Oscillatory dynamics in a discrete predator-prey model with distributed delays

This work aims to discuss a predator-prey system with distributed delay. Various conditions are presented to ensure the existence and global asymptotic stability of positive periodic solution of the involved model. The method is based on coincidence degree theory and the idea of Lyapunov function. At last, simulation results are presented to show the correctness of theoretical findings.


Introduction
It is well known that the qualitative analysis of predator-prey models is an interesting mathematical problem and has received great attention from both theoretical and mathematical biologists [1][2][3][4][5]. In particular, the periodic solutions are of great interest. During the past decades, a great deal of excellent results have been reported for a lot of different continuous or impulsive predator-prey models. For example, Zhang and Hou [6] investigated the four positive periodic solutions of a ratio-dependent predator-prey system with multiple exploited (or harvesting) terms. Liu and Yan [7] considered positive periodic solutions for a neutral delay ratiodependent predator-prey model with a Holling type II functional response. Liu [8] dealt with the impulsive periodic oscillation of a predator-prey model with Hassell-Varley-Holling functional response. For more related work, one can see [9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27]. Dunkel [28] pointed out that feedback control item in predator-prey models depends on the population number for certain time past and also depends on the average of the population number for a period of time past. In particular, time delay often occur in predator-prey models due to the impact of all the past life history of the predators and preys on their present birth rates. In many cases, the time delay will extend over the entire past due to the intra-species and inter-species competition. Then there is a distribution of delays over a period of time, thus the distributed delays should be incorporated in predator-prey models.
The functional response plays a key role in characterizing the interaction of predators and preys. Based on the experiments of different kinds of species, Holling [29] proposed three a1111111111 a1111111111 a1111111111 a1111111111 a1111111111 types of functional responses: (I) f 1 (u) = au, (II) f 2 ðuÞ ¼ au cþu ; (III) f 3 ðuÞ ¼ au 2 cþu 2 ; where u(t) represents the prey density at time t, c > 0 is the half-saturation constant, a > 0 denotes the search rate of the predator. Holling type II functional response is most typical of predators that specialized on one or a few prey [29][30][31][32][33]. So in this paper, Holling type II functional response is introduced in model (1).
Motivated by the viewpoint, we proposed the following predator-prey model with Holling II functional response and distributed delays where x i (t)(i = 1, 2) stands for the prey and predator density at time t, r 1 (t) denotes the intrinsic growth rate of prey at time t and r 2 (t) denotes the death rate of predator at time t, m > 0 stands for the half-saturation constant, k i : (−1, 0] ! (0, +1)(i = 1, 2, 3, 4) is continuous function such that R t À 1 k i ðsÞds ¼ 1; For the biological meaning of model (1), one can see [34].
As pointed out in [35][36][37][38][39][40][41][42], discrete time models are more better to describe the dynamical behaviors than continuous ones since the populations have non-overlapping generations. What's more, discrete-time systems can provide convenience for numerical simulations. Thus it is interesting to investigate discrete-time systems. The principle aim of this paper is to propose a discrete version of system (1) and analyze the effect of the periodicity of the ecological and environmental parameters on the dynamics of discrete time predator-prey model.

Existence of positive periodic solutions
First we given two notations: where ℓ(k) is a α-periodic sequence of real numbers defined for k 2 Z. Let X, Y be normed vector spaces, L: DomL � X ! Y be a linear mapping, N: X ! Y be a continuous mapping. The mapping L will be called a Fredholm mapping of index zero if dimKerL = codimImL < +1 and ImL be closed in Y. If L is a Fredholm mapping of index zero and there exist continuous projectors P: X ! X and Q: Y ! Y such that ImP = KerL, ImL = KerQ = Im(I − Q), it follows that LjDomL \ KerP: (I − P)X ! ImL is invertible. We denote the inverse of this map by K P . If O is an open bounded subset of X, the mapping N will be called L-compact on � O if QNð � OÞ is bounded and K P ðI À QÞN : � O ! X is compact. Since ImQ is isomorphic to KerL, there exists a isomorphism J: ImQ ! KerL.

Global asymptotic stability
Let the delays be zero, then (4) becomes Theorem 2 Assume that (H1) and (H2) are satisfied and furthermore suppose that there exist positive constants ν, σ 1 and σ 2 such that Then the positive ω-periodic solution of system (37) is globally asymptotically stable. Proof In view of Theorem 1, there exists a positive periodic solution fx � 1 ðkÞ; x � 2 ðkÞg of system (37). Make the change of variable It follows from (37) that where jjg i jj jjujj ði ¼ 1; 2Þ converges to zero as ||u|| ! 0. Define a function V by where σ 1 > 0 and σ 2 > 0 are given by (44) and (45) respectively. Calculating the difference of V along the solution of system (40) and (41), we have where It follows from the condition (38) that 9 � > 0 such that, if k is sufficiently large and ||u|| < �, then In view of Freedman [45], we can see that the trivial solutions of (40) and (41) is uniformly asymptotically stable and so is the solution {(x � (k), y � (k)) T } of (37). The proof is complete. Remark 1 In [34], Ye et al. investigated the periodic solution of a continuous predator-prey system with Holling type II functional response and infinite delays by applying continuation theorem in coincidence degree theory and some priori estimates on solutions, moreover, this paper does not involve the global asymptotic stability. In this paper, we study the existence of periodic solution of discrete predator-prey model with distributed delays by applying continuation theorem in coincidence degree theory and analyze the global asymptotic stability of periodic solution by Lyapunov function. Form this viewpoint, the results of this article supplement the previous studies of Ye et al. [18].

Numerical example
Example 1 Consider the model as follows:

Conclusions
Based on the previous works and some biological meanings of predators and preys, we propose a new discrete delayed predator-prey system. By using the continuation theorem in coincidence degree theory, we present a set of sufficient conditions to ensure to ensure the existence of positive periodic solution of the discrete delayed predator-prey system. In addition, we also discussed the global asymptotic stability of positive periodic solution for the considered system. The obtained theoretical findings have important significance in biological ecology. Considering the effect of random factor, it is meaningful for us to deal with the dynamics of stochastic predator-prey system. This topic will be our future research direction.