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This paper is concerned with a discrete predator-prey model with Holling II functional response and delays. Applying Gaines and Mawhin’s continuation theorem of coincidence degree theory and the method of Lyapunov function, we obtain some sufficient conditions for the existence global asymptotic stability of positive periodic solutions of the model.

In recent years, numerous studies have been carried out on predator-prey interactions using Lotka-Volterra type functional response [

As pointed out in [

In order to obtain our main results, we assume that

the following inequalities are satisfied:

The principle aim of this paper is to discuss the effect of the periodicity of the ecological and environmental parameters on the dynamics of discrete time predator-prey model with Holling II functional response and distributed delays.

The paper is organized as follows. In Section

Throughout the paper, we always use the notations below:

Let

Let

for each

Let

The proofs of Lemma

Define

For

Let

Suppose that (H1), (H2), and (H3)

Let

Now we are at the point to search for an appropriate open, bounded subset

Let

Let the delays be zero; then (

Assume that (H1) and (H2) are satisfied and furthermore suppose that there exist positive constants

Since the delays in system (

Define a function

In this section, we present some numerical results of system (

The time series graph of

The time series graph of

In this paper, a discrete predator-prey model with Holling II functional response and delays is investigated. With the aid of Gaines and Mawhin’s continuation theorem of coincidence degree theory and the method of Lyapunov function, we establish some sufficient conditions for the existence and global asymptotic stability of positive periodic solutions of the model. Since the time scales can unify the continuous and discrete situations, it is meaningful to investigate the predator-prey model with Holling II functional response and delays on time scales. We leave it for future work.

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work was supported by State Science and Technology Support Program (Grant no. 2013BAD15B02), General Scientific Research Projects in Hunan Province, Department of Education, China (Grant no. 14C0542), Graduate Student Research Innovation Project of Hunan Province (Grant no. CX2014B306), and Guangxi Experiment Centre of Science and Technology (Grant no. LGZXKF201112).