Volume 2013, Issue 1 256249

Research Article

Open Access

Persistence and Nonpersistence of a Nonautonomous Stochastic Mutualism System

Peiyan Xia,

Peiyan Xia

School of Basic Sciences, Changchun University of Technology, Changchun, Jilin 130021, China ccut.edu.cn

School of Mathematics and Statistics, Northeast Normal University, Changchun, Jilin 130024, China nenu.edu.cn

Search for more papers by this author

Xiaokun Zheng,

Xiaokun Zheng

School of Mathematics and Statistics, Northeast Normal University, Changchun, Jilin 130024, China nenu.edu.cn

Search for more papers by this author

Daqing Jiang,

Corresponding Author

Daqing Jiang

[email protected]

School of Mathematics and Statistics, Northeast Normal University, Changchun, Jilin 130024, China nenu.edu.cn

Search for more papers by this author

Peiyan Xia,

Peiyan Xia

School of Basic Sciences, Changchun University of Technology, Changchun, Jilin 130021, China ccut.edu.cn

School of Mathematics and Statistics, Northeast Normal University, Changchun, Jilin 130024, China nenu.edu.cn

Search for more papers by this author

Xiaokun Zheng,

Xiaokun Zheng

School of Mathematics and Statistics, Northeast Normal University, Changchun, Jilin 130024, China nenu.edu.cn

Search for more papers by this author

Daqing Jiang,

Corresponding Author

Daqing Jiang

[email protected]

School of Mathematics and Statistics, Northeast Normal University, Changchun, Jilin 130024, China nenu.edu.cn

Search for more papers by this author

First published: 19 February 2013

https://doi.org/10.1155/2013/256249

Citations: 15

Academic Editor: Jifeng Chu

Share a link

Email
Wechat
Bluesky

Abstract

In this paper, a two-species nonautonomous stochastic mutualism system is investigated. The intrinsic growth rates of the two species at time t are estimated by respectively. Viewing the different intensities of the noises σ_i(t), i = 1, 2 as two parameters at time t, we conclude that there exists a global positive solution and the pth moment of the solution is bounded. We also show that the system is permanent, including stochastic permanence, persistence in mean, and asymptotic boundedness in time average. Besides, we show that the large white noise will make the system nonpersistent. Finally, we establish sufficient criteria for the global attractivity of the system.

1. Introduction

For more than three decades, mutualism of multispecies has attracted the attention of both mathematicians and ecologists. By definition, in a mutualism of multispecies, the interaction is beneficial for the growth of other species. Lotka-Volterra mutualism systems have long been used as standard models to mathematically address questions related to this interaction. Among these, nonautonomous Lotka-Volterra mutualism models are studied by many authors, see [1–7] and references therein. The classical nonautonomous Lotka-Volterra mutualism system can be expressed as follows:

()

where x_i(t), i = 1,2, …, n is the density of the ith population at time t, r_i(t) > 0, i = 1,2, …, n is the intrinsic growth rate of the ith population at time t, r_i(t)/a_ii(t) > 0, i = 1,2, …, n is the carrying capacity at time t, and coefficient a_ij(t) > 0, i, j = 1,2, …, n describes the influence of the jth population upon the ith population at time t.

It is shown in [1] that if different conditions hold (see conditions (a)–(e) in [1]), then the solution of system (1) is bounded, permanent, extinct, and global attractive, respectively. However, when the intrinsic growth rate and coefficient a_ij(t) are periodic, it is shown in [3] that there exists positive periodic solution and almost periodic solutions are obtained.

