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

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

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: 𝑑𝐻=𝑟𝑑𝑡¹−𝑎₁𝑃−𝑏₁𝐻𝐻,𝑑𝑃=𝑟𝑑𝑡²−𝑎₂𝑃𝐻𝑃, (1.1) 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 𝐻_∗=𝑟₁𝑎₂𝑎₁𝑟₂+𝑎₂𝑏₁,𝑃_∗=𝑟₁𝑟₂𝑎₁𝑟₂+𝑎₂𝑏₁. (1.2) 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: 𝑑𝐻=𝑟𝑑𝑡¹−𝑎₁𝑃−𝑏₁𝐻𝐻−𝑐₁𝐻,𝑑𝑃=𝑟𝑑𝑡²−𝑎₂𝑃𝐻𝑃−𝑐₂𝑃, (1.3) 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.

2. 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 𝐻_1∗=𝑟₁−𝑐₁𝑎₂𝑎₁𝑟₂−𝑐₂+𝑎₂𝑏₁,𝑃_1∗=𝑟₁−𝑐₁𝑟₂−𝑐₂𝑎₁𝑟₂−𝑐₂+𝑎₂𝑏₁. (2.1) Obviously, (𝐻_1∗,𝑃_1∗) satisfies the equalities 𝑟₁−𝑐₁−𝑏₁𝐻_1∗−𝑎₁𝑃_1∗=0,𝑟₂−𝑐₂=𝑎₂𝑃_1∗𝐻_1∗. (2.2)

Our result about the local stability property of this equilibrium is stated as follows.

Theorem 2.1. The positive equilibrium (𝐻_1∗,𝑃_1∗) of system (1.3) is locally asymptotically stable.

Proof. The variational matrix 𝐽^∗(𝐻,𝑃) of the system (1.3) is given by 𝑟₁−𝑐₁−𝑎₁𝑃−2𝑏₁𝐻−𝑎₁𝐻𝑎₂𝑃²𝐻²𝑟₂−𝑐₂−2𝑎₂𝑃𝐻. (2.3) So, the characteristic equation for 𝐽^∗(𝐻^∗,𝑃^∗) is given by 𝜆²+𝑎𝜆+𝑏=0, where 𝑎𝑎=²𝑏₁𝑟₁−𝑐₁𝑎₁𝑟₂−𝑐₂+𝑎₂𝑏₁+𝑟₂−𝑐₂(>0),𝑏=𝑟₁−𝑐₁𝑟₂−𝑐₂(>0). (2.4) 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 𝑚₁≤liminf_𝑡→+∞𝐻(𝑡)≤limsup_𝑡→+∞𝐻(𝑡)≤𝑀₁,𝑚₂≤liminf_𝑡→+∞𝑃(𝑡)≤limsup_𝑡→+∞𝑃(𝑡)≤𝑀₂. (2.5)
Theorem 2.2 shows that lim_𝑡→+∞𝐻(𝑡)=𝐻_1∗>0,lim_𝑡→+∞𝑃(𝑡)=𝑃_1∗>0. (2.6) 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: 𝑉()𝐻𝐻,𝑃=ln𝐻_1∗+𝐻_1∗𝐻𝑎−1+¹𝐻_1∗𝑎₂𝑃ln𝑃_1∗+𝑃_1∗𝑃−1. (2.7) Obviously, 𝑉(𝐻,𝑃) is well defined and continuous for all 𝐻,𝑃>0. By simple computation, we have 𝜕𝑉=1𝜕𝐻𝐻𝐻1−^1∗𝐻,𝜕𝑉=𝑎𝜕𝑃¹𝐻_1∗𝑎₂𝑃𝑃1−^1∗𝑃. (2.8) 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 𝜕²𝑉𝜕𝐻²=1𝐻²−1+2𝐻_1∗𝐻,𝜕²𝑉𝜕𝜕𝑃𝜕𝐻=0,²𝑉𝜕𝑃²=𝑎₁𝐻_1∗𝑎₂𝑃²−1+2𝑃_1∗𝑃. (2.9) Therefore, 𝑑²𝑉|_{(𝐻1∗,𝑃1∗)}=1𝐻_1∗𝑑𝐻²+𝑎₁𝐻_1∗𝑎₂𝑃_1∗𝑑𝑃²>0. (2.10) 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_𝐻→0𝑉(𝐻,𝑃)=lim_𝑃→0𝑉(𝐻,𝑃)=lim_𝐻→+∞𝑉(𝐻,𝑃)=lim_𝑃→+∞𝑉(𝐻,𝑃)=+∞. (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, 𝑉(𝐻,𝑃)>𝑉𝐻_1∗,𝑃_1∗=0, (2.12) 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=1𝑑𝑡𝐻𝐻1−^1∗𝐻𝑟₁−𝑐₁−𝑏₁𝐻−𝑎₁𝑃𝑎𝐻+¹𝐻_1∗𝑎₂𝑃𝑃1−^1∗𝑃⋅𝑟₂−𝑐₂−𝑎₂𝑃𝐻𝑃=𝐻−𝐻_1∗𝐻𝑏₁𝐻_1∗+𝑎₁𝑃_1∗−𝑏₁𝐻−𝑎₁𝑃+𝑎₁𝐻_1∗𝑎₂⋅𝑃1−^1∗𝑃⋅𝑎₂𝑃_1∗𝐻_1∗−𝑎₂𝑃𝐻𝑏=−¹𝐻𝐻−𝐻_1∗²−𝑎₁𝑃𝑃−𝑃_1∗². (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 𝐑²₊. In fact, we can also prove this declare by using Bendixson-Dulac theorem.

