DYNAMICS OF A NONAUTONOMOUS SEMIRATIO-DEPENDENT PREDATOR-PREY SYSTEM WITH NONMONOTONIC FUNCTIONAL RESPONSES

A nonautonomous semiratio-dependent predator-prey system with nonmonotonic functional responses is investigated. For general nonautonomous case, positive invariance, permanence, and globally asymptotic stability for the system are studied. For the periodic (almost periodic) case, sufficient conditions for existence, uniqueness, and stability of a positive periodic (almost periodic) solution are obtained.


Introduction
The Lotka-Volterra system is a rudimentary model on mathematical ecology and has been studied extensively in [6, 8-16, 19, 24-26] and the references cited therein.But in the Lotka-Volterra model, the fact that there are upper limits to the rates of increase of both prey and predator is not recognized.In [17,18], Leslie introduced a predator-prey model where the "carrying capacity" of the predator's environment is proportional to the number of prey.Leslie stresses the above fact.In the case of continuous time, these considerations lead to the following autonomous model: which is known as the Leslie-Gower predator-prey model [20].If we assume that the predator consumes the prey according to the functional response f (H), then the system (1.1) formulates as the following: 2 A semiratio-dependent predator-prey system which is the so-called semiratio-dependent predator-prey system with the functional responses.Recently, Wang et al. [23] explored the dynamics of a class of the nonautonomous semiratio-dependent predator-prey systems with the functional responses where H and P stand for the population of the prey and the predator, respectively, f (t,H) is the so-called predator functional response to prey, which describes the uptake of substrate by the microorganisms in microbial dynamics or chemical kinetics, and ∂ f (t,H)/ ∂H > 0 for H > 0, which implies that f (t,H) is monotonic function with respect to H.However, there are experiments that indicate that nonmonotonic responses occur at the microbial level: when the nutrient concentration reaches a high level, an inhibitory effect on the specific growth rate may occur.This is often seen when microorganisms are used for waste decomposition or for water purification, see Bush and Cook [3].The so-called Monod-Haldane function has been proposed and used to model the inhibitory effect at high concentrations, see Andrews [1].Collings [4] also used this response function to model mite predator-prey interactions and called it a Holling IV function.In experiments on the uptake of phenol by pure culture of Pseudomonas putida growing on phenol in continuous culture, Sokol and Howell [22] proposed a simplified Monod-Haldane function of the form and found that it fits their experimental data significantly better than the Monod-Haldane function and is simpler since it involves only two parameters.We would like to call this function a simplified Monod-Haldane or Holling IV response function.So it is very interesting to study dynamics of a class of the so-called semiratio-dependent predator-prey systems with the nonmonotonic functional responses

t) P(t) H(t) P(t).
(1.6) For the ecological sense of the system (1.6) we refer to [5,21,23,27] and the references cited therein.The plan of this paper is as follows: in Section 2, for general case, we will explore some basic problems for (1.6), such as positive invariance, permanence, and globally asymptotic stability for the system (1.6).In Section 3, for periodic case, sufficient conditions for H.-F. Huo and W.-T. Li 3 existence, uniqueness, and stability of a positive periodic solution are obtained.Section 4 is devoted to the case when all parameters are almost periodic, sufficient conditions for existence, uniqueness and stability of an almost periodic solution are also obtained.The methods used here will be comparison theorems, coincidence degree theory, and Liapunov function.