From another point of view, environmental noise always exists in real life. It is an interesting problem, both mathematically and biologically, to determine how the structure of the model changes under the effect of a fluctuating environment. Many authors studied the biological models with stochastic perturbation, see [8–12] and references therein. In [8] Ji et al. discussed the following two-species stochastic mutualism system

()

where B_i(t), i = 1,2 are mutually independent one dimensional standard Brownian motions with B_i(0) = 0, i = 1,2, and σ_i, i = 1,2 are the intensities of white noise. It is shown in [8] that if a₁₁a₂₂ > a₁₂a₂₁ then there is a unique nonnegative solution of system (2). For small white noise there is a stationary distribution of (2) and it has ergodic property. Biologically, this implies that with small perturbation of environment, the stability of the two species varies with the intensity of white noise, and both species will survive.

However, almost all known stochastic models assume that the growth rate and the carrying capacity of the population are independent of time t. In contrast, the natural growth rates of many populations vary with t in real situation, for example, due to the seasonality. As a matter of fact, nonautonomous stochastic population systems have recently been studied by many authors, for example, [13–17].

In this paper we consider the system

()

where r_i(t), a_ij(t), σ_i(t), i, j = 1,2 are all continuous bounded nonnegative functions on [0, +∞). The objective of our study is to investigate the long-time behavior of system (3). As in [8], we mainly discuss when the system is persistent and when it is not under a fewer conditions. More specifically, we show that there is a positive solution of system (3) and its pth moment bounded in Section 2. In Section 3, we deduce the persistence of the system. If the white noise is not large such that

, we will prove that the solution of system (3) is a stochastic persistence. In addition, we show that every component of the solution is persistent in mean. We further deduce that every component of the solution of system (3) is an asymptotic boundedness in mean. In Section 4, we show that larger white noise will make system (3) nonpersistent. Finally, we study the global attractivity of system (3).

Throughout this paper, unless otherwise specified, let (Ω, {ℱ_t} _t≥0, P) be a complete probability space with a filtration {ℱ_t} _t≥0 satisfying the usual conditions (i.e., it is right continuous and ℱ₀ contains all P-null sets). Let

be the positive cone of R², namely,

. If x ∈ Rⁿ, its norm is denoted by

. If f(t) is a continuous bounded function on [0, +∞), we use the notation sup

()

2. Existence and Uniqueness of the Positive Solution

In population dynamics, the first concern is that the solution should be nonnegative. In order to do that a stochastic differential equation can have a unique global (i.e., no explosion at any finite time) solution for any given initial value, the coefficients of the equation are generally required to satisfy the linear growth condition and local Lipschitz condition (Mao [18]). However, the coefficients of system (3) do not satisfy the linear growth condition, though they are locally Lipschitz continuous, so the solution of system (3) may explode at a finite time. Following the way developed by Mao et al. [19], we show that there is a unique positive solution of (3).

Theorem 1. Assume that . Then, there is a unique positive solution x(t) = (x₁(t), x₂(t)) of system (3) on t ≥ 0 for any given initial value , and the solution will remain in with probability 1, namely, for all t ≥ 0 almost surely.

The proof of Theorem 1 is similar to [8]. But it is skilled in taking the value of ϵ. We show it here.

Proof. Since the coefficients of the equation are locally Lipschitz continuous, for any given initial value there is an unique local solution x(t) = (x₁(t), x₂(t)) on t ∈ [0, τ_e), where τ_e is the explosion time. To show that this solution is global, we need to show that τ_e = ∞ a.s. Let m₀ > 1 be sufficiently large for every component of x(0) lying within the interval [1/m₀, m₀]. For each integer m ≥ m₀, define the stopping time

()

where throughout this paper we set inf ∅ = ∞ (as usual ∅ denotes the empty set). Clearly, τ_m is increasing as m → ∞. Set τ_∞ = lim _m→∞ τ_m, whence τ_∞ ≤ τ_e a.s. If we can show that τ_∞ = ∞ a.s., then τ_e = ∞ a.s. and

a.s. for all t ≥ 0. In other words, to complete the proof, all we need to show is that τ_∞ = ∞ a.s. If this statement is false, there is a pair of constant T > 0 and ε ∈ (0,1) such that

()

Hence, there is an integer m₁ ≥ m₀ such that

()

We define

()

