Harvesting Control for a Stage-Structured Predator-Prey Model with Ivlev ’ s Functional Response and Impulsive Stocking on Prey

We investigate a delayed stage-structured Ivlev’s functional response predator-prey model with impulsive stocking on prey and continuous harvesting on predator. Sufficient conditions of the global attractivity of predator-extinction periodic solution and the permanence of the system are obtained. These results show that the behavior of impulsive stocking on prey plays an important role for the permanence of the system. We also prove that all solutions of the system are uniformly ultimately bounded. Our results provide reliable tactical basis for the biological resource management and enrich the theory of impulsive delay differential equations.


Introduction
Biological resources are renewable resources.Economic and biological aspects of renewable resources management have been considered by Clark [1].In recent years, the optimal management of renewable resources, which has a direct relationship to sustainable development, has been studied extensively by many authors [1,2].Generally speaking, the exploitation of a species should be determined by the economic and biological values of the population.It is the purpose of this paper to analyze the exploitation of the stage-structured predator-prey model with harvesting on mature predator population.
In the natural world, there are many species whose individual members have a life history that takes them through two stages-immature and mature.In [3], a stagestructured model of population growth consisting of immature and mature individuals was analyzed, where the stage-structured was modeled by introduction of a constant time delay.Other population growth and infectious disease models with time delays were considered in [3][4][5][6][7].For the above discussion, we investigate a delayed stage-structured Ivlev's functional response predator-prey model with impulsive stocking on the prey and continuous harvesting on the predator.It may be more appropriate to the biological resource management.We will obtain the sufficient conditions for the global attractivity of the predator-extinction periodic solution and the permanence of the system.Our results provide reliable tactical basis for the biological resource management, and enrich the theory of impulsive differential equations.

Model formulation
There were many works concerning predator-prey system, and many good results are obtained [3,[8][9][10][11].Especially, Kooij and Zegeling [12] investigated the predator-prey model with Ivlev's functional response.The basic predator-prey model is where x 1 (t) and x 2 (t) are densities of the prey and the predator, respectively, r > 0 is the intrinsic growth rate of the prey, a > 0 is the coefficient of intraspecific competition, b > 0 is the per capita rate of predation of the predator, d > 0 is the death rate of the predator, c > 0 denotes the product of the per capita rate of predation and the rate of conversing prey into the predator.If rc − da < 0, system (2.1) do not have any positive equilibrium point, and the only unique equilibrium point (r/a,0) is globally asymptotically stable, which implies that the predator population will go extinction.If the prey is stocked at constant rate, then system (2.1) becomes the following differential equation: 2) It can be easily derived that if μ > d(ad − rc)/c 2 , system (2.2) has a unique globally asymptotically stable positive equilibrium (d/c,(rdc − ad 2 + μc)/bcd).This implies that the behavior of stocking prey assures the permanence of system (2.2).While stage-structured models were analyzed in many literatures [3,8,9,[13][14][15][16][17][18][19], the following single-species stage-structured model was introduced by Aiello and Freedman [9]: where x(t), y(t) represent the immature and mature populations densities, respectively.τ represents a constant time to maturity, and β, r, and η 2 are positive constants.This model is derived as follows.We assume that at any time t > 0, birth into the immature population is proportional to the existing mature population with proportionality constant β.We assume that the death rate of immature population is proportional to the existing immature population with proportionality constant r.We also assume that the death rate of mature population is of a logistic nature, that is, proportional to the square of the population with proportionality constant η 2 .
According to the nature of biological resource management, developing (2.2) with (2.3) by introducing the stocking on prey at fixed moments and harvesting mature predator population throughout the whole year or continuously, and considering Ivlev's functional response, we consider the following impulsive delay differential equations: where x 1 (t) denotes the density of the prey, x 2 (t), x 3 (t) represent the immature and mature predator densities, respectively.τ 1 represents a constant time to maturity, a > 0 is the intrinsic growth rate of the prey, b > 0 is the coefficient of intraspecific competition, r, w, θ, d 3 , d 4 , k, c, and β are positive constants, and 0 < E < 1 is the effect of continuous harvesting on the predator.This model is derived as follows.We assume that at any time t > 0, birth into the immature predator population is proportional to the existing mature predator population with proportionality constant r.We then assume that the death rate of immature predator population is proportional to the existing immature predator population with proportionality constant w. w(w > d), d 3 are called the death coefficient of x 2 (t), x 3 (t), respectively.We assume that the death rate of mature predator populations are of a logistic nature, that is, proportional to the square of the population with proportionality constant d 4 .k > 0 is the rate of conversing the prey into the predator.Δx 1 (t) = x 1 (t + ) − x 1 (t), μ ≥ 0 is the stocking amount of the prey at t = nτ, n ∈ Z + and Z + = {1, 2,...}, τ is the period of the impulsive stocking on the prey.We will prove that the system (2.4) has a predator-extinction periodic solution.Further, it is globally attractive.Due to the stocking on the prey, the mature predator population will not go extinction for the continuous harvesting of mature predator population, that is, system (2.4) is permanent.In this paper, we always assume that the immature predator population cannot predate the prey population.
Because the first and third equations of (2.4) do not contain x 2 (t), we can simplify model (2.4) and restrict our attention to the following model: The initial conditions for (2.5) are (2.6)

