Periodic Measures of Mean-Field Stochastic Predator-Prey System

Huang, Zaitang; Cao, Junfei

doi:https://doi.org/10.1155/2018/3584354

Discrete Dynamics in Nature and Society

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Research Article | Open Access

Volume 2018 | Article ID 3584354 | https://doi.org/10.1155/2018/3584354

Periodic Measures of Mean-Field Stochastic Predator-Prey System

Zaitang Huang¹and Junfei Cao²

Academic Editor: Kousuke Kuto

Received11 Oct 2017

Revised12 Aug 2018

Accepted26 Aug 2018

Published06 Nov 2018

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 [2–8].

However, population systems in the real world are very often subject to environmental noise [9–15]. 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 [16–20], 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, 21–24], 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 [25–27].

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 . FurthermoreWhen , the last term is . Then, for , we have thatare -martingales and for .
Usingfor all , we getthen we haveNext, we prove thatFor , define by We will show that It is easy to see that we already proved . By the definition of weak convergence, is proven directly.
By Fatou’s theory, for , thenand it is easy to see that is proved. By similar way, we can prove
Then giveSo impliesand therefore -a.s.
By the law of large numbers, we have proved that -a.s. the projection of at is equal to . By Lemma 3.1 in [30], we proved the assertion of the Theorem 4.

4. Periodic Distribution

In the section, under some suitable assumptions, we prove that the solution of mean-field stochastic predator-prey systems possesses a strictly periodic distribution.

Lemma 5. Suppose that . LetwhereandThenfor all .

Proof. Applying Itô’s formula to , we getSuppose that and .
Defining and andThen, we haveCase 1 (). Substituting by , we haveSolving the above equation, we havewhere and are real number independent of , but possibly based on and , then we havesince , it impliesUsing Gronwall’s Inequality and Fatou’s theory, when , we obtain the result.
Case 2 ( and ). By (104), Then we getAs before, we have where and are real numbers not relying on , and but possibly relying on , and , then we getSimilarly the proof is as in Case 1, so we prove Case 2 directly.

Lemma 6. Suppose that the condition of Lemma 5 is satisfied. Let , , and . The following conditions hold:
(1) For all satisfying ,
(2) There exists a constant such that if and .
Then for all .

Proof. Let and . By (105) and (106), we can estimate and byandthen, for any , we havewhere and and are real number independent of and but possibly based on , and . When , we haveFurthermorewhere
LetFix , and if there are some satisfying . Define . Sincefor all , then and ; it implies contradiction (117).

Lemma 7. Suppose that , and . The following conditions hold:
(i) There are constants such that, for any and ,(ii) For all initial conditions satisfying , Then there exist constants such thatwhere is the solution of (1) with .

Proof. LetApplying Itô’s formula, we haveThenBy Lemma 5, we haveThereforeNext, applying Itô’s formula, we get Then By Lemma 5, we also haveThereforeLet , and we have Then Letand we getwhereBy and , for , for some and , we have Then, we getBy Gronwall’s inequality, we haveThen, we prove the assertion of Lemma 7.

Theorem 8. Suppose that and . Let and , if the following conditions hold:
(i) There is a real number such that for all , .
(ii) There is a probability measure on and some such that .
Then but for ; i.e., (4) possesses a periodic distribution.

Proof. denote the unique periodic solution of (1), where (see [1]) andLetand denote the solution of system (1) starting at and let Since is a compact set but not included in , because there exists the uniqueness of the solution of system (1), it is easy to see that is proven. Next, we prove that . Suppose that there exist some such that and . Then the line segment and the curve form a set which is invariant, it is easy see that contains the limit cycle and , due to the globally stable limit cycle, then we prove . By the same way, we prove that .
For , we take values satisfying the condition of Lemma 6 with and . Let ; fix and satisfying and . Now let be large enough such thatand From (127) and (131), it is shown that if and if , , and . By Lemma 6, it is easy see that if and . It implies that there are real numbers and such that whenever . Let denote the set on and and for andNext, we will prove that maps into ; hence, we show that there are real numbers and satisfying whenever .
Let ; suppose that and . By the proof of Theorem 3, it proves that for all and for Then, we have Therefore, we getDue to , we show that the first integrand is negative real number for large enough . The second integrand is not larger than andHence, we getand using , we have Clearly, when for any . Therefore, we choose satisfyingIt shows that whenever .
Next, we prove that is weakly continuous. Hence, if we have proved that is continuous, it is easy to see that is weakly continuous on . Then it implies that is continuous. For , we have Let and take such that for all . Fix , let and take such that for all . Then for ; it is easy to see that is continuous on and therefore the proof is completed.