By Itô′s formula, we have

()

where

()

According to Young inequality, note that

, where

, then,

()

Since

, we obtain

and

. Hence, K is a positive constant. Integrating both sides of (9) from 0 to τ_m∧T, we therefore obtain

()

Whence, taking expectations yields

()

Set Ω_m = {τ_m ≤ T} for m ≥ m₁ and by (7), P(Ω_m) ≥ ε. Note that for every ω ∈ Ω_m, there is x₁(τ_m, ω) or x₂(τ_m, ω) equals either m or 1/m, and therefore

()

where lim _m→∞ h(m) = ∞. It then follows from (13) that

()

where

is the indicator function of Ω_m. Letting m → ∞ leads to the contradiction

()

so we must have τ_∞ = ∞ a.s. This completes the proof of Theorem 1.

Remark 2. By Theorem 1, we observe that for any given initial value , there is a unique solution x(t) = (x₁(t), x₂(t)) of system (3) on t ≥ 0 and the solution will remain in with probability 1, no matter how large the intensities of white noise are. So, under the same assumption there is an global unique positive solution of the corresponding deterministic system of system (3).

Next, we show that the pth moment of the solution of system (3) is bounded in time average.

Theorem 3. Assume that . Then there exists a positive constant K(p) such that the solution x(t) of system (3) has the following property:

()

where c₁, c₂ satisfy

()

Proof. By Itô′s formula, we have

()

where

, and

()

where

. According to Young inequality, we obtain

()

Thus, we have

()

Since

, there exist two positive constants c₁, c₂ which satisfy

()

Therefore,

()

From (23) and the values of ϵ₁, ϵ₂, we obtain

()

which implies that

and

. Let

()

then we have

()

Hence, we get

()

By the comparison theorem, we get

()

which implies that there is a T₀ > 0, such that

()

Besides, note that

is continuous, then there is a

such that

()

Let

, then

()

3. Persistence

Theorem 1 shows that the solution of system (3) will remain in the positive cone if . Studying a population system, we pay more attention on whether the system is persistent. In this section, we first show that the solution is a stochastic permanence. Next we show that the solution is persistent in time average. Moreover, we show that the solution x(t) of system (3) is an asymptotic boundedness in time average.

3.1. Stochastic Permanence

Let y(t) be the solution of a randomized nonautonomous competitive equation:

()

where B_i(t), i = 1,2, …, n, are independent standard Brownian motions, y(0) = y₀ > 0 while y₀ is independent of B(t), and b_i(t), a_ij(t), σ_i(t) are all continuous bounded nonnegative functions on [0, +∞).

Lemma 4 (see [15].)Assume that , then for any given initial value , the solution y(t) of (36) has the properties

()

where H is a constant, θ is an arbitrary positive constant satisfying

()

Let N(t) be the solution of a randomized nonautonomous logistic equation

()

where B(t) is a 1-dimensional standard Brownian motion, N(0) = N₀ > 0, and N₀ is independent of B(t).

Lemma 5 (see [13].)Assume that a(t), b(t), and α(t) are bounded continuous functions defined on [0, ∞), a(t) > 0 and b(t) > 0. Then there exists a unique continuous positive solution of (36) for any initial value N(0) = N₀ > 0, which is global and represented by

()

From Lemma 4 we have the following.

Lemma 6. Assume that a^l − ((α^u) ²/2) > 0, then for any given initial value N(0) ∈ R₊, the solution N(t) of (36) has the properties

()

where H is a constant, θ is positive constant satisfying

()

Let

be the solution of

()

where B_i(t), i = 1,2, are independent standard Brownian motions, ϕ(0) = ϕ₀ > 0, and

, r_i(t), a_ii(t), σ_i(t), i = 1,2 are all continuous bounded nonnegative functions on [0, +∞). From Lemma 4 it is easy to know the following.

