Periodic solutions for an impulsive semi-ratio-dependent predator – prey model with patches and time delays

In this paper, we study the existence of a positive periodic solution for a two-species semi-ratio-dependent predator-prey system with time delays and impulses in a two-patch environment. By using the method of coincidence degree theorem, a set of easily verifiable conditions are obtained for the existence of at least one strictly positive periodic solution for the system. In particular, our result generalizes some known criteria.


Introduction
In recent years, with the application of the theory of differential equations in mathematical ecology, a lot of mathematical models have been proposed in the study of population dynamics [2-4, 6, 8-11, 15-23].One of the famous models for the dynamics of populations is the so-called semi-ratio-dependent predator-prey system with functional response [4,7,9,15,17], for example where x and y stand for the population of the prey and predator, respectively, g(x) is the predator functional response to prey.
In equation (1.1), it has been assumed that the prey grows logistically with growth rate a and carrying capacity a/b in the absence of predation.The predator consumes the prey according to the functional response g(x) and grows logistically with growth rate d and carrying B Email: lrxcsu@163.com 2 R. Liang capacity x(t)/ f proportional to the population size of prey.The parameter f is a measure of the food quality that the prey provides for conversion into predator birth.
The form of the predator equation in (1.1) was first proposed by Leslie [9].The functional response g(x) in (1.1) can be classified into five types, including the Leslie-Gower model, the Holling-Tanner model, the Holling type III model, the Ivlev's functional response and so on.For more detail see reference [18].
We note that any biological or environmental parameters are naturally subject to fluctuation in time.Cushing [2] pointed out that it is necessary and important to consider models with periodic ecological parameters or perturbations which may be naturally exposed (for example, those due to seasonal effects of weather, food supply, mating habits, hunting or harvesting seasons, etc.).Thus, the assumption of periodicity of the parameters is a way of incorporating the periodicity of the environment.On the other hand, dispersal between patches often occurs in natural ecological environments, and more realistic models should include the dispersal process [8,20,21].
We consider the following systems with initial conditions where x i (t) represents the prey population in the i-th patch (i = 1, 2), and x 3 (t) represents the predator population.D i (t) denotes the dispersal rate of the prey in the i-th patch (i = 1, 2).τ = max{τ 1 , τ 2 }.However, there are numerous examples of evolutionary systems which at certain instants in time are subject to rapid changes.In the simulations of such processes it is frequently convenient and valid to neglect the durations of rapid changes and to assume that the changes can be represented by state jumps.Appropriate mathematical models for processes of the type described above are so-called systems with impulsive effects, see [1].One note that the research on theory and applications of impulsive differential equations have been many nice works [3, 6, 10, 12-14, 22, 23].Because harvest of many a populations are not continuous, the harvest is an annual harvest pulse.To describe a system more accurately, we should consider to use the impulsive differential equation.If we consider the regularly harvest, then (1.2) is revised as the following form: where b ik x i (t k ) (i = 1, 2, 3) represents the population x i (t) at t k regular harvest pulse.Through this paper, for system (1.3) the following conditions are assumed.
(C 2 ) g(t, x) is a continuous ω-periodic function with respect to the first variable and is differentiable with respect to the second variable, and g(t; 0) = 0, g(t, x) > 0 for any t ∈ R, x > 0.
(C 3 ) There exists a positive constant c 0 such that g(t, x) ≤ c 0 for any t ∈ R, x > 0.
(C 4 ) −1 < b ik ≤ 0, i = 1, 2, 3 for all k ∈ N and there exists a positive integer q such that In the following, we shall use the notations.
Theorem 1.1.In addition to (C 1 )-(C 3 ), assume further that the following hold: Then system (1.2) has at least one positive ω-periodic solution with strictly positive components.
The proof in [4] shows that Theorem 1.1 has room for improvement.The organization of this paper is as follows.In the next section, we establish some simple criteria for the existence of a positive periodic solution of system (1.3).We also note that our results improve Theorem A as b ik ≡ 0, because our results do not need the condition (H 2 ).Finally, we give some applications to show our results.

Existence of periodic solution
In this section, by using continuation theorem which was proposed in [5] by Gaines and Mawhin, we will establish the existence conditions of at least one positive periodic solution of system (1.3).To do so, we need to make some preparations.
Let X, Z be real Banach spaces, L : Dom L ⊂ X → Z be a Fredholm mapping of index zero (index L = dim Ker L − codim Im L), and let P : X → X, Q : Z → Z be continuous projectors such that Im P = Ker L, Ker Q = Im L and X = Ker L ⊕ Ker P, Z = Im L ⊕ Im Q. Denote by L P the restriction of L to Dom L ∩ Ker P, K P : Im L → Ker P ∩ Dom L the inverse (to L P ), and For convenience, we first introduce Mawhin's continuation theorem [5] as follows.
Lemma 2.1.Let Ω ⊂ X be an open bounded set.Let L be a Fredholm mapping of index zero and N be L-compact on Ω. Assume Then Lx = Nx has at least one solution in Ω ∩ Dom L.
To prove the main conclusion by means of the continuation theorem, we need to introduce some functional spaces. Let where | • | denotes the Euclidean norm.Then X and Y are Banach spaces.Theorem 2.2.In addition to (C 1 )-(C 4 ), assume further that the following hold: Then system (1.3) has at least one positive ω-periodic solution. Proof.Let then system (1.3) can be translated to T is a positive ω-periodic solution of (1.3).Therefore, to complete the proof, we need only to prove that (2.2) has one ω-periodic solution.
Let L : Dom L ⊂ X → Y, u → (u , ∆u(t 1 ), . . ., ∆u(t q )), Periodic solutions for impulsive predator-prey model It is easy to show that P and Q are continuous projectors satisfying Furthermore, through an easy computation, we can find that the inverse K P : Im L → Ker P ∩ Dom L has the form , and Clearly, QN and K P (I − Q)N are continuous.Using Lemma 2.4 in [1], it is not difficult to show that QN( Ω), K p (I − Q)N( Ω) are relatively compact for any open bounded set Ω ⊂ X.
3) is translated to (1.2).In this case, the condition (C 6 ) is the same as (H 3 ) of Theorem 1.1, but we see that (H 2 ) is not needed here.Hence our result improves and generalizes the corresponding result of [4].

Applications
In this section, we will list some applications of our above results.
From Theorem 2.2 one obatins the following.
where all functions are defined as above.The system (3.3)without impulse has been considered in [15].
From Theorem 2.2 we get the following. ω