Corollary 9. Suppose that and , , and the following conditions hold:
(i) There exists a sequence satisfying the conditions of Theorem 8 for every and .
(ii) There is a sequence on such that (4) with and .
Then system (4) possesses a periodic distribution and(a);(b).

Proof. Let and converge to with . It is easy to see that converges to zero. Then, the first assertion of theorem is proved. Furthermore, we have and hence, solving for , we get Applying Chebychev’s inequality, we havewhereandSincethen we prove the second assertion of theorem that isBy [31, pp 142], we getTherefore, for then is a martingale. Denoting the stopping time,By Chebychev’s inequality, we get For ,and hence, we have since , it implies Therefore, the proof is completed.

5. Fluctuations

In the section, under some suitable assumptions, we study the fluctuations of the periodic in distribution processes for mean-field stochastic predator-prey systems when the white noise converges to zero.

Theorem 10. Suppose that condition of Theorem 8 is satisfied. The following conditions hold:(i)There exist real numbers and satisfying and .(ii)There is a probability measure on such thatThen system (171) exists a periodic distribution for all .

Furthermore, there exists a periodic solution of (171) with weakly on .

Proof. By the proof of Theorem 8, we prove only but, using Lemma 6, it is easy to show that there is a periodic solution such that . Then the first assertion of the theorem is proven.
Next, let . Due to and , is family of compact. By the same proof of Theorem 8, it is easy to see that there is a weakly convergent subsequence of approaching to a solution ofLet denote the period of ; we choose a subsequence satisfying approaches to a solution of (172), and exists. Based on Theorem in [18, pp 264], it is easy to see that that the map from to is continuous, where is the solution of (171) with at
Let ; it will show that the periodic distribution solution of (172) is . Denoting , where is defined as in Lemma 5 with , then we havefor all , then . The right hand side of (173) is negative constant if either or is large enough; i.e., there has some satisfying whenever ; it easily shows that the support of which is periodic is included infor every
Let and ; we getHence, (w.p.1) , and therefore and andThen, we havewhenever if and are sufficiently large. Because and are periodic, it easily shows that ; i.e., is deterministic. There exists only one periodic solution of system (1) with and . Therefore, the proof is completed.

Theorem 11. Suppose that and . Let are nonnegative constants. For and , and are large enough,where Then there exists a periodic solution of system (4).

Proof. According to , and Theorem 10. It will prove for .
Define Lyapunov functions whereand Next, we prove thatLet denote the generator of the diffusion . Then, we have Firstly, we consider the case which shows and henceIn case Now, for , and , we getif and are large enough. Then, we haveFurthermore, since and , we havefor all and . Thenif is small enough.
If , then we havewhereHence, there is a constant satisfyingFurthermore, letthenand it showsprovided is small enough such that .
Furthermore, we haveas and are large enough and is a suitable constant.
By (197) and (198), it has proven that there are constants and satisfyingif and are large enough and is small enough.
Next, our aim is to consider the remaining terms in ; it contains derivatives of .
Since ,where and Then, we have .
Furthermore, if is small enough. Then we have for , . Hence, we have Because are periodic and , hence it shows thatif and satisfy the condition above and are large enough. Therefore, the proof is completed.

Theorem 12. Suppose that condition of Theorem 11 is satisfied. If and are large enough, as , then converge weakly to which possesses the unique time-invariant distribution solution of

Proof. By Theorem 11, it can easily see that the sequence is tight for any sequence . Furthermore where defined in (194).
By Theorem in [29], if we can prove that (205) is no larger than one solution such that for some , converge weakly to the solution of (205). For any solution of (205), it can be described bywhere Let , then converge in to the Gaussian processwhere is a periodic. For all , because the law of is periodic, it shows that ; therefore the laws of are identical with ; it is easy to see that the laws of are identical with if system (205) shows the uniqueness periodic solution. Therefore, the proof is completed.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This research was supported by the National Natural Science Foundation of China (no. 11561009 and no. 41665006), Guangxi Natural Science Foundation (no. 2016GXNSFAA380240), the Guangdong Province Natural Science Foundation (no. 2015A030313896), the Characteristic Innovation Project (Natural Science) of Guangdong Province (no. 2016KTSCX094), and the Science and Technology Program Project of Guangzhou (no. 201707010230).

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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.

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