Research Article | Open Access
Effects of Dispersal for a Logistic Growth Population in Random Environments
We study a stochastic logistic model with diffusion between two patches in this paper. Using the definition of stationary distribution, we discuss the effect of dispersal in detail. If the species are able to have nontrivial stationary distributions when the patches are isolated, then they continue to do so for small diffusion rates. In addition, we use some examples and numerical experiments to reflect that diffusions are capable of both stabilizing and destabilizing a given ecosystem.
Dispersal is a ubiquitous phenomenon in the natural world. This phenomenon plays a very important role in understanding the ecological and evolutionary dynamics of populations. The theoretical studies of spatial distributions can be traced back as far as Skellam . Then many scholars have focused on the effects of spatial factors which play a crucial role in the study of stability. Some mathematical models dealt with a single population dispersing among patches (see [2–9] and references cited therein). The others dealt with competition or predator-prey interactions in patchy environments (see [10–16] and references cited therein). These models centered round local and global stability of equilibrium points, persistence, and extinction of populations.
Through the studies for the diffusion systems and the corresponding ones without diffusion, many authors have discussed the relationship between the existence of the equilibriums and their stability. Levin  showed that two unstable competitive patches can be stabilized by diffusion; Levin  also showed that diffusion can destabilize a stable system by using a prey-predator model; Allen  proved that a single species diffusion system remains weakly persistent if the strength of diffusion is small enough; Beretta and Takeuchi [5, 6] showed that small diffusion cannot change the global stability of the model. Takeuchi also proved that diffusion among patches will not destabilize single-population dynamics .
However, the most natural phenomena do not follow strictly deterministic laws, but rather oscillate randomly about some averages. That is to say populations in the real word are inevitably affected by various environmental noises which is an important phenomenon in ecosystems [17–19]. So we will consider a stochastic diffusion system which is composed of two patches and connected by diffusion. Then we want to know ‘‘how are the effects of dispersal under random environments?” According to the author's best knowledge, there are few results dealing with this problem, and stabilizing and/or destabilizing effects of dispersal remain largely unknown due to difficulties involved by random disturbances. Generally speaking, there does not have time independent equilibrium point for a stochastic system. Hence we will investigate the effects of dispersal by the concept of stationary distribution (some analogue which plays the role of the deterministic equilibrium point and reflects the stability to some extent). In this paper, we will show that diffusion cannot change the existence of stable stationary distribution for the stochastic model if the strength of diffusion is small enough. Moreover, small diffusion rates have some stabilizing effects, and large diffusion rates have some destabilizing effects on the stochastic model. That is, diffusions are capable of both stabilizing and destabilizing a given ecosystem.
2. Formulation of the Mathematical Model
The classical mathematical model describing the dynamics of a single species is the logistic model, governed by the following differential equation: This is a very popular model, and many scholars have considered various ecosystems based on this equation. If we take the dispersal phenomenon into consideration, a single population dispersing in two patches becomes where represents the population density of the species in th patch. and are the growth rate and self-competition coefficient of the population in the th patch. is a nonnegative diffusion coefficient for the species from th patch to th patch (). It is supposed that the net exchange from th patch to th patch is proportional to the difference of population densities in each patch as the usual assumption (see [2, 4–6, 20, 21]).
Taking the effect of randomly fluctuating environment into consideration, we incorporate white noises in deterministic models. We assume that fluctuations in the environments will manifest themselves mainly as fluctuations in the growth rates of the populations. We usually estimate them by average values plus error terms which follow normal distributions in practice. Let where , are mutually independent Brownian motions and and reflect the intensities of the white noises. Then, the corresponding Itô-type stochastic system which takes the dispersal phenomenon into consideration becomes Throughout this paper, unless otherwise specified, we let be a complete probability space. is a filtration defined on this space satisfying the usual conditions (It is right continuous, and contains all -null sets.).
3. Existence and Uniqueness of the Positive Solution for System (4)
Population densities and should be nonnegative by their biological significance. For this reason, we want to study system (4) in the region Now, we will show that is a positive invariant set.
Theorem 1. For any initial value , there is a unique solution to system (4) on , and the solution will remain in with probability 1.
Proof. Our proof is motivated by the works of Mao et al. . All the coefficients in system (4) are locally Lipschitz continuous; then for any given initial value , there is a unique maximal local solution on , where is an explosion time (see e.g. [23, 24]). In order to show this solution is global, we only need to prove . Let be so large that , lying within the interval . For each integer , define stopping times as follows: It is easy to see is increasing as . Set ; hence a.s. If we can prove a.s., then a.s. and a.s. for all . In other words, we only need to prove a.s.. For if this statement is false, then there are two constants and such that Consequently, there is an integer satisfying for all . Define a -function by The nonnegativity of this function can be seen from If , Itô's formula shows that There exists a constant such that on ; so we can obtain as long as . Integrating both sides from 0 to and then taking expectations yield Denote for , by (8), . Note that, for every , there is some such that equals either or , and is no less than either or . Consequently, It is follows from (13) that Letting leads to the contradiction So we must have a.s.
Theorem 1 shows that the solution of system (4) will remain in the positive cone . This nice positive invariant property provides us with a great opportunity to construct different types of the Lyapunov functions to discuss the stationary distribution for system (4) in in more detail.
