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Discrete Dynamics in Nature and Society
Volume 2018, Article ID 3584354, 21 pages
https://doi.org/10.1155/2018/3584354
Research Article

Periodic Measures of Mean-Field Stochastic Predator-Prey System

1School of Mathematics and Statistics, Guangxi Teachers Education University, Nanning, Guangxi 530023, China
2Department of Mathematics, Guangdong University of Education, Guangzhou 510310, China

Correspondence should be addressed to Junfei Cao; moc.361@htamoacfj

Received 11 October 2017; Revised 12 August 2018; Accepted 26 August 2018; Published 6 November 2018

Academic Editor: Kousuke Kuto

Copyright © 2018 Zaitang Huang and Junfei Cao. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This paper discusses the dynamics of the mean-field stochastic predator-prey system. We prove the existence and pathwise uniqueness of the solution for stochastic predator-prey systems in the mean-field limit. Then we show that the solution of the mean-field equation is a periodic measure. Finally, we study the fluctuations of the periodic in distribution processes when the white noise converges to zero.

1. Introduction

The predator-prey equations is one of the most famous population modelswhere denotes the prey population density and denotes the predator population density. The parameters , and are positive real numbers. By the results in Ruan and Xiao [1], they discuss all kinds of bifurcation phenomena. Recently, system (1) was studied extensively that it exhibits complex dynamical phenomena, including bifurcation, stability, and attractive [28].

However, population systems in the real world are very often subject to environmental noise [915]. According to the Markov jump approach, a classical stochastic predator-prey model can be described bywhere is independent Brownian motions. Biologic in population satisfies the following equation, predator-prey model:where and denote the population density of and in the th out of population, and are nonnegative real number modelling the diffusion between the prey population density and the predator population density, and are independent Brownian motions.

Under regularity conditions, for any fixed , converge in law when , and then system (3) becomeswhere and are Noise intensity functions and and are real number. According to the mathematical approach [1620], these systems can appear very standardized. However, many real world problems process the nature of mixing randomness and periodicity, e.g., due to change of temperatures on earth, harvesting seasons, seasonal economic data, individual lifecycle, and seasonal effects of weather [15]. Biological populations are very often subject to random perturbations that come in a more-or-less periodic way. A number of random periodic results have been studied in the literature [15, 16, 2124], but none of them covers (4). And my method can also be extended to other noise, for example, the telephone noise, Markovian switching, and Lévy jumps [2527].

In the paper, we investigate the dynamics of mean-field stochastic predator-prey system. First, based on martingale approach and Vlassov-Limits, we prove the existence and uniqueness of the solution for mean-field stochastic predator-prey systems, then, by Tihonov’s fixed point theorem and martingale techniques, we prove that the solution of the stochastic predator-prey systems in the mean-field limit is a strictly periodic law under some suitable assumptions. Finally, we study the fluctuations of the periodic in distribution processes when the white noise converges to zero.

2. Preliminaries

Throughout this paper, let be a complete probability space. Suppose that , and are positive constants, and satisfy a Lipschitz condition with constant , and and are bounded,

Definition 1 (see [28]). Let be fixed. A random -valued process is called a d-periodic (in distribution) with period if where is the Borel a-algebra in .

Let be an -valued Wiener process with . Note that, for any and ,with a nuclear operator andwhere

Theorem 2 (see [1]). If and , then system (1) has three equilibria: two hyperbolic saddles and and an unstable focus (or node) in the interior of the first quadrant. Moreover, system (1) has a unique limit cycle, which is stable.

3. Existence and Uniqueness

In the section, under some suitable assumptions, we prove the existence and uniqueness of the solution for mean-field stochastic predator-prey systems.

Theorem 3. For every , let denote a probability measure on such that

(i) Then there exists a unique global strong solution of system (3).

(ii) Then there exists a unique nonnegative solution of system (3) satisfying , , and for all

Proof. (i) To show that this solution is global, we prove that it does not explode in finite time.
LetFrom (3), we have For , define by and by Applying Itô’s formula, we haveThen, we getSince the law of is symmetric , we getApplying the Bellman inequality, we haveDue to and from [29], then converge weakly to when ; by Fatou’s lemma, we obtain thatThen we haveTherefore, we prove the first assertion of the theorem.
(ii) Next, we will prove that there exists a unique nonnegative solution of system (3).
Let for all ; denotes nonnegative real solution of (4). Set , and where Then denote the real solution of (4) with instead of .
LetApplying Itô’s formula, we getandThen, we getandWhen , then converge in law to in ; by dominated convergence, we have .
To prove the uniqueness of solution, it will prove that there are some and some such thatBy iteration method, next, we prove the pathwise uniqueness on .
Firstly. Applying Itô’s formula to , we getLet , , andThen, we haveFor all , by and Fatou’s theorem, when , we have , andSince , it shows that Let so thatThen from (31), we obtain thatNow, we consider the equationBy the Itô formula, we haveBy (34) and (35), as , we infer thatBy Gronwall’s inequality, we getFrom (33) and (35), by Gronwall’s inequality, we getHence, if for some , then
Note that if for some , then we haveand therefore there exists such thatSecondly. To prove , defining , then, for and ,and for and for all , let , then we haveBy Gronwall’s inequality and Fatou’s theorem, when , we getFinally. Let and denote two solutions of (4) on the same space. Ones already showed thatSetBy truncation technique, we getandSincethen we havewhere . Furthermore, we haveTherefore, we haveLetwe getwhereFor , let and . Then . By Gronwall’s inequality, we havewhereBy Gronwall’s inequality, for , we showed that is continuous, and by Hölder’s inequality, we getFor some , it is easily seen that there is a sequence such that when uniformly in iffandBy Chebychev’s inequality, we have, for ,whereHence, we have, for ,Since for , we get In summary, we proved the theorem on

Theorem 4. Suppose that Theorem 3 is satisfied. When , thenconverge to independent copies of solutions of (4) in .

Proof. Let be the law of , where is the measure Then is relatively compact and is a tight family. So, we haveLetNext, we will prove thatis a submartingale. It is easy to see thatBy martingale theory, we getis a martingale. Therefore, we haveis a submartingale. By martingale inequality [29], we getFurthermore, we have thatis a martingale, then we get thatis a submartingale. Furthermore, we have So, for every , it is easy to see thatandLetand if there does not exist such . For any and , it is easy to see that Let ; it is easy to see that are continuous bounded function from to and set . Letand for where Definingwhere are a sequence such that . By (78), we have . Furthermore