Multiple Periodic Solutions of Delayed Predator-Prey Systems with Type IV Functional Responses on Time Scales

With the help of a continuation theorem based on Gaines and Mawhin's coincidence degree, easily verifiable criteria are established for the existence of multiple positive periodic solutions of delayed predator-prey systems with type IV functional responses on time scales. Our results not only unify the existing ones but also widen the range of applications.


Introduction
As was pointed out by Berryman 1 , the dynamic relationship between predators and their prey 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.At first sight, these problems may appear to be simple mathematically.However, in fact, they are often very challenging and complicated.Also, Zhen and Ma 2 argued that the environmental fluctuation is important in an ecosystem, and more realistic models require the inclusion of the effect of environmental changes, especially environmental parameters which are time dependent and periodically changing e.g., seasonal changes, food supplies, etc. .Hence, just as pointed out by Freedman and Wu 3 and Kuang 4 , it would be of great interest and importance to study the existence of periodic solutions for systems with periodic delay.Much progress has been made in this direction see, e.g., 5-8 and the references cited therein .
In 1959, in order to describe behavior of different kinds of species, Holling 9 proposed three types of functional response functions.However, some authors 10 have also described a type IV functional response that is humped and that declines at high prey densities.This decline may occur due to prey group defense or prey toxicity.Recently, Chen 11 has studied the following periodic predator-prey system with a type IV functional response: where c, σ, a j , b j , and τ j j 1, 2 are continuous ω-periodic functions with c t ≥ 0, σ t ≥ 0, a j t ≥ 0, and τ j ≥ 0, ω 0 c t dt > 0, and ω 0 b j t dt > 0, n and a are positive constants.The growth functions b j may change sign, since the environment fluctuates randomly.Under bad conditions, b j may be negative.
Considering that discrete time models governed by difference equations are more appropriate than continuous ones when the populations have nonoverlapping generations, Zhang et al. 12 studied the following discrete time predator-prey system: On the other hand, recently, Bohner et al. 13 pointed out that it is unnecessary to explore the existence of periodic solutions of some continuous and discrete population models in separate ways.One can unify such studies in the sense of dynamic equation on general time scales.The theory of calculus on time scales, which has recently received a lot of attention, was initiated by Hilger in his Ph.D. Thesis in 1988 14 in order to unify continuous and discrete analysis.Although there has been much research activity concerning the oscillation nonoscillation of solutions and periodic solution of differential equation on time scales or measure chains see, e.g., 15-29 , there are few results dealing with multiple periodic solutions of predator-prey systems with time delay.Motivated by the above work, we consider the following system on time scales T: where, for i 1, 2, b i : T → R, c, a i , τ i , τ : T → R are ω-periodic functions.n and a are positive constants.
The main purpose of this paper is to derive a set of easily verifiable sufficient conditions for the existence of multiple positive periodic solutions of 1.3 .The method used here will be the coincidence degree theory developed by Gaines and Mawhin 30 .
In 1.3 , set z 1 t exp{x t }, z 2 t exp{y t }.If T R, then 1.3 reduces to 1.1 .Also, if T Z, then 1.3 becomes 1.2 .Thus, our results also show that it is unnecessary to explore the existence of periodic solutions of continuous and discrete population models in separate ways.One can unify such studies in the sense of dynamic equations on time scales.
The paper is arranged as follows.In Section 2, we present some preliminary results such as the calculus on time scales and the continuation theorem in coincidence degree theory.In Section 3, we prove our main result.

Preliminaries
In this section, we give a short introduction to the time scales calculus and recall the continuation theorem from coincidence degree theory.
First, let us present some foundational definitions and results from the calculus on time scales, for proofs and further explanation and results, we refer to the paper by Hilger 14 .
Definition 2.1.A time scale is an arbitrary nonempty closed subset T of the real numbers R. The set T inherits the standard topology of R. Definition 2.2.For t ∈ T, one defines the forward jump operator σ : T → T by while the backward jump operator ρ : T → T is defined by ρ t : sup{s ∈ T : s < t}.

2.2
In this definition we put inf ∅ sup T i.e., σ t t if T has a maximum t and sup ∅ inf T i.e., ρ t t if T has a minimum t , where ∅ denotes the empty set.If σ t > t, we say that t is right-scattered, while if ρ t < t we say that t is left-scattered.Points that are right-scattered and left-scattered at the same time are called isolated.Also, if t < sup T and σ t t, then t is called right-dense, and if t > inf T and ρ t t, then t is called left-dense.Points that are right-dense and left-dense at the same time are called dense.Definition 2.3.A function f : T → R is said to be rd-continuous if it is continuous at rightdense points in T and its left-sided limits exist finite at left-dense points in T. The set of rd-continuous functions is denoted by Definition 2.4.Suppose f : T → R is a function, and let t ∈ T. Then one defines f Δ t , the delta-derivative of f at t, to be the number provided it exists with the property that, given any ε > 0, there is a neighborhood U of t i.e., U t − δ, t δ ∩ T for some δ > 0 such that Thus, f is said to be delta-differentiable if its delta-derivative exists.The set of functions f : T → R that are delta-differentiable and whose delta-derivative are rd-continuous functions is denoted by For convenience, one now introduces some notations to be used throughout this paper.Let where g ∈ C rd T is an ω-periodic real function.
In order to obtain the existence of positive periodic solutions of 1.3 , for the reader's convenience, we will summarize in the following a few concepts and results from 30 that will be basic for this paper.
Let X, Z be normed vector spaces, L : Dom L ⊂ X → Z a linear mapping, and N : X → Z a continuous mapping.The mapping L will be called a Fredholm mapping of index zero if dim Ker L Codim Im L < ∞ and Im L is closed in Z.If L is a Fredholm mapping of index zero, there exist continuous projectors P : X → X and We denote the inverse of that map by K P .Let Ω be an open bounded subset of X; the mapping N will be called L-compact on Ω if QN Ω is bounded and Lemma 2.9 continuation theorem .Let L be a Fredholm mapping of index zero, and let N be L-compact on Ω. Suppose a for each λ ∈ 0, 1 , every solution x of Lx λNx is such that x / ∈ ∂Ω; b QNx / 0 for each x ∈ ∂Ω ∩ Ker L and deg{JQN, Ω ∩ Ker L, 0} / 0.

