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Dynamics of Lotka-Volterra population with jumps (LVWJ) have recently been established (see Bao et al., 2011, and Bao and Yuan, 2012). They provided some useful criteria on the existence of stationary distribution and some asymptotic properties for LVWJ. However, the uniqueness of stationary distribution for

Recently, Bao et al. [

Jump processes can suppress the explosion.

Under Assumption A1 (see Section

When the white noise and sudden noise are large, stochastic population dynamics (

When the white noise and sudden noise are small, stochastic population dynamics (

Under Assumption A1 (see Section

Under Assumptions A1 and A3 (see Section

where

Some characterizations of the stationary distribution are provided.

Throughout this paper, we let

We shall need a few more notations. Define

Denote transition probabilities of (

The transition probabilities can be thought of as operators on bounded Borel measurable functions

Following Bao et al. [

We know from Bao et al. [

With the notations introduced in the previous section, we can state one of our main results.

Under Assumption A1, the SDE model (

The proof of Theorem

(1) Under Assumption A1, for any

(2) Under Assumptions A1 and A3, for any

Define

Suppose that for the open, bounded domain

For simplicity, let

Consider the function

Under Assumption A1, the SDE model (

To prove the weak Feller property, by Theorem 5.1 (see Bhattacharya and Waymire [

We are now able to prove Theorem

The proof of this theorem is divided into two steps.

Suppose that condition (A1) holds. If

The proof is essentially the same as the proof of Theorem 5.1 of Has’minskiĭ [

With the lemma above, we can get the following long time behavior of population system (

If Assumptions A1 and A3 hold, for any

Let

In what follows, we consider some characterizations of the stationary distribution of (

Under Assumption A4, stochastic population dynamics (

Sufficiency: By Itô's lemma, we have

Necessity: the proof can be found in Theorem 4.6 of Bao et al. [

In the rest of this paper, we consider the nontrivial stationary distribution case.

The next theorem gives an explicit formula for the mean vector of the stationary distribution of (

If conditions A1 and A3 hold, then the mean vector

Consider the function

To get some further characterizations for the stationary distribution defined by (

If

According to the condition

Consequently,

To state our main result, we need the following conditions about the coefficients.

We now state the main result of this section.

Assume that Assumptions A1, A4, A5, and A6 hold. The stationary distribution

Let

Because Theorem

The next result presents the characterization of expectation of the stationary distribution.

If conditions A1, A5, and A6 hold, then the covariance matrix

By Theorem

By letting

Mao [

Under Assumptions A1, A5, and A6, for

By virtue of Theorem

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

The authors are grateful to the anonymous referees for their valuable comments and suggestions which led to improvements in this paper. Research of the authors was partially supported by National Natural Science Foundation of China (nos. 11171062, 11101077, 11071258, 11101054, and 11201062), the Fundamental Research Funds for the Central Universities, and the Innovation Program of Shanghai Municipal Education Commission (no. 12ZZ063).