We investigate the complex dynamics of a diffusive Holling-Tanner predation model with the Allee effect on prey analytically and numerically. We examine the existence of the positive equilibria and the related dynamical behaviors of the model and find that when the model is with weak Allee effect, the solutions are local and global stability for some conditions around the positive equilibrium. In contrast, when the model is with strong Allee effect, this may lead to the phenomenon of bistability; that is to say, there is a separatrix curve that separates the behavior of trajectories of the system, implying that the model is highly sensitive to the initial conditions. Furthermore, we give the conditions of Turing instability and determine the Turing space in the parameters space. Based on these results, we perform a series of numerical simulations and find that the model exhibits complex pattern replication: spots, spots-stripes mixtures, and stripes patterns. The results show that the impact of the Allee effect essentially increases the models spatiotemporal complexity.

Recently, there has been a great interest in studying nonlinear difference/differential equations and systems [

The dynamics of model (

On the other hand, in population dynamics, any mechanism that can lead to a positive relationship between a component of individual fitness and either the number or density of conspecifics constitutes what is usually called an Allee effect [

From an ecological point of view, the Allee effect has been denominated in different ways [

if

if

The most common mathematical form describing this phenomenon for a single species is given by

Furthermore, Boukal et al. [

For model (

Obviously, we have the following:

if

if

if

And we can get the following model with the Allee effect on prey:

There are some excellent works on a Holling-Tanner model considering the diffusion [

However, the research about the influence of Allee effect on pattern formation of diffusive Holling-Tanner model seems rare. The main purpose of this paper is to study dynamical behaviors of a Holling-Tanner predator-prey model with the Allee effect. We will determine how the Allee effect affects the dynamics of the model and focus on the stability of the positive steady state and bifurcation mechanism and patterns formation analysis of the model.

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

Now, we prove that all solutions are eventually bounded.

All the solutions of model (

Let

Define the function

Using the theory of differential inequality, for all

In fact, if

Next, we will investigate the existence of equilibria and their local and global stability with respect to model (

In this subsection, we consider the existence and stability of the equilibrium of model (

We note that model (

The point

As the Jacobian matrix of the point

Clearly, if

The singularities of model (

Then,

Moreover, it is easy to verify that model (

The Jacobian matrix of model (

The Jacobian matrix of model (

And model (

Suppose that

If

If

If

If

Let

We can see that the sign of

Thus, we can obtain

Hence, we obtain

In the following results, we study the stability of the positive equilibrium

Define

If

if

if

If

if

if

A Hopf bifurcation occurs at

Here, we only give the proof of the existence of Hopf bifurcation. It is easy to see that

the characteristic equation is

From the Poincaré-Andronov-Hopf Bifurcation Theorem [

Figure

The phase portrait of model (

The phase portrait of model (

The unique equilibrium point

a nonhyperbolic attractor node, if and only if

a nonhyperbolic repellor node, if and only if

a cusp point, if and only if

We have

Moreover,

The cusp point is shown in Figure

The phase portrait of model (

In this subsection, we consider the stability of the equilibrium of model (

It is easy to verify that model (

From (

Hence, we have the following results on the stability of the positive equilibrium

Define

If

if

if

If

if

if

A Hopf bifurcation occurs at

In the following theorem, we study the global behavior of the positive equilibrium

If

Construct the following Lyapunov function:

Substituting the value of

Note that

Hence, the positive equilibrium

Figure

The phase portrait of model (

The phase portrait of model (

The phase portrait of model (

In this section, we will investigate the dynamics of the spatial model (

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

In the presence of diffusion, we will introduce small perturbations

Following Malchow et al. [

Having substituted

A general solution of (

Correspondingly,

Summarizing the previous discussions, we can get the following theorem immediately.

(i) The positive equilibrium

(ii) If the positive equilibrium

(i) Using Routh-Hurwitz criteria, we can know that the positive equilibrium

(ii) We select the Lyapunov function for model (

Using Green’s first identity in the plane,

From the previous analysis, we note that

On the other hand, Turing instability sets in when at least one of the conditions is either

Thus, a sufficient condition for Turing instability is that

Summarizing the previous discussions, we can obtain the following theorem.

If

The Turing instability (or bifurcation) breaks spatial symmetry, leading to the formation of patterns that are stationary in time and oscillatory in space [

Turing bifurcation diagram for model (

In this subsection, we performed extensive numerical simulations of the spatially extended model (

The numerical integration of model (

More precisely, the concentrations

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

Spots pattern of

Figure

Stripes pattern of

Spot-stripe mixtures pattern of

In this paper, we are concerned with the complex dynamics in a diffusive Holling-Tanner predator-prey model with the Allee effect on prey. The value of this study lies in two folds. First, the local asymptotic stability conditions for coexisting equilibrium and conditions for Hopf bifurcation are described briefly for the model with the weak and strong Allee effects. Second, it gives the analysis of Turing instability which determines the Turing space in the spatial domain and meanwhile illustrates the Turing pattern formation close to the onset Turing bifurcation via numerical simulations, which shows that the model dynamics exhibits complex pattern replication.

We note that in the analyzed models, a big difference between the dynamics of model with strong or weak Allee effect exists. In the case of strong Allee effect, two positive equilibria can coexist for a subset of parameters with a varied dynamics but different to other Holling-Tanner models analyzed earlier [

Furthermore, we have investigated the conditions for the predator-prey model which experiences spatial patterns through diffusion-driven instability. We have derived the conditions of Turing instability in terms of our model parameters analytically. In addition, to get a deeper insight into the model’s dynamics behaviour, we select the different values of parameter

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