Lemma 7. Assume that , then for any given initial value , the solution ϕ(t) of (40) has the properties

()

where H_i, i = 1,2 are two constants, θ is positive constant satisfying

()

Lemma 8. Assume that , then for any given initial value , the solution x(t) of system (3) has the properties

()

where H_i, i = 1,2 are two constants, θ is positive constant satisfying

()

Proof. Equation (43) follows directly from the classical comparison theorem of stochastic differential equations (see [20]). Thus, we obtain

()

Definition 9. System (3) is said to be stochastically permanent if for any ε ∈ (0,1), there exists a pair of positive constants δ = δ(ϵ) and M = M(ϵ) such that for any initial value , the solution obeys

()

Theorem 10. Assume that , then system (3) is stochastically permanent.

The proof is a simple application of the Chebyshev inequality, we omit it.

3.2. Persistence in Time Average

Theorem 10 shows that if the white noise is not large, the solution of system (3) is survive with large probability. In this part, we show x(t) is persistence in mean.

Lemma 11. Assume that , then for any given initial value , the solution ϕ(t) of (40) has the properties

()

where z(t) = (z₁(t), z₂(t)) is the solution of

()

Proof. From Lemma 5, we know

()

Similarly, we have

()

Lemma 12. Assume that , then for any given initial value , the solution z(t) of (49) has the following properties

()

where

are the solutions of the two equations, respectively,

()

Proof. Let , are the solutions of SDE (53) and (54), respectively, with the positive initial value z(0). By Lemma 5, we know

()

Thus,

()

By the classical comparison theorem of ordinary differential equations, we know

()

Lemma 13. Assume that , then for any given initial value , the solution ϕ(t) of (40) has the properties

()

Proof. By Lemma 12, we know

()

So, we have

()

Let

then

()

Since σ_i(t), i = 1,2 are bounded, then

()

By the strong law of large numbers, we know

()

Thus,

()

Then from (60) we obtain

()

Lemma 14. Assume that , then for any given initial value , the solution ϕ(t) of (40) has the properties

()

Proof. By Itô′s formula, we have

()

Integrating both sides of this equation from 0 to t yields

()

By Lemma 13, we know that

()

Hence,

()

Definition 15. System (3) is said to be persistent in time average if

()

Theorem 16. Assume that and , then the solution x(t) of system (3) with any initial value has the following property:

()

and so system (3) is persistent in time average.

Proof. By Lemma 8, we know that

()

where ϕ(t) = (ϕ₁(t), ϕ₂(t)) is the solution of system (40). Moreover, by Lemma 14 we know that

()

Hence, by Lemma 13 we know that

()

3.3. Asymptotic Boundedness of Integral Average

Theorem 16 shows that every component of the solution x(t) of system (3) will survive forever in time average, if the white noise is not large. In this part, we further deduce that every component of x(t) of system (3) will be an asymptotic boundedness in time average. Before we give the result, we do some preparation work.

Lemma 17. Let f ∈ C[[0, ∞) × Ω, (0, ∞)], F(t) ∈ ((0, ∞) × Ω, R). If there exist positive constants λ₀ and λ such that

()

and lim _t→∞ (F(t)/t) = 0 a.s., then

()

Proof. The proof is similar to the proof of Lemma in [21]. Let

()

Since f ∈ C[[0, ∞) × Ω, (0, ∞)], φ(t) is differentiable on [0, ∞) and

()

Substituting dφ(t)/dt and φ(t) into (76), we obtain the following:

()

thus

()

Note that lim _t→∞(F(t)/t) = 0 a.s., then for 0 < ε < min {1, λ}, ∃T = T(ω) > 0 and Ω_ε ⊂ Ω such that P(Ω_ε) > 1 − ε and F(t) ≥ −εt, t ≥ T, ω ∈ Ω_ε. Then we have

()

Integrating inequality (82) from 0 to t results in the following:

()

This inequality can be rewritten into

()

Taking the logarithm of both sides and dividing both sides by t(>0) yields

()

Then,

()

Letting ε → ∞ yields

()

This finishes the proof of the Lemma.

Theorem 18. Assume that and , then the solution x(t) of system (3) with any initial value has the property

()

where

()

Proof. To prove the results, we only need to prove

()

By Itô′s formula, we have

()

First, we prove (91). Integrating both sides of (92) from 0 to t yields

()

where

. Since x_i(t) > 0, i = 1,2, hence

()

So we have

()

By Theorem 16, we know that

()

Obviously,

()

Hence, we have

()

Similarly, we have

()

Next, we prove that (90) is true. Taking integration both sides of (92) from 0 to t, we have

()

By Theorem 16 we know that

()

then for any ε > 0, there is a T(ω) > 0 such that

()

for t > T(ω). It follows from (100) that, for t > T(ω),

()

From Lemma 17, we have

()

Similarly, we have

()

Continuing this process, we obtain two sequences M_n, N_n (n = 1,2, …) such that

()

By induction, we can easily show that M_n+1 > M_n, N_n+1 > N_n, n = 1,2, …, that is, sequences {M_n, n = 1,2, …} and {N_n, n = 1,2, …} are nondecreasing. Moreover, note that (98) and (99), then the sequences {M_n, n = 1,2, …} and {N_n, n = 1,2, …}, have upper bounds. Therefore, there are two positive M, N such that

()

which together with (106) implies

()

Letting ε → 0 yields

()

Hence,

()

which is as required.

4. Nonpersistence

In this section, we discuss the dynamics of system (3) as the white noise is getting larger. We show that system (3) will be nonpersistent if the white noise is large, which does not happen in the deterministic system.

Definition 19. System (3) is said to be nonpersistent, if there are positive constants q₁, q₂ such that

()

Theorem 20. Assume that and , then system (3) is nonpersistent, where .

Proof. Since x_i(t) > 0, i = 1,2 and , from (93) we have

()

where

which together with

()

implies

()

If K₁ < 0, then there must be

()

Hence, system (3) is nonpersistent.

Theorem 21. Assume that and , then system (3) is nonpersistent, where .

Here we omit the proof of Theorem 21 which is similar to the proof of Theorem 20.

Remark 22. If , then the conditions in Theorems 20 and 21 are obviously satisfied, respectively. That is to say, the large white noise will lead to the population system being non-persistent.

5. Global Attractivity

In this section, we turn to establishing sufficient criteria for the global attractivity of stochastic system (3).

Definition 23. Let x(t), y(t) be two arbitrary solutions of system (3) with initial values , respectively. If

()

then we say system (3) is globally attractive.

Theorem 24. Assume that , then system (3) is globally attractive.

Proof. Let x(t), y(t) be two arbitrary solutions of system (3) with initial values . By the Itô′s formula, we have

()

Then,

()

Since

, there exist two positive constants c₁, c₂ which satisfy

()

Thus,

Consider a Lyapunov function V(t) defined by

()

A direct calculation of the right differential d⁺V(t) of V(t) along the ordinary differential equation (119) leads to

()

where

. Integrating both sides of (122) form 0 to t, we have

()

Let t → ∞, we obtain

()

Note that u(t) = x(t) − y(t). Clearly, u(t) ∈ C(R₊, R²) a.s. It is straightforward to see from (124) that

()

Next, we prove that

()

By Theorem 3 we obtain that the pth moment of the solution of system (3) is bounded, the following proof is similar to the proof of Theorem 6.2 in [15] and hence is omitted.

Acknowledgments

The work was supported by the Ph.D. Programs Foundation of Ministry of China (no. 200918), NSFC of China (no. 10971021), and Program for Changjiang Scholars and Innovative Research Team in University.

References

1 Redheffer R., Nonautonomous Lotka-Volterra systems. I, Journal of Differential Equations. (1996) 127, no. 2, 519–541, https://doi.org/10.1006/jdeq.1996.0081, 1389408, ZBL0856.34056.
10.1006/jdeq.1996.0081
Web of Science® Google Scholar
2 Redheffer R., Nonautonomous Lotka-Volterra systems. II, Journal of Differential Equations. (1996) 132, no. 1, 1–20, https://doi.org/10.1006/jdeq.1996.0168, 1418497, ZBL0864.34043.
10.1006/jdeq.1996.0168
Web of Science® Google Scholar
3 Tineo A., On the asymptotic behavior of some population models. II, Journal of Mathematical Analysis and Applications. (1996) 197, no. 1, 249–258, https://doi.org/10.1006/jmaa.1996.0018, 1371287, ZBL0872.34027.
10.1006/jmaa.1996.0018
Web of Science® Google Scholar
4 Teng Z. D., Nonautonomous Lotka-Volterra systems with delays, Journal of Differential Equations. (2002) 179, no. 2, 538–561, https://doi.org/10.1006/jdeq.2001.4044, 1885679, ZBL1013.34072.
10.1006/jdeq.2001.4044
Web of Science® Google Scholar
5 Wei F. Y. and Ke W., Global stability and asymptotically periodic solution for nonautonomous cooperative Lotka-Volterra diffusion system, Applied Mathematics and Computation. (2006) 182, no. 1, 161–165, https://doi.org/10.1016/j.amc.2006.03.044, 2292028, ZBL1113.92062.
10.1016/j.amc.2006.03.044
Web of Science® Google Scholar
6 Nakata Y. and Muroya Y., Permanence for nonautonomous Lotka-Volterra cooperative systems with delays, Nonlinear Analysis. Real World Applications. (2010) 11, no. 1, 528–534, https://doi.org/10.1016/j.nonrwa.2009.01.002, 2570572, ZBL1186.34119.
10.1016/j.nonrwa.2009.01.002
Web of Science® Google Scholar
7 Abdurahman X. and Teng Z., On the persistence of a nonautonomous n-species Lotka-Volterra cooperative system, Applied Mathematics and Computation. (2004) 152, no. 3, 885–895, https://doi.org/10.1016/S0096-3003(03)00605-2, 2062784.
10.1016/S0096-3003(03)00605-2
Web of Science® Google Scholar
8 Ji C. Y., Jiang D. Q., Liu H., and Yang Q. S., Existence, uniqueness and ergodicity of positive solution of mutualism system with stochastic perturbation, Mathematical Problems in Engineering. (2010) 2010, 18, 684926, https://doi.org/10.1155/2010/684926, 2670476, ZBL1204.34065.
10.1155/2010/684926
Web of Science® Google Scholar
9 Ji C. Y. and Jiang D. Q., Persistence and non-persistence of a mutualism system with stochastic perturbation, Discrete and Continuous Dynamical Systems A. (2012) 32, no. 3, 867–889, https://doi.org/10.3934/dcds.2012.32.867, 2851882, ZBL1233.92076.
10.3934/dcds.2012.32.867
Web of Science® Google Scholar
10 Mao X. R., Sabanis S., and Renshaw E., Asymptotic behaviour of the stochastic Lotka-Volterra model, Journal of Mathematical Analysis and Applications. (2003) 287, no. 1, 141–156, https://doi.org/10.1016/S0022-247X(03)00539-0, 2010262, ZBL1048.92027.
10.1016/S0022-247X(03)00539-0
Web of Science® Google Scholar
11 Mao X. R., Yuan C. G., and Zou J. Z., Stochastic differential delay equations of population dynamics, Journal of Mathematical Analysis and Applications. (2005) 304, no. 1, 296–320, https://doi.org/10.1016/j.jmaa.2004.09.027, 2124664, ZBL1062.92055.
10.1016/j.jmaa.2004.09.027
Web of Science® Google Scholar
12 Liu M. and Wang K., Survival analysis of a stochastic cooperation system in a polluted environment, Journal of Biological Systems. (2011) 19, no. 2, 183–204, https://doi.org/10.1142/S0218339011003877, 2819510, ZBL1228.92074.
