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The nonautonomous stochastic Gilpin-Ayala competition model driven by Lévy noise is considered. First, it is shown that this model has a global positive solution. Then, we discuss the asymptotic behavior of the solution including moment and pathwise estimation. Finally, sufficient conditions for extinction, nonpersistence in the mean, and weak persistence of the solution are established.

Population dynamics that describes and understands changes in populations over time is an important subfield of ecology. One of the famous models for population dynamics is the traditional Lotka-Volterra competition system which has received a lot of attention and has been studied extensively because of its theoretical and practical significance. It was suggested by Lotka [

On the other hand, population dynamics in the real world is inevitably affected by environment noise which is an important component in an ecosystem. So it is necessary and important to consider the corresponding stochastic population model. For the stochastic population models, we refer the reader to [

Hence, motivated by the above discussion, in this paper, we consider the nonautonomous stochastic Gilpin-Ayala competition models perturbed by Lévy noise; that is,

It is worth observing that the model (

Throughout this paper, let

If

The rest of the paper is arranged as follows. In Section

The solution of (

In what follows we always denote by

Consider a

The following conclusion is given by [

Let

Before stating the main results in this paper, we give the following assumptions for the jump-diffusion coefficient.

(H1) Assume that, for any

(H2) Assume further that for

Let the assumptions (H1) and (H2) hold. Then, for any given initial value

Since, by (

In the previous section, we show that the solution of (

Let conditions of Theorem

For any

If we note the inequality

Under the conditions of Theorem

In the following theorem, we prove that the average in time of the moment of the solution to (

Under the conditions of Theorem

If

If

For

Case

In the previous section we have discussed how the solution of system (

Assume that

Let the conditions of Theorem

Let

Integrating 0 into

Noting the limit

Under conditions of Theorem

One of the important problems in population biology is whether all species of one system will become extinct over time. There are many factors including natural and man-made factors which may cause the extinction of some species, such as low birth rate, high death rate, decreasing habitats, and aggravating living environment. Sometimes, small even large population of some species may be destroyed by some extraordinary perturbation. To proceed, we need some appropriate definitions of extinction and persistence. Hallam and Ma proposed the concepts of weak persistence [

Stochastic population dynamics (

The population

The population

Let conditions of Theorem

Applying Itô’s formula to

On the other hand, for every integer

Let conditions of Theorem

Let conditions of Theorem

The proofs of Theorems

In this section we provide a numerical example to substantiate the analytical findings for the stochastic model system reported in the previous sections.

Let

Solutions of (

In Figure

Solutions of systems (

In view of Theorem

This paper is concerned with a nonautonomous stochastic Gilpin-Ayala competition model with jumps. We show that the model (

This work was substantially supported by the National Natural Sciences Foundation of China (nos. 11071259, 11371374) and Research Fund for the Doctoral Program of Higher Education of China (no. 20110162110060).