Let 𝐹(𝐻,𝑃)≐𝑟₁−𝑎₁𝑃−𝑏₁𝐻𝐻−𝑐₁𝐺()≐𝑟𝐻,𝐻,𝑃₂−𝑎₂𝑃𝐻𝑃−𝑐₂𝐵()≐1𝑃,𝐻,𝑃.𝐻𝑃 (2.14) Obviously, 𝐹(𝐻,𝑃), 𝐺(𝐻,𝑃), and 𝐵(𝐻,𝑃)∈𝐂¹(𝐑²₊).

Calculating from above equations we get 𝜕𝐵𝐹+𝜕𝐻𝜕𝐵𝐺𝑏𝜕𝑃=−¹𝑃−𝑎₂𝐻²<0,(𝐻,𝑃)∈𝐑²₊. (2.15)

According to Bendixson-Dulac theorem, we know that there is no limit cycle of system (1.3) in 𝐑²₊.

3. The Influence of Harvesting

We will discuss this topic on three aspects.

(1) The case of only harvesting prey species
In this case, 𝐻_1∗=𝑟₁−𝑐₁𝑎₂𝑎₁𝑟₂+𝑎₂𝑏₁,𝑃_1∗=𝑟₁−𝑐₁𝑟₂𝑎₁𝑟₂+𝑎₂𝑏₁. (3.1) Obviously, 𝐻_1∗, 𝑃_1∗ are all continuous differentiable function of parameter 𝑐₁ and 𝑑𝐻_1∗𝑑𝑐₁=−𝑎₂𝑎₁𝑟₂+𝑎₂𝑏₁<0,𝑑𝑃_1∗𝑑𝑐₁=−𝑟₂𝑎₁𝑟₂+𝑎₂𝑏₁<0. (3.2) The above inequalities show that 𝐻_1∗ and 𝑃_1∗ 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, 𝐻_1∗=𝑟₁𝑎₂𝑎₁𝑟₂−𝑐₂+𝑎₂𝑏₁,𝑃_1∗=𝑟₁𝑟₂−𝑐₂𝑎₁𝑟₂−𝑐₂+𝑎₂𝑏₁. (3.3) that is, 𝐻_1∗, 𝑃_1∗ are all continuous differentiable function of parameter 𝑐₂. Noticing that 𝑑𝐻_1∗𝑑𝑐₂=𝑟₁𝑎₂𝑎₁𝑎₁𝑟₂−𝑐₂+𝑎₂𝑏₁²>0,𝑑𝑃_1∗𝑑𝑐₂=−𝑟₁𝑎₂𝑏₁𝑎₁𝑟₂−𝑐₂+𝑎₂𝑏₁²<0. (3.4) It is easy to see that 𝐻_1∗ is the strictly increasing function of parameter 𝑐₂, while 𝑃_1∗ 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 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, 𝐻_1∗=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 