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

Research Article

Periodic Measures of Mean-Field Stochastic Predator-Prey System

Zaitang Huang¹ and Junfei Cao²

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; jfcaomath@163.com

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 models

where 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 by

where 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) becomes

where 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 and

where

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.
Let

From (3), we have

For , define by

and by

Applying Itô’s formula, we have

Then, we get

Since the law of is symmetric , we get

Applying the Bellman inequality, we have

Due to and from [29], then converge weakly to when ; by Fatou’s lemma, we obtain that

Then we have

Therefore, 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 .
Let

Applying Itô’s formula, we get

and

Then, we get

and

When , 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 that

By iteration method, next, we prove the pathwise uniqueness on .
Firstly. Applying Itô’s formula to , we get

Let , , and

Then, we have

For all , by and Fatou’s theorem, when , we have , and

Since , it shows that

Let so that

Then from (31), we obtain that

Now, we consider the equation

By the Itô formula, we have

By (34) and (35), as , we infer that

By Gronwall’s inequality, we get

From (33) and (35), by Gronwall’s inequality, we get

Hence, if for some , then
Note that if for some , then we have

and therefore there exists such that

Secondly. To prove , defining , then, for and ,

and for and for all , let , then we have

By Gronwall’s inequality and Fatou’s theorem, when , we get

Finally. Let and denote two solutions of (4) on the same space. Ones already showed that

Set

By truncation technique, we get

and

Since

then we have

where . Furthermore, we have

Therefore, we have

Let

we get

where

For , let and . Then . By Gronwall’s inequality, we have

where

By Gronwall’s inequality, for , we showed that

is continuous, and by Hölder’s inequality, we get

For some , it is easily seen that there is a sequence such that when uniformly in iff

and

By Chebychev’s inequality, we have, for ,

where

Hence, we have, for ,

Since for , we get

In summary, we proved the theorem on

Theorem 4. Suppose that Theorem 3 is satisfied. When , then

converge 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 have

Let

Next, we will prove that

is a submartingale. It is easy to see that

By martingale theory, we get

is a martingale. Therefore, we have

is a submartingale. By martingale inequality [29], we get

Furthermore, we have that

is a martingale, then we get that

is a submartingale. Furthermore, we have

So, for every , it is easy to see that

and

Let

and 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 . Let

and for

where

Defining

where are a sequence such that . By (78), we have . Furthermore

When , the last term is . Then, for , we have that

are -martingales and for .
Using

for all , we get

then we have

Next, we prove that

For , 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 , then

and it is easy to see that is proved. By similar way, we can prove
Then give

So implies

and 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 . Let

where

and

Then

for all .

Proof. Applying Itô’s formula to , we get

Suppose that and .
Defining and

and

Then, we have

Case 1 (). Substituting by , we have

Solving the above equation, we have

where and are real number independent of , but possibly based on and , then we have

since , it implies

Using Gronwall’s Inequality and Fatou’s theory, when , we obtain the result.
Case 2 ( and ). By (104),

Then we get

As before, we have

where and are real numbers not relying on , and but possibly relying on , and , then we get

Similarly 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 by

and

then, for any , we have

where and and are real number independent of and but possibly based on , and . When , we have

Furthermore

where
Let

Fix , and if there are some satisfying . Define . Since

for 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 that

where is the solution of (1) with .

Proof. Let

Applying Itô’s formula, we have

Then

By Lemma 5, we have

Therefore

Next, applying Itô’s formula, we get

Then

By Lemma 5, we also have

Therefore

Let , and we have Then

Let

and we get

where

By and , for , for some and , we have

Then, we get

By Gronwall’s inequality, we have

Then, 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]) and

Let

and

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 that

and

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

and

Next, 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 get

Due to , we show that the first integrand is negative real number for large enough . The second integrand is not larger than and

Hence, we get

and using , we have

Clearly, when for any . Therefore, we choose satisfying

It 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 have

where

and

Since

then we prove the second assertion of theorem that is

By [31, pp 142], we get

Therefore, 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 that

Then 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 of

Let 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 have

for 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 in

for every
Let and ; we get

Hence, (w.p.1) , and therefore and

and

Then, we have

whenever 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

where

and

Next, we prove that

Let denote the generator of the diffusion . Then, we have

Firstly, we consider the case which shows and hence

In case

Now, for , and , we get

if and are large enough. Then, we have

Furthermore, since and , we have

for all and . Then

if is small enough.
If , then we have

where

Hence, there is a constant satisfying

Furthermore, let

then

and it shows

provided is small enough such that .
Furthermore, we have

as and are large enough and is a suitable constant.
By (197) and (198), it has proven that there are constants and satisfying

if 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 that

if 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 by

where

Let , then converge in to the Gaussian process

where 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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