 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 < c i < r i , i 1, 2 has no influence on the persistent property of the system.
Proof of Theorem 2.2.We will adapt the idea of Korobeinikov 3 to prove Theorem 2.2.More precisely, we construct the following Lyapunov function: Obviously, V H, P is well defined and continuous for all H, P > 0. By simple computation, we have Equation 2.8 shows that the positive equilibrium H 1 * , P 1 * is the only extremum of the function V H, P in the positive quadrant.Noting that Therefore, Above analysis shows that H 1 * , P 1 * is the only minimum extremum of the function V H, P in the positive quadrant.One could easily verify that lim

2.11
From 2.8 and 2.11 , we can see that the positive equilibrium H 1 * , P 1 * is the global minimum, that is, for all H, P > 0.
Calculating the derivative of V along the solution of the system 1.3 , by using equalities 2.2 , we have dV dt

2.13
Obviously, dV/dt < 0 strictly for all H, P > 0 except the positive equilibrium H 1 * , P 1 * , where dV/dt 0. Thus, V H, P satisfies Lyapunov's asymptotic stability theorem, and the positive equilibrium H 1 * , P 1 * of system 1.3 is globally stable.This ends the proof of Theorem 2.2.
Remark 2.4.With the restriction 0 < c i < r i , 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 c i , 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
Obviously, F H, P , G H, P , and B H, Calculating from above equations we get

2.15
According to Bendixson-Dulac theorem, we know that there is no limit cycle of system 1.3 in R 2 .

The Influence of Harvesting
We will discuss this topic on three aspects. (

1) The case of only harvesting prey species
In this case, Obviously, H 1 * , P 1 * are all continuous differentiable function of parameter c 1 and The above inequalities show that H 1 * and P 1 * are both the strictly decreasing function of c 1 , 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, that is, H 1 * , P 1 * are all continuous differentiable function of parameter c 2 .Noticing that It is easy to see that H 1 * is the strictly increasing function of parameter c 2 , while P 1 * is the strictly decreasing function of c 2 , 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 H 1 * and P 1 * are all continuous differential functions of parameters c i , i 1, 2. Though we had made the assumption 0 < c i < r i , 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 parameters c i such that after the harvesting of predator and prey, the densities of prey species as t → ∞ still has no change?That is,

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 case H 1 * H * , that is, Solving the above equality, we obtain It means that with the suitable capture efforts c i 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 Obviously, P 1 * is the continuous differential function of parameter c 1 and 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 P 1 * .

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.Let p 1 is the price per unit biomass of the prey H, p 2 is the price per unit biomass of the predator P , q 1 is the fishing cost per unit effort of the prey H, q 2 is the fishing cost per unit effort of the predator P .
Then, the economic rent revenue at any time is given by where N 1 def p 1 H − q 1 c 1 , N 2 def p 2 P − q 2 c 2 , that is, N 1 and N 2 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 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.
that is, 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., c 1 0 and the predator harvesting remains operational if p 2 P − q 2 > 0.
Then, from 4.4 , we have Substituting it into 4.3 , it follows that Again, substituting 4.7 and 4.8 into 4.2 leads to So, if r 1 > a 2 q 2 /p 2 and r 2 > a 2 b 1 q 2 /r 1 p 2 − a 1 q 2 hold together, we have the bionomic equilibrium H 1∞ , P 1∞ , 0, c 2∞ .Case 4.2.If q 2 p 2 > P, 4.10 that is, 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., c 2 0 and the predator harvesting remains operational if p 1 H − q 1 > 0.
that is, 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, c 1 0, c 2 0. In this case, there is no bionomic equilibrium.Case 4.4.If q 2 p 2 < P, q 1 p 1 < H, 4.17 that is, 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 Substituting above equalities into 4.2 and 4.3 , it is easy to obtain hold together, we have the bionomic equilibrium H 1∞ , P 1∞ , c 1∞ , c 2∞ .
It is obviously that the bionomic equilibrium may exist if the intrinsic growth rates of two species exceed some value.

Optimal Harvesting Policy
In order to determine the optimal harvesting policy, we consider the present value J of a continuous time-stream of revenue where δ denotes the instantaneous annual rate of discount and c i t i 1, 2 are the control variables, which are subject to the assumption 0 < c i t < r i , i 1, 2. Now, our objective is to maximize J subject to the state equations 1.3 by invoking Pontryagin's maximal principle.
The Hamiltonian for the problem is given at first where λ 1 t and λ 2 t are the adjoint variables.
Obviously, the control variables c 1 and c 2 appear linearly in the Hamiltonian function H. So, the conditions are necessary for the singular control to be optimal.Then, we have that is Therefore, the shadow prices e δt λ i t do not vary with time in the optimal equilibrium.Hence they remain bounded as t → ∞.By the maximal principle, the adjoint variables satisfy dλ 1 /dt −∂H/∂H and dλ 2 /dt − ∂H/∂P , for all t ≥ 0, that is.

5.7
From 5.7 , we may find the positive values of H δ , P δ .Substituting H δ , P δ into 1.3 , we get the equations as follows:

5.8
Then, we may have the optimal equilibrium effort levels c 1δ and c 2δ , if 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.

6.3
From the above results, one could easily see that there is only one result H δ 1.121042647, P δ 0.6323992784 meeting the condition H δ > q 1 p 1 0.5 , P δ > q 2 p 2 0.4 .

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.
3 is as follows: