Multiple Periodic Solutions of a Ratio-Dependent Predator-Prey Discrete Model

It is known that one of important factors impacted on a predator-prey system is the functional response. Holling proposed three types of functional response functions, namely, Holling I, Holling II, and Holling III, which are all monotonously nondescending 1 . But for some predator-prey systems, when the prey density reaches a high level, the growth of predator may be inhibited; that is, to say, the predator’s functional response is not monotonously increasing. In order to describe such kind of biological phenomena, Andrews proposed the so-called Holling IV functional response function 2


Introduction
It is known that one of important factors impacted on a predator-prey system is the functional response.Holling proposed three types of functional response functions, namely, Holling I, Holling II, and Holling III, which are all monotonously nondescending 1 .But for some predator-prey systems, when the prey density reaches a high level, the growth of predator may be inhibited; that is, to say, the predator's functional response is not monotonously increasing.In order to describe such kind of biological phenomena, Andrews proposed the so-called Holling IV functional response function 2 where x t and y t represent predator and prey densities, respectively.In 1.2 , the functional response function g IV x cx/ m 2 x 2 is a special case of Holling IV functional response.The functional response functions mentioned previously only depend on the prey x.But some biologists have argued that the functional response should be ratio dependent or semi-ratio dependent in many situations.Based on biological and physiological evidences, Arditi and Ginzburg first proposed the ratio-dependent predator-prey model 12 where the functional response function g r x, y cx/y / m x/y is ratio dependent.Many researchers have putted up a great lot of works on the ratio-dependent or semi-ratiodependent predator-prey system 13-19 .Recently, some researchers incorporated the ratio-dependent theory and the inhibitory effect on the specific growth rate into the predator-prey model 3, 7, 11, 15 .Ding et al. considered a semi-ratio-dependent predator-prey system with nonmonotonic functional response and time delay 11 ; they obtained some sufficient conditions for the existence and global stability of a positive periodic solution to this system.Hu and Xia considered a functional response function 7, 15 : 1.4 With the functional response function, Xia and Han proposed the following periodic ratiodependent model with nonmonotone functional response 15 : where a t , b t , c t , d t , and h t are all positive periodic continuous functions with period ω > 0, m is a positive real constant, and K s : R → R is a delay kernel function.Based on Mawhins coincidence degree, they obtained some sufficient conditions for the existence of two positive periodic solutions of the ratio-dependent model 1.5 .
It is well known that discrete population models are more appropriate than the continuous models when the populations do not overlap among generations.Therefore, many scholars have studied some discrete population models 3, 4, 14, 16-19 .For example, Lu and Wang considered the following discrete semi-ratio-dependent predator-prey system with Holling type IV functional response and time delay 3 : They proved that the system 1.6 is permanent and globally attractive under some appropriate conditions.Furthermore, they also obtained some sufficient conditions which guarantee the existence and global attractivity of positive periodic solution.
Motivated by the mentioned previously, this paper is to investigate the existence of multiple periodic solutions of the following discrete ratio-dependent model with nonmonotone functional response: for n ∈ Z 0 , where a, d : Z 0 → R, b, c, h : Z 0 → R , and τ : Z 0 → Z 0 are all ω-periodic sequences, ω is a positive integer, m is a positive real constant, and where Z, Z 0 , Z , R, R 0 , and R denote the sets of all integers, nonnegative integers, positive integers, real numbers, nonnegative real numbers, and positive real numbers, respectively.The model 1.7 is created from the continuous-time system 1.5 by employing the semidiscretization technique.

Preliminaries
For convenience, we will use the following notations in the discussion: where f is a ω-periodic sequence of real numbers defined for k ∈ Z.
In the system 1.7 , the time delay kernel sequence K l satisfies then G l is uniformly convergent with respect to l ∈ I ω , and it satisfies ω−1 l 0 G l 1.
Lemma 2.1.x * n , y * n is a positive ω-periodic solution of system 1.7 if and only if u * 1 n , u * 2 n ln x * n /y * n , ln y * n is a ω-periodic solution of the following system 2.3 : where a n , b n , c n , d n , h n , and τ n are the same as those in model 1.7 .
Proof.Let u 1 n , u 2 n ln x n /y n , ln y n ; then the system 1.7 can be rewritten as

2.4
Therefore, x * n , y * n is a positive ω-periodic solution of system 1.7 if and only if

2.6
Because G l ∞ k 0 K l kω is uniformly convergent with respect to l ∈ I ω , so we have

2.7
Therefore, u * 1 n , u * 2 n is a ω-periodic solution of the system 2.3 if and only if it is a ωperiodic solution of the system 2.4 .This completes the proof.From 1.8 , the initial conditions associated with 2.3 are of the form where Throughout this paper, we assume that Under the assumption H1 , there exist the following six positive numbers: .