case 𝐻_1∗=𝐻_∗, that is, 𝑟₁−𝑐₁𝑎₂𝑎₁𝑟₂−𝑐₂+𝑎₂𝑏₁=𝑟₁𝑎₂𝑎₁𝑟₂+𝑎₂𝑏₁. (3.5) Solving the above equality, we obtain 𝑐₂=𝑎₁𝑟₂+𝑎₂𝑏₁𝑟₁𝑎₁⋅𝑐₁. (3.6) 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 𝑃_1∗=𝑟²₁𝑎₁𝑟₂−𝑟₁𝑐₁𝑎₁𝑟₂+𝑎₂𝑏₁𝑎₁𝑟₁−𝑐₁𝑎₁𝑟₂+𝑎₂𝑏₁. (3.7) Obviously, 𝑃_1∗ is the continuous differential function of parameter 𝑐₁ and 𝑑𝑃_1∗𝑑𝑐₁𝑏=−¹𝑎₂𝑟²₁𝑎₁𝑟₁−𝑐₁²𝑎₁𝑟₂+𝑎₂𝑏₁<0, (3.8) 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∗).

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.

Let 𝑝₁ 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 𝑁=TR−TC=𝑝₁𝐻−𝑞₁𝑐₁+𝑝₂𝑃−𝑞₂𝑐=₂^def𝑁₁+𝑁₂, (4.1) where 𝑁=₁^def(𝑝₁𝐻−𝑞₁)𝑐₁, 𝑁=₂^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 𝑑𝐻=𝑟𝑑𝑡¹−𝑎₁𝑃−𝑏₁𝐻𝐻−𝑐₁𝐻,𝑑𝑃=𝑟𝑑𝑡²−𝑎₂𝑃𝐻𝑃−𝑐₂𝑝𝑃,𝑁=₁𝐻−𝑞₁𝑐₁+𝑝₂𝑃−𝑞₂𝑐₂=0. (4.2)(4.3)(4.4)

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 𝑞₁𝑝₁>𝐻, (4.5) that is, 𝑝₁𝐻−𝑞₁<0 (4.6) 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 𝑃_1∞=𝑞₂𝑝₂. (4.7) Substituting it into (4.3), it follows that 𝐻_1∞=𝑟₁𝑝₂−𝑎₁𝑞₂𝑏₁𝑝₂. (4.8) Again, substituting (4.7) and (4.8) into (4.2) leads to 𝑐_2∞=𝑟₂−𝑎₂𝑝_1∞𝐻_1∞=𝑟₂−𝑎₂𝑏₁𝑞₂𝑟₁𝑝₂−𝑎₁𝑞₂. (4.9)
So, if 𝑟₁>𝑎₂(𝑞₂/𝑝₂) and 𝑟₂>(𝑎₂𝑏₁𝑞₂/𝑟₁𝑝₂−𝑎₁𝑞₂) hold together, we have the bionomic equilibrium [𝐻_1∞,𝑃_1∞,0,𝑐_2∞].

