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In this work, we establish a new model of one prey and two predators with antipredator behavior. The basic properties on the positivity and boundedness of solutions and the existence of equilibria are established. Through analyzing the global dynamics, we find that there exist some values of the parameters such that one of the predators can be driven into being extinct by another. Furthermore, the coexistence of the three species is investigated which shows that the antipredator behavior makes the species coexist by periodic oscillation. The results give a new insight into the influence of antipredator behavior in nature selection.

Antipredator behavior is very common in the nature. This behavior usually developed by the evolutionary adaptation of the prey and predator. And the prey with the antipredator behavior can help itself to struggle against its predators. The antipredator behavior has already been widely studied by the ecologist [

All of the model mentioned above assumed that the antipredator behavior only happened in two species; one is the prey and the other the predator. However, the antipredator behavior always influences more than one predator and can also make the predators’ competition become more complex [

To the best of our knowledge, the model that consists of one prey and two predators with antipredator behavior is rarely discussed. Thus following the idea of [

The coordinate transformation

The remainder of this paper is organized as follows. In the next section we analyze the basic properties of of system (

We first analyze the positivity and boundedness of system (

All orbits of system (

Since the three coordinate planes of

All of the orbits with positive initial conditions are positive, which means

The boundedness will be proved by using the Poincaré Compactification method [

In the local chart

The invariant plane

In the local chart

System (

Then we continue to analyze the stability of equilibrium

As

For the local chart

Then we use the local charts

It is easy to check that the boundedness conditions

Then we analyze the boundary equilibria of system (

The following statements hold.

(1) If

(2) If

(3) If

(4) If

The Jacobin matrix of system (

The following statements hold.

(1)

(2)

(2.1)

(2.2)

(2.3)

(3)

(3.1)

(3.2)

(3.3)

Next, we analyze the positive equilibria. Solving the following equations

If

Each positive equilibrium of

Let

So

In this section we mainly talk about the competitive exclusion conditions of system (

Assuming the boundedness conditions

hold and all of the equal signs are not established at the same time then all orbits of system (

It follows from Theorem

Let

Thus, we can get

Let

Using the same method as above theorem and taking the function

Assuming the boundedness condition

hold and all of the equal signs are not established at the same time then all orbits of system (

Note that the existence condition of the equilibria

In this section we mainly consider the dynamics when the existence condition of

For system (

Since

Then using sole existence theorem of solution to ordinary differential equations we can get this conclusion.

From the above theorem we know all the trajectories of system (

Furthermore, we can obtain the following results for the restricted system (

Assuming

We use the Hopf bifurcation method that is mentioned in the [

It is easy to get the Jacobin matrix evaluated at the equilibrium

Moreover, we get

We fix the parameter

Transforming this equilibrium to the origin, we take the following transformation:

Then system (

Then, we obtain the system

This result implies that on the plane system where the

Furthermore, we can obtain a weaker condition which allows

For system (

It can be proved by utilizing the Poincaré Bendixson theorem [

It follows from Theorems

It follows from (

Therefore, it follows from the Poincaré Bendixson theorem that there must exist at least one limit cycle in

Numerical simulations are done to illustrate the dynamic behaviors of systems (

The two-dimension phase diagram of the stable solution of system (

The two-dimension phase diagram of the stable solution of system (

The simulated solutions of

The simulated solutions of

In this work we consider a new model of one prey and two predators with the antipredator behavior. To analyze the model, we first prove the positivity and boundedness of system (

Our results lead to a new insight into natural selection mechanism, especially how the antipredator behavior influences the natural selection. From Theorems

In the second section we have talked about the dynamic when the equilibria

No data were used to support this study.

The authors declare that they have no conflicts of interest.