Permanence and Periodic Solution of Predator-Prey System with Holling Type Functional Response and Impulses

In biomathematics, many mathematical models have been established to describe the relationships between species and the outer environment, and the connections between different species. Among the relationships between the species living in the same outer environment, the predator-prey theory plays an important and fundamental role. The dynamic relationship between predators and their prey has long been one of the dominant theses in both ecology and mathematical ecology. Many excellent works have been done for the Lotka-Volterra type predator-prey system, for example, see [1–12]. In many situations, especially when predators have to search for food, a suitable general predatorprey theory is based on the so called ratio-dependent theory. Accordingly, researchers have proposed many ratio-dependent response functions. In [13], Holling suggested that there are three functional responses of the predator which usually are called Holling type I, Holling type II, and Holling type III. The type III response is typical of predators showing learning behavior, and


Introduction
In biomathematics, many mathematical models have been established to describe the relationships between species and the outer environment, and the connections between different species.Among the relationships between the species living in the same outer environment, the predator-prey theory plays an important and fundamental role.The dynamic relationship between predators and their prey has long been one of the dominant theses in both ecology and mathematical ecology.Many excellent works have been done for the Lotka-Volterra type predator-prey system, for example, see [1][2][3][4][5][6][7][8][9][10][11][12].In many situations, especially when predators have to search for food, a suitable general predatorprey theory is based on the so called ratio-dependent theory.Accordingly, researchers have proposed many ratio-dependent response functions.In [13], Holling suggested that there are three functional responses of the predator which usually are called Holling type I, Holling type II, and Holling type III.The type III response is typical of predators showing learning behavior, and ϕ(x) = μx 2 1 + ρx 2  (1.1) is usually a suitable response function.In [14], Wang and Li investigated the global existence of positive periodic solutions and permanent property of the ratio-dependent predator-prey system with Holling type III functional response and delay, which takes the form x 2 + Ay 2 (t) , where the functional response ϕ(u) = cu 2 /(1 + Au 2 ), u = x/ y; a(t), b(t), c(t), e(t), and d(t) are all positive periodic continuous functions, and A > 0, τ ≥ 0 are real constants.They found that the criteria for the permanence is exactly the same as that for the existence of the positive periodic solutions of (1.2).Other results about ratio-dependent functional response can be found in [11,[15][16][17][18].
On the other hand, biological species may undergo discrete changes of relatively short duration at a fixed time.Moreover, continuous changes in environment parameters such as temperature or rainfall can also create discontinuous outbreaks in pest population.Systems with short-term perturbations are often naturally described by impulsive differential equations.
The theory of impulsive differential equations is now being recognized to be not only richer than the corresponding theory of differential equations without impulses, but also representing a more natural framework for mathematical modeling of many real world phenomena [19][20][21][22][23][24].Thus the wide applications naturally motivate a deeper theoretical study of the subject.
Recently, some impulsive equations have been introduced in population dynamics in relation to population ecology, chemotherapeutic treatment of disease, impulsive birth, see [25][26][27][28][29] and the references therein.
In this paper, we consider the nonautonomous ratio-dependent predator-prey system with Holling type III functional response and impulsive effect Throughout this paper, we always assume system (1.3) satisfies the following conditions.
Denote, by PC(J,R), the set of function ρ : J → R which are continuous for t ∈ J, t = t k are continuous from the left for t ∈ J, and have discontinuities of the first kind at the point t k .Denote, by PC 1 (J,R), the set of function ρ : J → R with a derivative (dρ/dt) ∈ PC(J,R).The solution of the system (1.3) is a piecewise continuous function u = col(x(t), y(t)) : With the model (1.3), we can take into account the possible exterior effects under which the population densities change very rapidly.For instance, impulsive effect of the pest population density is possible after its partial destruction by catching, poisoning with chemicals used in agriculture (−1 < g k < 0), or after its increase because of migration of the outside pest population (g k > 0).An impulsive increase of the predator population density is possible by artificially breeding the species or releasing some species (p > 0).
We will also investigate the asymptotic behavior of nonnegative solution for system (1.3).Note that according to biological interpretation of the solutions x(t) and y(t), they must be nonnegative.Our results extend the ideas in [30].The organization of this paper is as follows.In the next section, we present necessary preliminaries and consider the dynamics of a single species model.We obtain the sufficient and necessary condition for permanence in Section 3. In Section 4, we discuss the existence and attractivity of the periodic solution of system (1.3).

Preliminary lemmas
In this section, we first consider the nonlinear single species model where ; γ i is a positive constant; there exists an integer q > 0 such that h k+q = h k , t k+q = t k + ω, and In order to explore the existence of periodic solutions of (2.1), for the reader's, we first summarize below a few concepts and results from [31] that will be used in this section.
Let X, Z be a normed space, L : DomL ⊂ X → Z be a linear mapping, and N : X → Z be a continuous mapping.The mapping L will be called a Fredholm mapping of index if dimKer L = codimIm L < +∞ and ImL is closed in Z.If L is a Fredholm mapping of index zero, there exist continuous projectors P : X → X and Q : Z → Z such that Im P = Ker L, Ker Q = Im P. It follows that L| DomL∩Ker P → Im L is invertible.We denote the inverse of that map by K P .If Ω is an open bounded subset of X, the mapping N will be called Lcompact on Ω if QN(Ω) is bounded and K P (I − Q)N : Ω → X is compact.Since Im Q is isomorphic to KerL, there exists an isomorphism J : ImQ → Ker L. Lemma 2.1 [31].Let Ω ⊂ X be an open bounded set and L be a Fredholm mapping of index zero.Assume that N : X → Z is a continuous operator and L-compact on Ω. Suppose (a) Lx = λNx for all λ ∈ (0,1) and
(2) Making the change of variable x(t) = exp u(t), (2.1) is transformed into (2.7) Weibing Wang et al. 5 Let Then X and Y are Banach spaces.
Lemma 2.4.Consider the equation where α(t) is a continuous ω-periodic function, p is a positive constant, and there is an integer q > 0 such that t k+q = t k + ω.Assume that ω 0 α(t)dt < 0, then (2.28) has a unique positive, globally asymptotically stable ω-periodic solution.
It is easy to show that the following function where is a unique positive, globally asymptotically stable ω-periodic solution of (2.28).