model with nonmonotonic functional response

By using a continuation theorem based on coincidence degree theory, some new sufficient conditions are obtained for the existence of positive periodic solutions of the following neutral delay predator-prey model with nonmonotonic functional response: x 0 (t) = x(t)(r(t) −a(t)x(t −�(t)) −b(t)x 0 (t −�(t))) −g(x(t))y(t), y 0 (t) = y(t)(−d(t) +µ(t)g(x(t −�(t))). Moreover, an example is employed to illustrate the main results.


Introduction
In a classic study of population dynamics, the predator-prey models have been studied extensively.We refer the reader to [1−5] and the references cited therein.Up to the present, most authors just studied systems with monotonic functional response, such as [6,7].However, the actual living environments of species are not always like this due to the ecological effects of human activities and industry, e.g., the location of manufacturing industries and pollution of the atmosphere, rivers, and soil etc.In view of such kinds of situations, Fan and Quan [8] investigated the existence and uniqueness of limit cycle of such a type of predator-prey system, in which the predator would decrease its grasping ability while the prey has group defence ability, namely, where Φ(0) = 0, lim x→∞ Φ(x) < 0, Ψ(x), Φ(x) ∈ C 1 [0, +∞), Ψ(0) = 0, and ∃k > 0, such that (x − k)Ψ ′ (x) < 0 and lim x→∞ Ψ(x) = 0, µ, D are positive constants.For a special case of this system, in view of time delay effect, Ruan [9] and Xiao [10] considered the bifurcation and stability of the following predator-prey model with nonmonotonic functional response x ′ (t) = x(t)[a − bx(t)] − cx(t)y(t) m 2 +x 2 (t) , y ′ (t) = y(t)[−d + µx(t−τ ) m 2 +x 2 (t−τ ) ]. (1.1) where x(t) and y(t) represent predator and prey densities respectively, a, b, m, µ and d are all positive constants, and τ is a nonnegative constant.Furthermore, Fan and Wang [11] established verifiable criteria for the global existence of positive periodic solutions of a more general delayed predator-prey model with nonmonotonic functional response with periodic coefficients of the In particular, Kuang [12] studied the local stability and oscillation of the following neutral delay Gause-type predator-prey system: (1. (1.4) where x(t) and y(t) represent predator and prey densities respectively, r(t), a(t), b(t), d(t), and µ(t) are all positive periodic continuous functions with period ω > 0, σ(t), τ (t) are ω-periodic continuous functions, the function g satisfying the following conditions: where C n is the nth order continuous function space, n = 1, 2.
As pointed out by Kuang [13], it would be of interest to study the existence of periodic solutions for periodic systems with time delay.The periodic solutions play the same role as is played by the equilibria in autonomous systems.In addition, in view of the fact that many predator-prey systems display sustained fluctuations, it is thus desirable to construct predatorprey models capable of producing periodic solutions.To our knowledge, no such work has been done on the global existence of positive periodic solutions of (1.4).Motivated by this, our aim in this paper is, using the coincidence degree theory developed by Gaines and Mawhin [14], to derive a set of easily verifiable sufficient conditions for the existence of positive periodic solutions of system (1.4).For convenience, we will use the following notations In this paper, we always make the following assumptions for system (1.4). where and 2 The existence of a positive periodic solution In this section, we shall study the existence of at least one positive periodic solution of system (1.4).The method to be used in this paper involves the applications of the continuation theorem of the coincidence degree.For the readers' convenience, we introduce some concepts and results concerning the coincidence degree as follows.
Let X, Z be real Banach spaces, L : DomL ⊂ X → Z be a linear mapping, and N : X → Z be a continuous mapping.The mapping L is called a Fredholm mapping of index zero if If L is a Fredholm mapping of index zero and there exist continuous projectors P : X → X, We denote the inverse of that map by K P .
If Ω be an open bounded subset of X, the mapping N will be called L-compact on Ω if Since ImQ is isomorphic to KerL, there exists an isomorphism J : ImQ → KerL.
EJQTDE, 2012 No. 48, p. 4 Lemma 2.2 (See [11]).Suppose (H 3 ) holds, the algebraic equations has a unique positive solution if and only if, there exist two positive constants u 1 and u 2 such that and Theorem 2.1.Assume that (H 1 ) − (H 4 ) hold.Then system (1.4) has at least one ω-periodic solution with strictly positive components.
Proof.Consider the following system: ). ( where all functions are defined as ones in system (1.4).It is easy to see that if system (2.1) has one ω-periodic solution (u * 1 (t), u * 2 (t)) T , then (x * (t), y * (t)) T = (e u * 1 (t) , e u * 2 (t) ) T is a positive ω-periodic solution of system (1.4).Therefore, to complete the proof it suffices to show that system (2.1) has one ω-periodic solution. Take and define Then X and Z are Banach spaces when they are endowed with the norms and EJQTDE, 2012 No. 48, p. 5   With these notations system (2.1) can be written in the form Therefore L is a Fredholm mapping of index zero.Now define two projectors P : X → X and Q : Z → Z as Then P and Q are continuous projectors such that Furthermore, the generalized inverse (to L) K P : ImL → DomL ∩ KerP exists and has the form Then QN : X → Z and K P (I − Q)N : X → X can be read as and ω 0 r(s) − c(s)e u 1 (s−σ(s)) − h(e u 1 (s) )e u 2 (s) ds Suppose that (u 1 (t), u 2 (t)) T ∈ X is a solution of (2.2) for a certain λ ∈ (0, 1).Integrating (2.2) over the interval [0, ω] leads to In view of (2.2),(2.5)and (H 1 ), one can find (2.7) Let t = ϕ(p) be the inverse function of p = t − σ(t).It is easy to see that c(ϕ(p)) and σ ′ (ϕ(p)) are all ω-periodic functions.Furthermore, it follows from (2.5) and (H 1 ) that According to the mean value theorem of differential calculus, we see that there exists ξ ∈ [0, ω] e u 1 (ξ) + c(t)e u 1 (ξ−σ(ξ)) ≤ 2r.
Remark 2.1.It is easy to see that (H 3 ) is also the necessary condition for the existence of positive ω-periodic solutions of system (1.4).
Remark 2.2.The time delays σ(t) and τ (t) have influence on the existence of positive periodic solutions to system (1.4).

2 ) ω 0 −
d(s) + µ(s)g(e u 1 (s−τ (s)) ) ds     Obviously, QN and K P (I − Q)N are continuous by the Lebesgue theorem, and it is not difficult to show that K P (I − Q)N ( Ω) is compact for any open bounded Ω ⊂ X by using Arzela-Ascoli theorem.Moreover, QN ( Ω) is clearly bounded.Thus, N is L−compact on Ω for any open bounded set Ω ⊂ X.Now we reach the position to search for an appropriate open bounded subset Ω for the application of Lemma 2.1.Corresponding to operator equation Lu = λN u, λ ∈ (0, 1), we have constant M > 0 such that h(x) ≤ M, for x ∈ [0, +∞).