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This paper is concerned with a stochastic delay logistic model with jumps. Sufficient and necessary conditions for extinction are obtained as well as stochastic permanence. Numerical simulations are introduced to support the theoretical analysis results. The results show that the jump process can affect the properties of the population model significantly, which conforms to biological significance.

Recently, Freedman and Wu [

As we know, stochastic population models have recently been investigated by many authors (see, e.g., [

On the other hand, the population may suffer from sudden environmental shocks, for example, massive diseases like avian influenza and SARS, earthquakes, hurricanes, epidemics, and so forth. Bao et al. [

Based on the fact that model (

For model (

For each

There exists a positive constant

For the simplicity, we define the following notations:

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

The classical existence and uniqueness result for solutions of a stochastic differential delay equation with jumps requires the coefficient functions to satisfy a local Lipschitz condition and a linear growth condition (see, e.g., [

The following inequalities hold:

Let assumptions (A1)–(A3) hold. For any given initial value

Since the coefficients of the equation are locally Lipschitz continuous, for any given initial value

Let assumptions (A1)–(A3) hold. If

Now applying Itô’s formula to (

Population size

Let assumptions (A1)–(A3) hold. If

First, we prove that for arbitrary

Let

Next, we claim that for arbitrary

Obviously,

Obviously, if assumptions (A1)–(A3) hold,

In line with

If

In this section, we will use the Euler scheme (see, e.g., [

Here, we choose

The horizontal axis and the vertical axis in this represent the time

In this paper, we investigate the permanence and extinction of a stochastic delay logistic model with jumps. Sufficient and necessary conditions for extinction are established as well as stochastic permanence.

Besides, some interesting topics deserve further consideration. One may propose some more realistic but complex models, such as introducing the colored noise into the model [

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

This paper is supported by the National Natural Science Foundation of China (11271101), the NNSF of Shandong Province in China (ZR2010AQ021), and the Scientific Research Foundation of Harbin Institute of Technology at Weihai (HIT (WH) 201319).