General nonautonomous case
In this section, we will address such properties as boundedness of solutions, permanence, and globally asymptotic stability of system (1.6).In the following discussion, we always assume that r i (t), i = 1,2, a i (t), i = 1,2, and b 1 (t) are all continuous and bounded above and below by positive constants. Let For a bounded continuous function g(t) on R, we use the following notations: Lemma 2.1.Both the nonnegative and positive cones of R 2 are positively invariant for system (1.6).
Proof.Note that system (1.6) is equivalent to (2. 2) The assertion of the lemma follows immediately for all t ≥ t 0 .The proof is complete.
In the remainder of this paper, for biological reasons, we only consider solutions (H(t), P(t)) with H(t 0 ) > 0 and P(t 0 ) > 0.
Definition 2.2.The solution of system (1.6) is said to be ultimately bounded if there exists B > 0 such that for every solution (H(t),P(t)) of system (1.6), there exists T > 0 such that (H(t),P(t)) ≤ B, for all t ≥ t 0 + T, where B is independent of particular solution while T may depend on the solution.
then the set Γ defined by is positively invariant with respect to system (1.6),where and ≥ 0 is sufficiently small so that m 1 > 0.
Proof.Let (H(t),P(t)) be the solution of system (1.6) through (H(t 0 ),P(t 0 )) with From the first equation in (1.6) and the positivity of the solutions of (1.6), it follows that (2.8) A standard comparison argument shows that which together with the second equation in (1.6) produces and hence H.-F. Huo and W.-T. Li 5 Similarly, the first equation of system (1.6) also yields and therefore, Moreover, by the second equation of system (1.6), we have which implies Thus, Γ is positive invariant for system (1.6), and the proof is complete.
For the predator population, we can also have some estimates.
Lemma 2.6.Assume that (2.4) holds, then Proof.Since lim sup t→∞ H(t) ≤ M 0 1 , for any sufficient small > 0, there is some t 1 > t 0 such that for t ≥ t 1 , 6 A semiratio-dependent predator-prey system Then, from the second equation of system (1.6), it follows that (2.20) Hence, by using the comparison theorem of ordinary differential equations and the arbitrariness of , we have By a similar argument, we can easily show that The proof is complete.
Lemmas 2.5 and 2.6 immediately lead to the following.
From the proofs of Lemmas 2.5 and 2.6, we can easily obtain the following ultimate boundedness of Γ with > 0 sufficiently small.Theorem 2.8.Assume that (2.4) holds, then the set Γ with > 0 defined by (2.5) is an ultimately bounded region of system (1.6).Definition 2.9.A bounded nonnegative solution (H * (t),P * (t)) of (1.6) is said to be globally asymptotically stable (or globally attractive) if for any other solution (H(t),P(t)) of (1.6) with positive initial values the following holds: (2.23) Remark 2.10.In general, if the above property holds for any two solutions with positive initial values, then we say system (1.6) is globally asymptotically stable.One can easily show that if system (1.6) has a bounded positive solution which is globally asymptotically stable, then system (1.6) is globally asymptotically stable, and vice versa.
The following lemma is from Barbalat [2], and will be employed in establishing the globally asymptotic stability of system (1.6).

Lemma 2.11. let h be a real number and let f be a nonnegative function defined on [h,+∞) such that f is integrable on [h,+∞) and is uniformly continuous on
H.-F. Huo and W.-T. Li 7 Theorem 2.12.Let (H * (t),P * (t)) be a bounded positive solution of system (1.6).Assume that (2.4) holds, and ) Calculating the right derivative of V (t) along the solution of system (1.6), we derive for t ≥ t 0 + T 1 that (2.26) From (2.24) it follows that there exists a positive constant μ > 0 such that (2.27) 8 A semiratio-dependent predator-prey system Integrating on both sides of (2.27) from t 0 + T 1 to t produces (2.28) (2.30) The proof is complete.