Case 4.2. If 𝑞₂𝑝₂>𝑃, (4.10) that is, 𝑝₂𝑃−𝑞₂<0 (4.11) 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 𝐻_1∞=𝑞₁𝑝₁. (4.12) Substituting (4.12) into (4.3), we obtain 𝑃_1∞=𝑟₂𝑞₁𝑎₂𝑝₁. (4.13) Again, substituting (4.12) and (4.13) into (4.2) leads to 𝑐_1∞=𝑟₁−𝑎₁𝑃_1∞−𝑏₁𝐻_1∞=𝑟₁−𝑎₁𝑟₂−𝑎₂𝑏₁𝑞₁𝑎₂𝑝₁. (4.14) So, if 𝑟₁>((𝑎₁𝑟₂−𝑎₂𝑏₁)𝑞₁/𝑎₂𝑝₁) hold, we have the bionomic equilibrium [𝐻_1∞,𝑃_1∞,𝑐_1∞,0].

Case 4.3. If 𝑞₂𝑝₂𝑞>𝑃,¹𝑝₁>𝐻, (4.15) that is, 𝑝₂𝑃−𝑞₂<0,𝑝₁𝐻−𝑞₁<0 (4.16) 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 𝑞₂𝑝₂𝑞<𝑃,¹𝑝₁<𝐻, (4.17) that is, 𝑝₂𝑃−𝑞₂>0,𝑝₁𝐻−𝑞₁>0 (4.18) 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 𝐻_1∞=𝑞₁𝑝₁,𝑃_1∞=𝑞₂𝑝₂. (4.19)
Substituting above equalities into (4.2) and (4.3), it is easy to obtain 𝑐_1∞=𝑟₁−𝑎₁𝑝₂𝑞₁−𝑏₁𝑝₁𝑞₂𝑝₁𝑝₂,𝑐_2∞=𝑟₂−𝑎₂𝑝₁𝑞₂𝑝₂𝑞₁. (4.20) So, if 𝑟₁>𝑎₁𝑝₂𝑞₁−𝑏₁𝑝₁𝑞₂𝑝₁𝑝₂,𝑟₂>𝑎₂𝑝₁𝑞₂𝑝₂𝑞₁ (4.21) 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.

5. Optimal Harvesting Policy

In order to determine the optimal harvesting policy, we consider the present value 𝐽 of a continuous time-stream of revenue 𝐽=^∞₀𝑒^−𝛿𝑡𝑝₁𝐻−𝑞₁𝑐₁(𝑡)+𝑝₂𝑃−𝑞₂𝑐₂(𝑡)𝑑𝑡, (5.1) 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 𝐻=𝑒^−𝛿𝑡𝑝₁𝐻−𝑞₁𝑐₁(𝑡)+𝑝₂𝑃−𝑞₂𝑐₂(𝑡)+𝜆₁𝑟₁−𝑎₁𝑃−𝑏₁𝐻𝐻−𝑐₁𝐻+𝜆₂𝑟₂−𝑎₂𝑃𝐻𝑃−𝑐₂𝑃, (5.2) where 𝜆₁(𝑡) and 𝜆₂(𝑡) are the adjoint variables.

Obviously, the control variables 𝑐₁ and 𝑐₂ appear linearly in the Hamiltonian function 𝐻. So, the conditions 𝜕𝐻𝜕𝑐_𝑖=0,𝑖=1,2 (5.3) are necessary for the singular control to be optimal.

