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By using a continuation theorem based on coincidence degree theory, we establish some easily verifiable criteria for the existence of positive periodic solutions for neutral delay ratio-dependent predator-prey model with Holling-Tanner functional response

The dynamic relationship between the predator and the prey has long been and will continue to be one of the dominant themes in population dynamics due to its universal existence and importance in nature [

Nowadays attention have been paid by many authors to Holling-Tanner predator-prey model (see [

Recently, there is a growing explicit biological and physiological evidence [

However, time delays of one type or another have been incorporated into biological models by many researchers; we refer to the monographs of Cushing [

Recently, Saha and Chakrabarti [

In addition, based on the investigation on laboratory populations of Daphnia magna, Smith [

In this paper, motivated by the above work, we will consider the following neutral delay ratio-dependent predator-prey model with Holling-Tanner functional response:

As pointed out by Freedman and Wu [

For convenience, we will use the following notations:

In this paper, we always make the following assumptions for system (

Our aim in this paper is, by using the coincidence degree theory developed by Gaines and Mawhin [

In this section, we will study the existence of at least one positive periodic solution of system (

Let

The mapping

If

The mapping

Since

Let

for each

for each

We are now in a position to state and prove our main result.

Assume that (H_{1})–(H_{4}) hold. Then system (

Consider the following system:

Take

In order to apply Lemma

Corresponding to the operator equation _{2}), (_{1}), and (H_{2}) that
_{2}), yields
_{3}), imply that, for any _{4}), we obtain
_{4}), implies that
_{3}), implies that
_{3}) that
_{4}), the algebraic equations

We now take

Taking

From the proof of Theorem

Assume that (H_{1}), (H_{4}) hold, and

Next consider the following neutral ratio-dependent predator-prey system with state-dependent delays:

Assume that (H_{1})–(H_{4}) hold. Then system (

The proof is similar to that of Theorem

In this paper, we have discussed the combined effects of periodicity of the ecological and environmental parameters and time delays due to the negative feedback of the predator density and gestations on the dynamics of a neutral delay ratio-dependent predator-prey model. By using Gaines and Mawhin's continuation theorem of coincidence degree theory, we have established sufficient conditions for the existence of positive periodic solutions to a neutral delay ratio-dependent predator-prey model with Holling-Tanner functional response. By Theorem

We note that

From the results in this paper, we can find that the neutral term effects are quite significant.

This work was supported by the Natural Science Foundation of China (no. 11001157), Tianyuan Mathematics Fund of China (no. 10826080) and the Youth Science Foundation of Shanxi Province (no. 2009021001-1, no. 2010021001-1).