10.1142/S0218339011003877
Web of Science® Google Scholar
13 Jiang D. Q. and Shi N. Z., A note on nonautonomous logistic equation with random perturbation, Journal of Mathematical Analysis and Applications. (2005) 303, no. 1, 164–172, https://doi.org/10.1016/j.jmaa.2004.08.027, 2113874, ZBL1076.34062.
10.1016/j.jmaa.2004.08.027
Web of Science® Google Scholar
14 Jiang D. Q., Shi N. Z., and Li X. Y., Global stability and stochastic permanence of a non-autonomous logistic equation with random perturbation, Journal of Mathematical Analysis and Applications. (2008) 340, no. 1, 588–597, https://doi.org/10.1016/j.jmaa.2007.08.014, 2376180, ZBL1140.60032.
10.1016/j.jmaa.2007.08.014
Web of Science® Google Scholar
15 Li X. Y. and Mao X. R., Population dynamical behavior of non-autonomous Lotka-Volterra competitive system with random perturbation, Discrete and Continuous Dynamical Systems A. (2009) 24, no. 2, 523–545, https://doi.org/10.3934/dcds.2009.24.523, 2486589, ZBL1161.92048.
10.3934/dcds.2009.24.523
CAS Web of Science® Google Scholar
16 Liu M. and Wang K., Persistence and extinction in stochastic non-autonomous logistic systems, Journal of Mathematical Analysis and Applications. (2011) 375, no. 2, 443–457, https://doi.org/10.1016/j.jmaa.2010.09.058, 2735535, ZBL1214.34045.
10.1016/j.jmaa.2010.09.058
Web of Science® Google Scholar
17 Liu M. and Wang K., Asymptotic properties and simulations of a stochastic logistic model under regime switching II, Mathematical and Computer Modelling. (2012) 55, no. 3-4, 405–418, https://doi.org/10.1016/j.mcm.2011.08.019, 2887385.
10.1016/j.mcm.2011.08.019
Google Scholar
18 Mao X. R., Stochastic Differential Equations and Applications, 1997, Horwood, Chichester, UK.
Google Scholar
19 Mao X., Marion G., and Renshaw E., Environmental Brownian noise suppresses explosions in population dynamics, Stochastic Processes and their Applications. (2002) 97, no. 1, 95–110, https://doi.org/10.1016/S0304-4149(01)00126-0, 1870962, ZBL1058.60046.
10.1016/S0304-4149(01)00126-0
Web of Science® Google Scholar
20 Wantanabe I. N., Stochastic Differential Equations and Diffusion Processes, 1981, North-Holland, Amsterdam, The Netherlands.
Google Scholar
21 Liu H. P. and Ma Z. E., The threshold of survival for system of two species in a polluted environment, Journal of Mathematical Biology. (1991) 30, no. 1, 49–61, https://doi.org/10.1007/BF00168006, 1130788.
10.1007/BF00168006
CAS PubMed Web of Science® Google Scholar

Citing Literature

All articles

Persistence and Nonpersistence of a Nonautonomous Stochastic Mutualism System

Abstract

1. Introduction

2. Existence and Uniqueness of the Positive Solution

3. Persistence

3.1. Stochastic Permanence

3.2. Persistence in Time Average

3.3. Asymptotic Boundedness of Integral Average

4. Nonpersistence

5. Global Attractivity

Acknowledgments

References

Citing Literature

References

Information

About Wiley Online Library

Help & Support

Opportunities

Connect with Wiley

Persistence and Nonpersistence of a Nonautonomous Stochastic Mutualism System

Abstract

1. Introduction

2. Existence and Uniqueness of the Positive Solution

3. Persistence

3.1. Stochastic Permanence

3.2. Persistence in Time Average

3.3. Asymptotic Boundedness of Integral Average

4. Nonpersistence

5. Global Attractivity

Acknowledgments

References

Citing Literature

References

Related

Information