Then, we have 𝜆₁=𝑒^−𝛿𝑡𝑝₁−𝑞₁𝐻,𝜆₂=𝑒^−𝛿𝑡𝑝₂−𝑞₂𝑃, (5.4) that is 𝜆₁𝑒^𝛿𝑡=𝑝₁−𝑞₁𝐻,𝜆₂𝑒^𝛿𝑡=𝑝₂−𝑞₂𝑃. (5.5) Therefore, the shadow prices 𝑒^𝛿𝑡𝜆_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 𝑑𝜆₁/𝑑𝑡=−𝜕𝐻/𝜕𝐻 and 𝑑𝜆₂/𝑑𝑡=−(𝜕𝐻/𝜕𝑃), for all 𝑡≥0, that is. 𝑑𝜆₁𝑒𝑑𝑡=−^−𝛿𝑡𝑝₁𝑐₁+𝜆₁𝑟₁−𝑐₁𝑎₁𝑝₁−2𝑏₁𝐻−𝜆₂𝑎₂𝑃𝐻²,𝑑𝜆₂𝑒𝑑𝑡=−^−𝛿𝑡𝑝₂𝑐₂+𝜆₂𝑟₂−𝑐₂−2𝑎₂𝑃𝐻−𝜆₁𝑎₁𝐻. (5.6) Substituting (5.4) and (5.5) into (5.6), we get 2𝑎₂𝑝₂𝑃²+𝑎₁𝑝₁𝐻²𝑃+𝑝₂𝛿−𝑟₂𝑝₂−𝑎₁𝑞₁𝐻𝑃−𝑎₂𝑞₂𝑃−𝑞₂𝛿𝐻=0,2𝑏₁𝑝₁𝐻³+𝑎₁𝑝₁𝑃𝐻²+𝑝₁𝛿−𝑟₁𝑝₁−𝑏₁𝑞₁𝐻²−𝑞₁𝛿𝐻−𝑎₂𝑝₂𝑃+𝑎₂𝑞₂=0. (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: 𝑐_1𝛿=𝑟₁−𝑎₁𝑃_𝛿−𝑏₁𝐻_𝛿,𝑐_2𝛿=𝑟₂−𝑎₂𝑃_𝛿𝐻_𝛿. (5.8) Then, we may have the optimal equilibrium effort levels 𝑐_1𝛿 and 𝑐_2𝛿, if 𝑟₁>𝑎₁𝑃_𝛿+𝑏₁𝐻_𝛿,𝑟₂>𝑎₂𝑃_𝛿𝐻_𝛿. (5.9)

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 𝑟₁=1.6,𝑎₁=0.7,𝑏₁=0.3,𝑝₁=0.4,𝑞₁𝑟=0.2,₂=1,𝑎₂=0.25,𝛿=1.25,𝑝₂=0.5,𝑞₂=0.2. (6.1)

From Section 6, 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 0.25𝑃²+0.28𝐻²𝑃−0.015𝐻𝑃−0.05𝑃−0.25𝐻=0,0.24𝐻³+0.28𝑃𝐻²−0.2𝐻²−0.25𝐻−0.125𝑃+0.05=0. (6.2)

Solving (6.2) by using Maple, we get the results as follow: {},{},{},{},{},{}.𝐻=0.03875658339,𝑃=0.3212762700𝐻=0.3442951974,𝑃=−0.5444632570𝐻=1.121042647,𝑃=0.6323992784𝐻=0.8269704200+1.125716729𝐼,𝑃=0.1137766719−1.326783285𝐼𝐻=0.8269704200−1.125716729𝐼,𝑃=0.1137766719+1.326783285𝐼𝐻=−0.7529121920+0.04647885793𝐼,𝑃=−0.2550446252−0.7947461329𝐼 (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 condition𝐻_𝛿>𝑞₁𝑝₁(=0.5),𝑃_𝛿>𝑞₂𝑝₂(=0.4). (6.4) In this case, the corresponding system of (1.3) is as follows: 𝑑𝐻=()𝑑𝑡1.6−0.7𝑃−0.3𝐻𝐻−𝑐₁𝐻,𝑑𝑃=𝑃𝑑𝑡1−0.25𝐻𝑃−𝑐₂𝑃. (6.5)

Substituting (𝐻_𝛿=1.121042647, 𝑃_𝛿=0.6323992784) into (6.5), we obtain the optimal harvesting efforts 𝑐_1𝛿=0.821008<𝑟₁(=1.6),𝑐_2𝛿=0.858971<𝑟₂(=1). (6.6)

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.

Acknowledgments

The author is grateful to the anonymous referees for their excellent suggestions, which greatly improved the presentation of the paper. Also, this work was supported by the Technology Innovation Platform Project of Fujian Province (2009J1007) and the Start-up Foundation of Science and Technology of Mingjiang University (YKQ08001).