The Dynamics of an Impulsive Predator-Prey System with Stage Structure and Holling Type III Functional Response

and Applied Analysis 3 The solution of (4), denote by V = (x(t), y 1 (t), y 2 (t)) T, is a piecewise continuous function V : R + → R 3 + , V(t) is continuous on ((n − 1)T, (n + l − 1)T] × R + and ((n + l − 1) T, nT] × R 3 + (n ∈ N, 0 ≤ l ≤ 1) . (7) Obviously the existence and uniqueness of solutions of (4) is guaranteed by the smoothness properties off, which denotes the map defined by the right hand of system (4). The following lemmas are useful for the proof of themain results. Lemma 2 (see [5, 7]). Consider the following differential equation: ?̇? (t) = ax (t − τ) − bx (t) − cx 2 (t) , (8) where a, b, c, and τ are positive constants and x(t) > 0 for t ∈ [−τ, 0]. We have the following: (i) if a < b, then lim t→+∞ x(t) = 0; (ii) if a > b, then lim t→+∞ x(t) = (a − b)/c. Lemma 3 (see [14]). Consider the following system: ?̇? (t) = − wu (t) , t ̸ = (n + l − 1) T, t ̸ = nT, u (t + ) = u (t) + ru (t) (1 − u (t) K ) , t = (n + l − 1) T, u (t + ) = (1 − p) u (t) , t = nT. (9) Then, system (9) has a positive periodic solution u(t) with period T, which is globally stable, where u ∗ (t) = { { { { { { { { { {


Introduction
In recent decades, with the increase of population and the development of science and technology, human accelerated the exploitation of natural resources. Many harvesting predator-prey models have been studied. The dynamic relationship between predators and their preys has long been and will continue to be one of the dominant themes in both ecology and mathematical ecology due to its universal existence and importance; see [1][2][3][4][5][6] and the references cited therein. On the other hand, in the real world, many species usually go through two or even more life stages as they proceed from birth to death. Thus, it is practical to introduce the stage structure into predator-prey models; see [7][8][9][10][11][12]. For example, Wei and Wang [13] considered the following predator-prey system with stage structure: where ( ), 1 ( ), and 2 ( ) denote the densities of prey population and immature and mature individual predators at time , respectively. The meanings of all parameters may refer to [13]. Authors obtained the sufficient conditions of the persistence for system (1). However, it is well known that many evolution processes are characterized by the fact that at certain moments their stage changes abruptly. For example, for IPM strategy on ecosystem, the predators are released periodically every time , and periodic catching or spraying pesticide is also applied. Hence, the predator and prey experience a change of state abruptly. Consequently, it is natural to assume that these processes act in the form of impulse. Impulsive methods have been applied in almost every field of applied sciences. For example, many population models assume that the populations are born throughout the year, whereas it is often the case that many species give birth only during a single period of the year; that is, births occur in regular pulses. Hence, the authors Z. Xiang, D. Long, and X. Y. Song gave a single population logistic model with birth pulse and impulsive harvesting at different moments as follows: where ( ) represents the density of the resource population at time . Parameter is the intrinsic growth rate, the positive constant is referred to as the environmental carrying capacity, and parameter is the death rate of resource population. Parameter denotes the harvest rate of resource population. For more details of the biological meaning of system (2), we can refer to [14][15][16][17].
In addition, in the description of dynamical interactions on predators and their preys, a crucial element of all models is the classic definition of a predator's functional response. A functional response of the predator to the prey density refers to the change in the density of prey attached per unit time per predator as the prey density changes. For example, Zhang et al. [18] suggested a function called Holling type III. It is monotonic in the first quadrant, that is, if the prey population increases, then the consumption rate of prey per predator will increase too. And √ is the halfsaturation constant. The field of research on the dynamics of impulsive predator-prey model with functional response seems to be a new increasingly interesting area, which draws many scholars' attention. Moreover, by the picture of ocean's food-chain, we know that small fish can prey on fish larvae as a predator; also it can be eaten by the higher predator as a prey (see [19]). Hence, according to the nature of biological resource management, it is interesting to investigate impulsive harvesting on prey and mature predator (e.g., the small fish and higher predator in the ocean's food-chain) simultaneously at some fixed time.
Based on the above discussion, we consider the stagestructured predator-prey model with Holling type III functional response, birth pulse, and impulsive harvesting at different moments as follows: where ( ) denotes the density of the prey and 1 ( ), 2 ( ) represent the immature and mature predator densities, respectively. Parameters , , , , 1 , 2 , 3 , and 4 are positive constants, where is the intrinsic growth rate of the prey, denotes the capacity rate, concerned with the maintaining of the evolution of the population, represents the predation rate of predator, is the conversion rate that translated into predator population increase, 1 , 2 , and 3 denote the death rate of prey, immature predator, and mature predator, respectively, and 4 is the intrinsicspecific competition rate of the mature predator. Parameter represents a constant time from immaturity to maturity. Parameters 1 , 2 denote the harvesting rates of prey and mature predator at = , ∈ + , respectively, + = {1, 2, . . .}, and is the period of the impulsive effect.
It is well known that, in the sustainable development of natural resources, it is very important to study the sustainable survival of species. So, in this paper, we aim to investigate the global attractivity of predator-extinction periodic solution and the permanence of system (4). From the biological point of view, we only consider (4) in the biological meaning region This paper is organized as follows. Firstly, some preliminaries are given in Section 2. In Section 3, the sufficient conditions for the global attractivity of predator-extinction periodic solution are obtained. The permanence of system (4) is investigated in Section 4. In Section 5, we present some examples and simulations to illustrate our results. At last, a brief conclusion is given in Section 6.

Preliminaries
In this section, some definitions and lemmas are introduced. Let to be the map defined by the right hand of system (4). Let : Definition 1. Let ∈ 0 , and ( , V) ∈ (( −1) , ( + −1) ]× 3 + and (( + −1) , ]× 3 + , the upper right derivative of ( , V) with respect to the impulsive differential system (4) is defined as Abstract and Applied Analysis 3 The solution of (4), denote by V = ( ( ), 1 ( ), 2 ( )) , is a piecewise continuous function V : Obviously the existence and uniqueness of solutions of (4) is guaranteed by the smoothness properties of , which denotes the map defined by the right hand of system (4).
The following lemmas are useful for the proof of the main results.

Permanence of System (4)
In the real world, from the principle of ecosystem balance and saving resources, we only need to control the predator under Abstract and Applied Analysis 5 the economic threshold level and not to eradicate the predator totally. Thus, we focus on the permanence of system (4). First, we give the definition of permanence. (4) is said to be persistent if there exist positive constants and such that every positive solution ( ( ), 1 ( ), 2 ( )) of system (4) satisfies ≤ ( ), 1 ( ), 2 ( ) ≤ for sufficiently large enough. (4) is permanent, if the following conditions hold:

Examples and Numerical Simulations
In this section, we give some examples and numerical simulations to show the effectiveness of the main results.

Conclusion
In this paper, a stage-structured predator-prey model with Holling type III functional response, birth pulse, and impulsive harvesting at different moments is proposed. By using comparison theorem of impulsive differential equations and some analysis techniques, the sufficient conditions ensuring the predator-extinction periodic solution and the permanence of system (4) are obtained. Theorem 6 implies that reducing 1 and appropriately and (A 2 ) can be propitious to the global attractivity of the predator-extinction periodic solution ( * ( ), 0, 0). That is, if the mature predator is caught excessively or the immature predator is stocked too few, it will lose the merits of exploitative mature predator population. Similarly, we can see from Theorem 8 that reducing 1 and appropriately and (A 4 ) are in favor of the permanence of system (4). It implies that reasonable harvesting on mature predator and appropriate protecting on immature predator play an important role in the permanence of system (4). Moreover, by Theorems 6 and 8, we believe that there exists a sharp threshold, which is beneficial for people to make the best use of biological resources but will not break the biological balance. It will be interesting for us to continue to study the optimal harvesting policy of system (4) in the near future.