4. Stationary Distribution for System (4)
Assumption B. There exists a bounded domain with regular boundary, having the following properties.(B1) In the domain and some neighborhood thereof, the smallest eigenvalue of the diffusion matrix is bounded away from zero.(B2) for all , where is a family of countable compact subsets such that ; is the mean time at which a path issuing from reaches the set .
Lemma 2 (see ). If (B) holds, then the Markov process has a stable stationary distribution confined on .
To validate (B1), it suffices to prove is uniformly elliptical in , where + ; that is, there is a positive number such that (see Chapter 3 of  and Rayleigh's principle in ). To verify (B2), it suffices to show that there exists some neighborhood and a nonnegative -function such that, for any , is negative (see).
The deterministic system (2) has two equilibrium points, namely, and . Takeuchi  has proved that this single species diffusion model has a positive and globally stable equilibrium point for any diffusion rate; the results obtained in his paper show that no diffusion rate; can change the global stability of the deterministic model.
Suppose is the equilibrium points of system (2). Then, they meet the following equations: These relations will be useful in the proof of the next theorem. Next, we will show the conditions under which system (4) exists on a stable stationary distribution and discuss the effect of diffusion on the stochastic system.
Proof. Define , where is a positive constant to be determined later. is a positive definite function for all . By Itô's formula, we can calculate Then we have Choosing , we can obtain where Then we have If we denote , , then It is well known that is an elliptic-curve when that is, Now we take to be the intersection of () and with . So, for , is negative, which implies condition (B2) is satisfied. Besides, If is bounded away from , that is, which can be satisfied when then there is a constant such that which implies condition (B1) is also satisfied. Therefore, system (4) has a stable stationary distribution confined on . These together with the positive invariant property of complete our proof.
Lemma 4 (see [32, Corollary 1]). Equation has a nontrivial stationary distribution if and only if .
Remark 5. Suppose there is no diffusion; that is, . Then condition (21) is always satisfied, and the corresponding equations, have stationary distributions when , . This is in agreement with the results in the literature  (see Lemma 4).
Remark 6. Suppose condition (24) is satisfied. Then the species has a nontrivial stationary distribution in all patches if the patches are isolated; that is, the diffusion among patches is neglected, and the species is confined to each patch. Condition (21) can be satisfied when we choose , sufficiently small. Then we have a conclusion that diffusion cannot change the existence of stable stationary distribution for stochastic system if the strength of diffusion rate is small enough.
Remark 7. An immediate consequence of condition (22) is that environmental noises are against the stationary distribution for stochastic system. If , ; that is, only species in the 2nd patch have a nontrivial stationary distribution when there is no diffusion. But we can choose sufficiently small such that conditions in Theorem 3 are satisfied, and there exists a stationary distribution for . This implies small diffusion rate has some stabilizing effects on stochastic system. However, if we choose sufficiently large, then the conditions of Theorem 3 are destroyed which implies large diffusion rate also has some destabilizing effects on stochastic models.
5. Examples and Numerical Simulation
Now we will give three examples to explain both the stabilizing and destabilizing effects of diffusion on the population dynamics. The data we used here are only some hypothetical data which are used to explain the effect of diffusion. We use the Milsteins Higher Order Method mentioned in  to numerically simulate (4): where and are the Gaussian random variables .
It is very difficult to choose parameters in the system from realistic estimation. The estimation of the parameters can be derived by some statistical methods and filtering theory which are linked to statistical problems and filtering problems. Therefore, we will only use some hypothetical parameters to verify the theoretical effects in this section.
Example 8. For system (38), we let , , , , , and . Note that and ; so for system (38), species in the 1st patch has a Dirac delta distribution with mass concentrated in 0, and species in the 2nd patch has a nontrivial distribution. (see literature .) Numerical simulations of (38) are showed in Figures 1(a) and 1(b).
Example 9. For system (4), we let , , , , , , , and . Its corresponding deterministic system (2) has a globally asymptotically stable equilibrium point . We also have So, from Theorem 3, we obtain that there is a stationary distribution with respect to for system (4) with initial value (see Figures 2(a), 2(b), 2(c), and 2(d)). The stabilizing effect of small diffusion rate can be seen clearly from this example.
Example 10. For system (4), we let , , , , , , , and . It is clear that conditions of Theorem 3 are destroyed and numerical simulations of this example showed in Figures 3(a) and 3(b). The destabilizing effect of large diffusion rate can be seen clearly from this example.
6. Concluding Remarks
The main objective of this paper is to study the effects of dispersal on stationary distribution for a stochastic logistic diffusion system. We show that the dispersal stabilizes the system when the dispersal rate is small, and destabilizes the system, when the dispersal rate is large. Our results show that small dispersal rate cannot change the existence of stationary distribution for the stochastic model such as it cannot change the global stability of the deterministic model. Though diffusions have stabilizing effects, our examples show that dispersal may also have the side effects which result in destabilization. This suggests that dispersal among patches should be regulated. Their ecological implications are that neither no diffusion nor unlimited diffusion may serve the interest of stabilizing the given ecosystem in random environments! This observation may be useful in planning and controlling of ecosystems.
The authors would like to thank the editor and the referees for their suggestions which improved the presentation of this paper. This paper was partially supported by Grants from the National Natural Science Foundation of China (no. 11171081), (no. 11171056), (no. 11126219), (no. 11001032), and (no. 11101183), Shandong Provincial Natural Science Foundation of China (Grant no. ZR2011AM004), and Natural Scientific Research Innovation Foundation in Harbin Institute of Technology (HIT.NSRIF.2011094).
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