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The Stationary Distribution of Competitive Lotka-Volterra Population Systems with Jumps
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 and asymptotic pathwise estimation are still unknown for LVWJ. One of our aims in this paper is to show the uniqueness of stationary distribution and asymptotic pathwise estimation for LVWJ. Moreover, some characterizations for stationary distribution are provided.
Recently, Bao et al.  introduced a jump process into the underlying population dynamics to characterize sudden environmental shocks, for example, earthquakes, hurricanes, epidemics, and so forth. They assume that population sizes follow the following stochastic differential equations: where stands for the left limit of , , , , and . Here represents the effect of interspecies (if ) or intraspecies (if ) interaction and is an -dimensional Brownian motion. is a Poisson counting measure with characteristic measure on a measurable subset of with and . Besides, we suppose that and are independent and for , are nonnegative constants. If , , , , , then the model (1) is termed as the facultative Lotka-Volterra model with jumps. If , , , , , then the model (1) is termed as the competitive Lotka-Volterra model with jumps. Bao et al. [1, 2] reveal some important and nice properties. (i)Jump processes can suppress the explosion.(ii)Under Assumption A1 (see Section 2), there exists an invariant measure for the solution of population model (1).(iii)When the white noise and sudden noise are large, stochastic population dynamics (1) tend to be extinct. To be precisely, under Assumptions A1 and A2 (see Section 2), the population dynamics (1) tend to be extinct.(iv)When the white noise and sudden noise are small, stochastic population dynamics (1) is stochastically permanent and has a unique stationary distribution. It is a natural question that when the white noise and sudden noise are small do the population dynamics (1) have a unique stationary distribution for ? If yes, can we give some characterizations of the stationary distribution? In this paper, we shall show that the population dynamics have the following properties. (i)Under Assumption A1 (see Section 2), there exists a unique stationary distribution and it has ergodic property.(ii)Under Assumptions A1 and A3 (see Section 2), the average in time of any of the path is bound with probability one. Besides, we have where is the stationary distribution and is a positive constant.(iii)Some characterizations of the stationary distribution are provided.
Throughout this paper, we let be a complete probability space with a filtration satisfying the usual conditions (i.e., it is right continuous and contains all -null sets). Let denote the Euclidean norm in . If is a vector or matrix, its transpose is denoted by . If is a matrix, its trace norm is denoted by . Let be -dimensional real Euclidean space and be the set ; : the trace norm of matrix ; that is, .
We shall need a few more notations. Define , .
Denote transition probabilities of (1) by
The transition probabilities can be thought of as operators on bounded Borel measurable functions via
Assumption A1. For any and with , , , , , and are bounded functions; , , and , .
Assumption A2. For and , the coefficients of determined by (1) obey the following condition:
Assumption A3. There exists a constant such that for some , , , we have
We know from Bao et al.  that Assumption A1 is a basic condition which guarantees that for any initial condition , (1) has a unique global positive solution; namely, . Besides, throughout this paper, we let be a generic positive constant whose values may vary at its different appearances.
3. Main Results
With the notations introduced in the previous section, we can state one of our main results.
Theorem 1. Under Assumption A1, the SDE model (1) has a unique stationary distribution and it has ergodic property.
Lemma 2 (Bao et al. ). (1) Under Assumption A1, for any , there is a constant such that .
(2) Under Assumptions A1 and A3, for any , there is a constant such that .
Define to be the first time the diffusion exists in domain : Similar to Proposition 7.2 in Karatzas and Shreve [3, pages 364-365], we have the following lemma.
Lemma 3. Suppose that for the open, bounded domain and for some , if the following condition: holds, then , for any .
Proof. For simplicity, let , , and let , , .
Consider the function ; , where the positive constants , shall be determined later. Then, we have Note that the inequality for all ; then, Consequently, if Assumption (i) holds, then we can choose and a sufficiently large such that for any , we have With the boundedness of the function and its derivatives on , Dnykin’s formula implies, , , the following: By letting , we obtain
Lemma 4. Under Assumption A1, the SDE model (1) has the (weak) Feller property.
Proof. To prove the weak Feller property, by Theorem 5.1 (see Bhattacharya and Waymire [4, pages 643–645]), we only need to show that for every bounded Lipschitzian function on , if is well defined, we have
The proof of the limit above is essentially the same as that of Lemma 3.2 in Tong et al. . Consequently, we only need to prove that is well defined. For any , let . By Assumption A1, we have
By Lemma 3, we have , which implies that is well defined.
We are now able to prove Theorem 1.
Proof of Theorem 1. The proof of this theorem is divided into two steps.
Step 1. (Existence). Bao et al.  show that (1) exists as an invariant measure:
Step 2. (Uniqueness). By Lemma 4 and the same discussion as that of Theorem 3.1 in Tong et al. , one can easily prove that is the unique station distribution.