Periodic case
Throughout this section, we will assume that r i (t), a i (t), i = 1,2, b 1 (t), are positive ωperiod functions, that is, system (1.6) is ω-period system and will study the existence and stability of a positive periodic solution of (1.6).
Lemma 3.1 (Brouwer fixed point theorem).Let σ be a continuous operator that maps a closed, bounded, convex subset Ω ⊂ R n into itself.Then Ω contains at least one fixed point of the operator σ, that is, there exists an x * ∈ Ω such that σ(x * ) = x * .
Next, we will employ an alternative approach, that is, a continuation theorem in coincidence degree theorem, to establish some different criteria for the same problem.To this end, we need some preparation as follows.
Let X and Z be two Banach spaces.Consider an operator equation where L : DomL ∩ X → Z is a linear operator and λ is a parameter.Let P and Q denote two projectors such that In the sequel, we will use the following result of Gaines and Mawhin [7, page 40].
Lemma 3.3.Let X and Z be two Banach spaces and let L be a Fredholm mapping of index zero.Assume that N : Then the equation Lx = Nx has at least one solution in Ω ∩ DomL.
Recall that a linear mapping L : DomL ∩ X → Z with KerL = L −1 (0) and ImL = L(Dom L) will be called a Fredholm mapping if the following two conditions hold: (i) KerL has a finite dimension; (ii) ImL is closed and has a finite codimension.Recall also that the codimension of ImL is the dimension of Z/ Im L, that is, the dimension of the cokernel coker L of L.
When L is a Fredholm mapping, its index is the integer IndL = dimkerL − codim Im L.
We will say that a mapping N is L-compact on Ω if the mapping QN : Ω → Z is continuous, QN(Ω) is bounded, and K p (I − Q)N : Ω → X is compact, that is, it is continuous and K p (I − Q)N(Ω) is relatively compact, where K p : ImL → DomL ∩ Ker P is a inverse of the restriction L p of L to DomL ∩ Ker P, so that LK p = I and K p L = I − P.
For convenience, we will introduce the notation where u is a periodic continuous function with period ω.
Theorem 3.4.If the system of algebraic equations has a unique solution (v * 1 ,v * 2 ) T ∈ intR 2 + with v * i > 0, i = 1,2, then system (1.6) has at least one positive ω-periodic solution. Proof.Since the solution of system (1.6) remains positive for t ≥ 0, we can let and derive that In order to use Lemma 2.1 to system (1.6), we take and denote Then X and Z are Banach spaces when they are endowed with the norms • .Set Hence, L is a Fredholm mapping of index zero.Furthermore, H.-F. Huo and W.-T. Li 11 the generalized inverse (to L) K p : ImL → Ker P ∩ domL has the form (3.17) 12 A semiratio-dependent predator-prey system Suppose that x(t) = (x 1 ,x 2 ) ∈ X is a solution of system (3.17) for a certain λ ∈ (0,1).By integrating (3.17) over the interval [0,ω], we obtain Hence From (3.17), (3.19), and (3.20), we obtain ( Note that (x 1 (t),x 2 (t)) T ∈ X, then there exists ξ i ,η i ∈ [0,ω], i = 1,2 such that By (3.19) and (3.22), we obtain ) In addition, from (3.22) and system (3.17),we obtain H.-F. Huo and W.-T. Li 13 Then x 1 (t) ≤ max ln ( In view of (3.20) and (3.22), we have that is , we obtain that and so (3.34)Clearly, M i , i = 1,2, are independent of λ.Under the assumption in Theorem 3.4, it is easy to show that the system of algebraic equations 14 A semiratio-dependent predator-prey system has a unique solution and define It is clear that Ω satisfies the condition (a) of Lemma 2.1.When x is a constant vector in R 2 with x = M. Then By now we know that Ω verifies all the requirements of Lemma 3.3 and then system (3.11) has at least one ω-periodic solution.By the medium of (3.10), we derive that system (1.6) has at least one positive ω-periodic solution.The proof is complete.

Almost periodic case
The assumption of almost periodicity of system (1.6) is a way of incorporating the time dependent variability of the environment, especially when the various components of the environment are periodic with not necessary commensurate period.Mathematically, system (1.6) will denote a generation of an autonomous and periodic system.Therefore, throughout this section, in addition to the assumptions in Section 2, we further assume that r i (t), a i (t is positively invariant and ultimately bounded region of system (4.2),where M i , m i , i = 1,2, are defined in (2.6).
In order to prove the main result of this section, we will first introduce a useful lemma.Consider the ordinary differential equation where D is an open set in R n , f (t,x) is almost periodic in t uniformly with respect to x ∈ D.
To discuss the existence of an almost periodic solution of (4.4), we consider the product system of (4.4) , where μ > 0 is a constant.Moreover, suppose that system (4.4) has a solution that remains in a compact set S ⊂ D for all t ≥ t 0 ≥ 0. Then system (4.4) has an unique almost periodic solution in S, which is uniformly asymptotically stable in D.

. 16 )
Clearly, QN and K p (I − Q)N are continuous and, moreover,QN(Ω), K p (I − Q)N(Ω) are relatively compact for any open bounded set Ω ⊂ X.Hence, N is L-compact on Ω, here Ω is any open bounded set in X.Now we reach the position to search for an appropriate open bounded subset Ω for the application of Lemma 2.1.Corresponding to equation Lx = λNx, λ ∈ (0,1), we have