Global Stability of a Stage-Structured Predator-Prey Model with Stochastic Perturbation

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.


Introduction
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 [1][2][3][4][5][6][7][8][9].In most of the cases, the study is based on interactions between homogeneous populations.However, in the nature, most of the species must go through two life stages from birth to death.In [6][7][8][9], some stage-structured models of population growth consisting of immature and mature individuals were discussed.In particular, [9] considered a predator-prey model with two populations, that is, predator and their food prey.In their model, only prey species is divided into two life stages, the immature prey and the mature prey.And the predators only consume the immature prey species.In [10], Chinese firebellied newt is described as an example, which is unable to prey on the mature Rana chensinensis and can only prey on the immature one.So, they consider the following nonlinear ordinary differential equations: In [11], the authors have studied the dynamical properties of deterministic model (1) and the stochastic behavior of the corresponding stochastic model (19), which was first introduced by Beretta et al. [12] and Shaikhet [13].Then, they obtained stochastic stability condition in mean square sense by utilizing Lyapunov function.
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 [5], Beddington-DeAngelis type [7], and Watt type [14].On the other hand, population is inevitably affected by environmental noise in nature [15,16].May [17] also showed that, due to environmental fluctuation, the birth rate, the death rate, competition coefficients, and other parameters usually show random fluctuation to a certain extent that should be stochastic.Therefore, many authors have taken stochastic perturbation into deterministic models and shown the effect of environmental variability on population dynamics in mathematical ecology [18][19][20][21][22][23][24][25][26].For example, [26] considered the following stochastic stage-structured predator-prey model: In this paper, the authors mainly utilize Itô's formula, the theory of stochastic differential equations, and Lyapunov functions to investigate the global stability of the positive equilibrium of model (2).Motivated by the above works, in this paper, we will consider the following stochastic stage-structured predatorprey model: where  3) which is the solution of the algebraic equations with initial conditions   (0) > 0,  = 1, 2, 3. Noting that if  1 =  2 =  3 = 0, then model (3) becomes the following corresponding deterministic stage-structured predator-prey systems: Therefore, in this paper, we only need to establish the sufficient conditions for global asymptotic stability of system (3).And, in this paper, we will also use Itô' formula, the theory of stochastic differential equations, and Lyapunov functions to study the global stability of the positive equilibrium point of stochastic system (3).The paper is organized as follows.In Section 2, we study the existence and uniqueness of global positive solution of system (3).In Section 3, sufficient conditions for global asymptotic stability of system (3) are established.Then, we introduce some simulation figures to illustrate the main result in Section 4. In the last section, we give the conclusions.

Existence and Uniqueness of Solution
In this section, we will show that the solution of system (3) is positive and global.We give the following theorem.

Globally Asymptotically Stable
For the sake of convenience, denote as well as Then, the positive equilibrium position ( * 1 ,  * 2 ,  * 3 ) of model ( 3) is globally asymptotically stable with probability one; that is, for any positive initial data ( 1 (0),  2 (0),  3 (0)), the solution of system (3) has the property that almost surely.
Proof.From the stability theory of stochastic differential equations, we only need to find a suitable Lyapunov function () satisfying () ≤ 0 and the identity holds if and only if  =  * [30], where  = () is the solution of the following stochastic differential equation: * is the positive equilibrium position of ( 19), and For  ∈  3 + , define where We can rewrite (3) as follows: Applying Itô formula to model (23), we can get that Clearly, the conditions of Theorem 2 and the above inequality denote () < 0 along all trajectories in  3 + except ( * 1 ,  * 2 ,  * 3 ).Then, we get the desired assertion immediately.For deterministic system (5), we have that the following theorem holds.
then the equilibrium position ( * 1 ,  * 2 ,  * 3 ) of system (5) is globally asymptotically stable.Remark 4. From Theorems 2 and 3, we can see that if the positive equilibrium of the deterministic system is globally stable and the noise perturbation is not very large, then the stochastic system will keep the nice property.

Numerical Simulations
In this section, we will utilize the Milstein method mentioned in Higham [31] to consolidate the analytical findings.Here, we consider the discretization equations of model (3) as follows: