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The reaction-diffusion Holling-Tanner prey-predator model considering the Allee effect on predator, under zero-flux boundary conditions, is discussed. Some properties of the solutions, such as dissipation and persistence, are obtained. Local and global stability of the positive equilibrium and Turing instability are studied. With the help of the numerical simulations, the rich Turing patterns, including holes, stripes, and spots patterns, are obtained.

The Holling-Tanner prey-predator model is an important and interesting predator-prey model in both biological and mathematical sense [

Set

The dynamics of the reaction-diffusion Holling-Tanner prey-predator model has proven quite interesting and received intensive study by both ecologists and mathematicians in many articles, see, for example, [

On the other hand, in population dynamics, Allee effect [

Based on the previous discussion, in the present paper we adopt the reaction-diffusion Holling-Tanner prey-predator model with Allee effect on predator.

If we assume that the predator population is subject to an Allee effect, taking into account zero-flux boundary conditions, model (

The corresponding kinetic equation to model (

The plan of the paper is as follows. Section

In this section, we give some properties of the solutions, and these results will be often used later.

For any solution

Let

As a result, for any

Model (

In the following, we will show that model (

If

The proof is based on comparison principles. From (

Let

Since

Similarly, by the second equation in model (

Let

In this section, we will devote consideration to the stability of the positive equilibrium for model (

Clearly, model (

For the sake of simplicity, we rewrite model (

Let

The linearization of model (

From [

For each

So, the local stability of the positive equilibrium

Assume that

The stability of the positive equilibrium

In view of

In view of the relation

Therefore, the eigenvalues of the matrix

In the following, we prove that there exists

Let

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

Consequently, the spectrum of

In the following, we shall prove that the positive equilibrium

Suppose that

We adopt the Lyapunov function

Then,

It is obvious that

Hence,

In this section, we will investigate Turing instability and bifurcation for our model problem. We will also study pattern formation of the predator-prey solutions.

Mathematically speaking, an equilibrium is Turing instability (diffusion-driven instability) means that it is an asymptotically stable equilibrium of model (

Now, the conditions for the positive equilibrium to be stable for the ODE are given by

However, the inequality

Summarizing the previous analysis and calculations, we have the following results.

Assume that the positive equilibrium

then the positive equilibrium

In Figure

Turing bifurcation diagram for model (

In this section, we perform extensive numerical simulations of the spatially extended model (

The numerical integration of model (

In the numerical simulations, different types of dynamics are observed, and it is found that the distributions of predator and prey are always of the same type. Consequently, we can restrict our analysis of pattern formation to one distribution. In this section, we show the distribution of prey

Figure

Holes pattern formation for model (

Figure

Stripes pattern formation for model (

Figure

Spots pattern formation for model (

Ecologically speaking, spots pattern shows that the prey population is driven by predators to a high level in those regions, while holes pattern shows that the prey population is driven by predators to a very low level in those regions.

In this paper, we have studied the dynamics of a reaction-diffusion Holling-Tanner prey-predator model where the predator population is subject to Allee effect under the zero-flux boundary conditions. The value of this study lies in threefolds. First, it investigates qualitative properties of solutions to this reaction-diffusion model. Second, it gives local and global stability of the positive equilibrium of the model. Third, it rigorously proves the Turing instability and illustrates three categories of Turing patterns close to the onset Turing bifurcation, which shows that the model dynamics exhibits complex pattern replication.

It is seen that if Allee effect constant

Comparing Figures