Lemma 5. Suppose that condition (A1) holds. If is a function integrable with respect to the measure , then
Proof. The proof is essentially the same as the proof of Theorem 5.1 of Has’minskiĭ . We omit its proof.
With the lemma above, we can get the following long time behavior of population system (1).
Theorem 6. If Assumptions A1 and A3 hold, for any , then we have
Proof. Let and for some , where . Obviously, and are integrable with stationary distribution . Hence, by Lemma 5, we get According to Lemma 2, it follows that From (18) and (20), (19), and (21), we get By letting , it follows that Namely, and are integrable with respect to the unique invariant measure . Again by Lemma 5, the proof is complete.
4. Characterizations of Stationary Distribution
In what follows, we consider some characterizations of the stationary distribution of (1). By Theorem 4.6 of Bao et al. , for , , if then stochastic population dynamics (1) are exponential extinctions; that is the unique stationary distribution . We shall show that condition (24) also is necessary condition for exponential extinction if we replace the bounded coefficients with uniformly bounded coefficients under Assumption A1. Hence, we impose a stronger assumption on coefficients of population dynamics (1).
Assumption A4. For any and with , , , , , and are uniformly bounded functions on time , , and , .
Proof. Sufficiency: By Itô's lemma, we have
where , . Under Assumption A4, the sharp bracket process is as follows:
By the strong law of large numbers of martingales, we have
If stochastic population dynamics are exponential extinctions a.s., then , which implies the assertion.
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.
Assumption A5. We have Next, we consider the mean vector and covariance matrix of the stationary distribution. If we assume that Assumptions A1 and A3 hold, then Theorem 6 implies that both mean vector and covariance matrix of the stationary distribution; namely, are well defined.
The next theorem gives an explicit formula for the mean vector of the stationary distribution of (1).
Theorem 8. If conditions A1 and A3 hold, then the mean vector of the stationary distribution follows
Proof. Consider the function . By the proof of Theorem 7, it immediately obtains this result.
To get some further characterizations for the stationary distribution defined by (15), we consider its Laplace transform: where stands for the real part of complex . Recall that a.s., which implies that the stationary distribution is concentrated on . By the bounded property of the distribution function, the Laplace transform of stationary distribution is well defined on .
Lemma 9. If and the process has a unique stationary distribution , then
Proof. According to the condition , then function can be rewritten as
where is infinitesimal; that is, . In virtue of equality (33), we have
Consequently, Note that . For any , there exists a sufficiently large constant , such that for any , we have Consequently, As for the second term, it is not hard to show that . According to the ergodic lemma, we have
To state our main result, we need the following conditions about the coefficients.
Assumption A6. There exist constants , , , such that
We now state the main result of this section.
Theorem 10. Assume that Assumptions A1, A4, A5, and A6 hold. The stationary distribution defined by (15) satisfies, for any complex number , with , the following:
Proof. Let , , . By Itô’s formula, we have where stands for the continuous part of the process . In integral form, we get where By Theorem 6 and Assumption A4, it is easy to see the sharp bracket process: Hence, by the strong law of large numbers of martingales, Therefore, by dividing both sides by of equality (42) and then letting , one gets, by Lemmas 5 and 9 and Theorem 6, that the assertion (40) holds.
Because Theorem 10 holds for all , one can get a series equality with different values of . In particular, If we let on the both sides of (40), then we immediately obtain a characterization for covariance matrix of the stationary distribution.
The next result presents the characterization of expectation of the stationary distribution.
Corollary 11. If conditions A1, A5, and A6 hold, then the covariance matrix of the stationary distribution obeys
Proof. By Theorem 10, the stationary distribution satisfies
By letting , we get This completes the proof.
Remark 12. Mao  obtains the result (46) for the facultative model without jumps with nonsingular noise coefficients and Tong et al.  obtain the result (46) for the facultative model without jumps with singular noise coefficients.
Corollary 13. Under Assumptions A1, A5, and A6, for , we have
Proof. By virtue of Theorem 6, the th moment of stationary distribution is well defined. Expanding the exponential function into the power series and comparing the terms with the same powers of , one can obtain the equality (49) immediately.
Conflict of Interests
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).
R. N. Bhattacharya and E. C. Waymire, Stochastic Processes with Applications, Wiley Series in Probability and Mathematical Statistics: Applied Probability and Statistics, John Wiley & Sons, New York, NY, USA, 1990.View at: MathSciNet
R. Z. Has'minskiĭ, Stochastic Stability of Differential Equations, vol. 7 of Monographs and Textbooks on Mechanics of Solids and Fluids: Mechanics and Analysis, Sijthoff & Noordhoff, Alphen aan den Rijn, The Netherlands, 1980.View at: MathSciNet