6
Discrete Dynamics in Nature and Society Obviously, In this paper, we adopt coincidence degree theory to prove the existence of multiple positive periodic solutions of 1.7 .We first summarize some concepts and results from the book by Gaines and Mawhin 20 .Let X and Y be normed vector spaces.Define In our proof of the existence, we also need the following two lemmas.

Existence of Two Positive Periodic Solutions
We are ready to state and prove our main theorem.
Proof.It is easy to see that if the system 2.3 has a ω-periodic solution u * 1 n , u * 2 n , then 2 n is a positive ω-periodic solution to the system 1.7 .Therefore, to complete the proof, it suffices to show that the system 2.3 has at least two ω-periodic solutions.
We take and define the norm of X and Y u max Then X and Y are Banach spaces when they are endowed with the previous norm • .
For any u u 1 , u 2 ∈ X, because of its periodicity, it is easy to verify that and dim ker L codim Im L 2. Therefore, L is a Fredholm mapping of index zero.
Define two mappings P and Q as

3.5
It is easy to prove that P and Q are two projectors such that Im P ker L and Im L ker Q Im I − Q .Furthermore, by a simple computation, we find that the inverse K p of L p : Im L → Dom L ∩ ker P has the form Corresponding to the operator equation 2.11 , we get the following system: where λ ∈ 0, 1 .Suppose that u 1 n , u 2 n ∈ X is an arbitrary solution of system 3.8 for a constant λ ∈ 0, 1 .Summing 3.8 over I ω , we obtain From system 3.8 , we have

3.11
By using 3.9 and 3.10 , we obtain Obviously, there exist ξ i ,η i ∈ I ω , such that 3.14 From 3.10 , it follows that By using Lemma 2.3 and 3.12 , we obtain 3.17 In particular, we have The assumption H1 implies that h exp |a| |d| a d ω > 2md.So we have ln l − < u 1 ξ 1 < ln l .

3.20
From 3.10 , we also have

3.22
By using Lemma 2.3 and 3.12 again, we have

3.23
In particular, we have Therefore,

3.26
From 3.12 and 3.20 , we have

3.27
Similarly, from 3.12 and 3.26 , we have

3.35
Notice that Under the conditions H1 and H2 , we can obtain two distinct solutions of QN u 1 , u 2 0

3.37
After choosing a constant C > 0 such that we can define two bounded open subsets of X as follows:

3.39
It follows from 2.10 and 3.38 that u − ∈ Ω 1 and u ∈ Ω 2 .Because of ln v − < ln v , it is easy to see that Ω 1 ∩Ω 2 is empty, and Ω i satisfies the condition a in Lemma 2.2 for i 1, 2.Moreover, QNu / 0 for u ∈ ∂Ω i ker L ∂Ω i R 2 .This shows that the condition b in Lemma 2.2 is satisfied.
Because Im Q ker L, we can take the isomorphic J as the identity mapping, then we have

3.41
Similarly, we can obtain that

3.42
So the condition c in Lemma 2.2 is also satisfied.By now we know that Ω i i 1, 2 satisfies all the requirements of Lemma 2.2.Hence the system 2.3 has at least two ω-periodic solutions.This completes the proof.

An Example
In the system 1.Therefore, the conditions H1 and H2 are satisfied.From Theorem 3.1, the system 1.7 has at least two 3-periodic solutions.

Conclusion
In 3 , Lu and Wang investigated a discrete time semi-ratio-dependent predator-prey system 1.6 with Holling type IV functional response and time delay.They established sufficient conditions which guarantee the existence and global attractivity of a positive periodic solution of the system.In this paper, a ratio-dependent predator-prey discrete-time model with discrete distributed delays and nonmonotone functional response is investigated.By using the continuation theorem of Mawhins coincidence degree theory, we prove that the system 1.7 has at least two positive periodic solutions under conditions H1 and H2 .As 3 , we would like to know the local stability of the two positive periodic solutions of system 1.7 , which is our future work.

Lemma 2 . 2
continuation theorem 20 .Let L be a Fredholm mapping of index zero and let N be L-compact on Ω. Suppose that a for each λ ∈ 0, 1 , x ∈ ∂Ω ∩ Dom L, Lx / λNx; b for each x ∈ ∂Ω ∩ ker L, QNx / 0; c deg JQN, Ω ∩ ker L, 0 / 0. Then the operator equation Lx Nx has at least one solution in Dom L ∩ Ω. Lemma 2.3 see 14 .If u : Z → R is a ω-periodic sequence, then for any fixed n 1 ,n 2 ∈ I ω , one has

7 and
K p I − Q N are continuous by the Lebesgues convergence theorem.Moreover, by Arzela Ascolis theorem, QN Ω and K p I − Q N Ω are relatively compact for the open bounded set Ω ⊂ X.Therefore, N is L-compact on Ω for the open bounded set Ω ⊂ X.