Some important lemmas
The solution of (2.4), denoted by x(t) = (x 1 (t),x 2 (t),x 3 (t)) T , is a piecewise continuous function x : R + → R 3 + , x(t) is continuous on (nτ,(n + 1)τ], n ∈ Z + and x(nτ + ) = lim t→nτ + x(t) exists.Obviously the global existence and uniqueness of the solutions of (2.4) are guaranteed by the smoothness properties of f , which denote the mapping defined by right-hand side of system (2.4) (see Lakshmikantham et al. [20] and Baȋnov and Simeonov [21]).For the continuity of the initial conditions, we require Before we have the the main results, we need to give some lemmas which will be used in the next.
Then any solution of system (2.4) is strictly positive.
By induction and similar method to the proof of [22, Theorem 1], we can show that x 2 (t) > 0 for all t ≥ 0. This completes the proof.
Consider the following delay equation: we assume that a 1 ,a 2 ,τ > 0; x(t) > 0 for −τ ≤ t ≤ 0. The following result for system (3.12) can be easily obtained from Lemma 3.4.
Similarly, we can obtain the following important lemma for our next work.

Global attractivity
In this section, we will obtain the sufficient condition of the global attractivity of the predator-extinction periodic solution of system (2.4).

Permanence
The next work is to investigate the permanence of the system (2.4).Before starting our theorem, we give the definition of permanence.
Definition 5.1.System (2.4) is said to be permanent if there are constants m, M > 0 (independent of initial value) and a finite time T 0 such that for all solutions (x 1 (t),x 2 (t),x 3 (t)) with all initial values x 1 (0 + ) > 0, x 2 (0 + ) > 0, Here T 0 may depend on the initial values (x 1 (0 + ), x 2 (0 + ),(x 3 (0 + )). (5.1) Then there is a positive constant q such that each positive solution (x 1 (t),x 3 (t)) of (2.5) satisfies for t large enough.Where x * 3 is determined by the following equation: Proof.The second equation of (2.5) can be rewritten as x 3 (u)du. ( Let us consider any positive solution (x 1 (t),x 3 (t)) of system (2.5).According to (5.4), V (t) can be defined as By calculating the derivative of V (t) along the solution of (2.5), we have Due to Lemma 3.3, (5.6) can be written K. Liu and L. Chen 11 we can easily know that there exists a sufficiently small ε > 0 such that (5.9) We claim that for any t 0 > 0, it is impossible that x 3 (t) < x * 3 for all t > t 0 .Suppose that the claim is not valid.Then there is a t 0 > 0 such that x 3 (t) < x *   3 for all t > t 0 .It follows from the first equation of (2.5) that for all t > t 0 , (5.10) Consider the following comparison impulsive system for all t > t 0 : (5.11)By Lemma 3.6, for t ∈ (nτ,(n + 1)τ], we obtain is the unique positive periodic solution of (5.11) which is globally asymptotically stable, where v * = a/2b By the comparison theorem for impulsive differential equation [21], we know that there exists t 1 (> t 0 + τ 1 ) such that the following inequality holds for t ≥ t 1 : (5.13) Thus for all t ≥ t 1 .We make notation as σ Δ = v * − b/2a − ε for convenience.From (5.9), we have (5.15)By (5.6) and (5.14), we have for all t > t 1 .Set x 3 (t). (5.17) We will show that x 3 (t) ≥ x m 3 for all t ≥ t 1 .Suppose the contrary.Then there is a T 0 > 0 such that x 3 (t) ≥ x m 3 for t 1 ≤ t ≤ t 1 + τ 1 + T 0 , x 3 (t 1 + τ 1 + T 0 ) = x m 3 and x 3 (t 1 + τ 1 + T 0 ) < 0. Hence, the first equation of systems (2.5) and (5.14) imply that This is a contradiction.Thus, x 3 (t) ≥ x m 3 for all t > t 1 .As a consequence, (5.9) and (5.16) lead to for all t > t 1 .This implies that as t → ∞, V (t) → ∞.It is a contradiction to V (t) ≤ M(1 + rτ 1 e −wτ1 + kβ(1 − e −θM )).Hence, the claim is true.By the claim, we are left to consider two cases.First, x 3 (t) ≥ x * 3 for all t large enough.Second, x 3 (t) oscillates about x * 3 for t large enough.Define where q 1 = x * 3 e −(d3+E+d4M)τ1 .We will show that x 3 (t) ≥ q for all t large enough.The conclusion is evident in first case.For the second case, let t * > 0 and ξ > 0 satisfy x 3 (t * ) = x 3 (t * + ξ) = x * 3 and x 3 (t) < x * 3 for all t * < t < t * + ξ, where t * is sufficiently large such that x 3 (t) > σ for t * < t < t * + ξ. (5.21) x 3 (t) is uniformly continuous.The positive solutions of (2.5) are ultimately bounded and x 3 (t) is not affected by impulses.Hence, there is a T(0 < t < τ 1 and T is independent of the choice of t * ) such that x 3 (t) > x * 3 /3 for t * < t < t * + T. If ξ < T, there is nothing to prove.Let us consider the case T < ξ < τ 1 .Since x 3 (t) > −(d 3 + E + d 4 M)x 3 (t) and x 3 (t * ) = x * 3 , it is clear that x 3 (t) ≥ q 1 for t ∈ [t * ,t * + τ 1 ].Then, proceed exactly as the proof for the K. Liu and L. Chen 13 above claim.We see that x 3 (t) ≥ q 1 for t ∈ [t * + τ 1 ,t * + ξ] because the kind of interval t ∈ [t * ,t * + ξ] is chosen in an arbitrary way (we only need t * to be large).We conclude that x 3 (t) ≥ q for all large t.In view of our above discussion, the choice of q is independent of the positive solution, and we prove that any positive solution of (2.5) satisfies x 3 (t) ≥ q for all sufficiently large t.This completes the proof of the theorem.
then the system (2.4) is permanent.
Proof.Denote (x 1 (t),x 2 (t),x 3 (t)) is any solution of system (2.4).From the first equation of system (2.5) and Theorem 5.2, we have where δ = r(p − e −wτ1 M)/w − ε.By Theorem 5.2 and the above discussion, system (2.4) is permanent.The proof of Theorem 5.3 is complete.

Discussion
According to the fact of biological resource management, in this paper, a delayed stagestructured Ivlev's functional response predator-prey system with impulsive stocking on the prey and continuous harvesting on the predator is considered.We get the condition under which the predator-extinction periodic solution of system (2.4) is globally attractive, and obtained the condition for the permanent of system (2.4).From Theorems 4.1 and 5.3, we can easily guess there must exist a threshold μ * .If μ < μ * , the predatorextinction periodic solution ( x 1 (t),0,0) of (2.4) is globally attractive.If μ > μ * , system (2.4) is permanent.Or from Theorems 4.1 and 5.2, we can easily guess that there must exist a threshold E * .If E > E * , the predator-extinction periodic solution ( x 1 (t),0,0) of (2.4) is globally attractive.If E < E * , system (2.4) is permanent.The results show that the behavior of impulsive stocking on the prey plays an important role for the permanence of system (2.4), that is, it can prevent the predator from dying out.This can meet in biological balance protection.But there are some interesting problems: how does the impulsive stocking on prey affect the dynamical behavior of system (2.4)?What are the optimal harvesting policy of the system (2.4)?We will continue to study these problems in the future.