2.6
Then the equation Lx Nx has at least one solution lying in Dom L ∩ Ω.
Now, we give a lemma which will be useful in our following proof.The proofs of the lemmas can be found in 13 .Lemma 2.10.Let t 1 , t 2 ∈ I ω and t ∈ T.

Existence of Periodic Solutions
The goal of this section is to establish sufficient conditions on the existence of periodic solution for system 1.3 , where, for i 1, 2, b i , c, a i , τ i , τ are rd-continuous functions.Firstly, we always assume that For further convenience, we define the following six positive numbers:

3.2
We now come to the main result of this paper.
Theorem 3.1.In addition to H 1 , assume further that holds and the system 1.3 has at least two ω-periodic solutions.
Proof.In order to apply Lemma 2.9 continuation theorem to 1. for any x, y ∈ X or Z .Then X, Z are both Banach spaces when they are endowed with the above norm • .For x y ∈ X, we define

3.4
Then, it follows that and P, Q are continuous projectors such that Therefore, L is a Fredholm mapping of index zero.Furthermore, the generalized inverse to L K P : Im L → Ker P ∩ Dom L reads

3.8
Obviously, QN and for any open bounded set Ω ⊂ Z by using the Arzela-Ascoli theorem.Moreover, QN Ω is clearly bounded.Thus, N is L-compact on Ω with any open bounded set Ω ⊂ Z. Now we reach the position to search for an appropriate open bounded subset Ω for the application of the continuation theorem Lemma 2.9 .Corresponding to the operator equation Lx λNx, Ly λNy, λ ∈ 0, 1 , we have 3.9 Suppose that x t , y t T ∈ X is a solution of system 3.9 for a certain λ ∈ 0, 1 .Integrating 3.2 over the set I ω , we obtain Δt.

3.11
It follows from 3.9 -3.11 that

3.13
Note that x t , y t T ∈ X, then there exist ξ i , η i ∈ I ω , i 1, 2, such that

3.14
Then, By 3.11 and 3.14 , we have

3.16
According to 3.12 , 3.16 , and Lemma 2.10, we derive 3.17 In particular, we have

3.21
From 3.12 and 3.20 and Lemma 2.10, one has This, combined with 3.10 and 3.14 , gives

3.24
It follows from 3.23 that This, together with 3.13 and Lemma 2.10, yields Moreover, because of H 2 , it follows from 3.24 that

3.27
This, together with 3.13 and Lemma 2.10 again, yields

3.28
It

3.35
Moreover, just the same as the authors in 11, 12 point out, Theorem 3.1 will remain valid if some or all terms are replaced by terms with discrete time delays, distributed delays finite or infinite , state-dependent delays, or deviating arguments.That is to say that time delays of any type or the deviating arguments can have no effect on the existence of positive periodic solutions.
follows from 3.26 and 3.28 that ± , ln u ± , M 1 , and M are independent of λ.Now, let us consider QNz 1 with z 1x, y T ∈ R. Note that By virtue of H 1 and H 2 , we can show that QNz 1 0 has two distinct solutions z 11 Then both Ω 1 and Ω 2 are bounded open subsets of X.It follows from 3.2 and 3.31 that z 11 ∈ Ω 1 and z 12 ∈ Ω 2 .With the help of 3.2 , 3.20 -3.22 , 3.29 , and 3.31 , it is easy to see that Ω 1 ∩ Ω 2 ∅, and Ω i satisfy the requirement a in Lemma 2.9 for i 1, 2.Moreover, QNz / 0 for z ∈ ∂Ω i ∩ Ker L. A direct calculation shows that So far, we have proved that Ω i verifies all the requirements in Lemma 2.9.Hence 1.3 has at least two ω-periodic solutions.This completes the proof.Remark 3.2.In 1.3 , set z 1 t exp{x t }, z 2 t exp{y t }.When T R, then 1.3 reduces to 1.1 .Also, if T Z, then 1.3 becomes 1.2 .Hence, our result unifies the main results of 11, Theorem 2.2 and 12, Theorem 2.1 .Moreover, our result will also be useful whenT hZ {hk | k ∈ Z},where h > 0 and h / 1; however, 11, Theorem 2.2 and 12, Theorem 2.1 are not applicable.In this case, we have -periodic sequence of positive real numbers, g with ω > 1 and t ∈ T.Remark 3.3.According to the above proof, we also can obtain that Theorem 3.1 is true for the following general system:x Δ t b 1 t − a 1 t exp x t − τ 1 t − c t exp y t − τ t exp{2x t − τ 2 t }/nexp{x t − τ 2 t } a , y Δ t −b 2 t a 2 t exp{x t − τ 3 t } exp{2x t − τ 4 t }/n exp{x t − τ 4 t } a .
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