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We investigated the dynamics of a diffusive ratio-dependent Holling-Tanner predator-prey model with Smith growth subject to zero-flux boundary condition. Some qualitative properties, including the dissipation, persistence, and local and global stability of positive constant solution, are discussed. Moreover, we give the refined a priori estimates of positive solutions and derive some results for the existence and nonexistence of nonconstant positive steady state.

In order to precisely describe the real ecological interactions between species such as mite and spider mite, lynx and hare, sparrow and sparrow hawk, and some other species [

The dynamics of model (

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

For model (

On the other hand, an implicit assumption contained in the logistic equation

Based on the above discussions, in this paper, we rigorously consider the radio-dependent Holling-Tanner model with Smith growth that takes the form of

Also considering the spatial dispersal and environmental heterogeneity, in this paper, we study the following generalized reaction-diffusion system for model (

The stationary problem of model (

Simple computation shows that if

The rest of the paper is organized as follows. In Section

In this section, the dissipation and persistence properties are studied for solution of model (

All the solutions of model (

The nonnegativity of the solution of model (

Note that

Let

As a result, for any

Let

The spatial model (

If

Let

Let

Due to

Similarly, by the second equation of model (

Let

In this subsection, we shall analyze the asymptotical stability of the positive constant solution

Let

Let

Let

Let

then

where

Assume that

Define

For each

In view of (

In the following, we prove that there exists

Let

By the Routh-Hurwitz criterion, it follows that the two roots

Let

This subsection is devoted to the global stability of the constant solution

Assume that the following hold:

In order to give the proof, we need to construct a Lyapunov function. Define

Set

By virtue of Theorems

As a result, we have

In this section, we will deduce a priori estimates of positive upper and lower bounds for positive solution of model (

In order to obtain the desired bound, we recall the following two lemmas which are due to Lin et al. [

Assume that

Let

Assume that

If

Assume that

If

For convenience, let us denote the constants

For any positive solution

Assume that

Since

Let

Let

By Lemma

Since

Note that

Let

Let

By the

In view of

In this subsection, we shall discuss the existence of the positive nonconstant solution of model (

Unless otherwise specified, in this subsection, we always require that

Let

Then

If

On the other hand, using the decomposition (

Let

Assume that, for all

To compute

If

Assume that

By Theorem

By virtue of Theorems

Set

Since

Clearly,

On the contrary, by the choice of

From (

The authors would like to thank the anonymous referee for the very helpful suggestions and comments which led to improvements of our original paper. And this work is supported by the Cooperative Project of Yulin City (2011).