This paper is concerned with a new predator-prey model with stage structure on prey, in which the immature prey and the mature prey are preyed on by predator. We think that the model is more realistic and interesting than the one in which only the immature prey or the mature prey is consumed by predator. Our work shows that the stochastic model and its corresponding deterministic system have a unique global positive solution and the positive solution is global asymptotic stability for each model. If the positive equilibrium point of the deterministic system is globally stable, then the stochastic model will preserve the nice property provided that the noise is sufficiently small. Results are analyzed with the help of graphical illustrations.

Within the past decades, the dynamic behavior between predator and their prey has received considerable interest due to their wide applications in ecology and mathematical ecology. There is a great deal of attention for predator-prey models from many scholars [

On one hand, the predators functional response, that is, the rate of prey consumption by an average predator, is one of the important components which can impact the relationship between predator and prey in population dynamics. There are many functional responses such as Holling type [

Motivated by the above works, in this paper, we will consider the following stochastic stage-structured predator-prey model:

The initial condition of model (

The paper is organized as follows. In Section

In this section, we will show that the solution of system (

For any given initial value

Since the coefficients of (

For the sake of convenience, denote

If

Then, the positive equilibrium position

From the stability theory of stochastic differential equations, we only need to find a suitable Lyapunov function

Let

For deterministic system (

If

From Theorems

In this section, we will utilize the Milstein method mentioned in Higham [

Here, we consider the discretization equations of model (

Set

Solution of system (

In this paper, we investigated two stage-structured predator-prey systems: deterministic one and stochastic one. In the models, we suppose that both immature prey and mature prey are consumed by predator. The model is more realistic and complicated than the one in which only the immature prey or mature prey is preyed on by predator. For each system, we established the sufficient conditions for global asymptotic stability. From the results and simulation figures, we can see that if the positive equilibrium position of the corresponding deterministic model is globally stable and the noise is sufficiently small, then the stochastic system will preserve the nice property. The result is useful and important for ecological balance. Up to our knowledge, the present work is the first attempt to study such stochastic model with stage structure on prey.

The authors declare that there is no conflict of interests regarding the publication of this paper.

The authors would like to thank the referee and editor for their valuable comments and suggestions that greatly improved the presentation of this paper. This work was supported by the Program for New Century Excellent Talents in University (NCET-10-0097), the NSFC Tianyuan Foundation (Grant no. 11226256), and the Zhejiang Provincial Natural Science Foundation of China (Grant no. LY13A010010).