Effects of Dispersal for a Logistic Growth Population in Random Environments

and Applied Analysis 3 = [− 0.5k 1 x 1.5 1 + 0.5k 1 x 1 + 0.5 (r 1 − ε 12 − 0.25σ 2 1 ) x 0.5 1 − 0.5 (r 1 − ε 12 − 0.5σ 2 1 )] dt + [− 0.5k 2 x 1.5 2 + 0.5k 2 x 2 + 0.5 (r 2 − ε 21 − 0.25σ 2 2 ) x 0.5 2 − 0.5 (r 2 − ε 21 − 0.5σ 2 2 )] dt + [0.5 (x −0.5 1 − x −1 1 ) ε 12 x 2 + 0.5 (x −0.5 2 − x −1 2 ) ε 21 x 1 ] dt + 0.5 (x 0.5 1 − 1) σ 1 dB 1 (t) + 0.5 (x 0.5 2 − 1) σ 2 dB 2 (t) . (11) There exists a constant N such that f(x) = x − x < N on t > 0; so we can obtain dV ≤ [− 0.5k 1 x 1.5 1 + 0.5 (k 1 + Nε 12 ) x 1 + 0.5 (r 1 − ε 12 − 0.25σ 2 1 ) x 0.5 1 − 0.5 (r 1 − ε 12 − 0.5σ 2 1 ) ] dt + [− 0.5k 2 x 1.5 2 + 0.5 (k 2 + Nε 21 ) x 2 + 0.5 (r 2 − ε 21 − 0.25σ 2 2 ) x 0.5 2 − 0.5 (r 2 − ε 21 − 0.5σ 2 2 ) ] dt + 0.5 (x 0.5 1 − 1) σ 1 dB 1 (t) + 0.5 (x 0.5 2 − 1) σ 2 dB 2 (t) ≤ Mdt + 0.5 (x0.5 1 − 1) σ 1 dB 1 (t) + 0.5 (x 0.5 2 − 1) σ 2 dB 2 (t) (12) as long as (x 1 , x 2 ) ∈ R 2 + . Integrating both sides from 0 to τ k ∧ T and then taking expectations yield EV (x 1 (τ k ∧ T) , x 2 (τ k ∧ T)) ≤ V (x 1 (0) , x 2 (0)) + ME (τ k ∧ T) ≤ V (x 1 (0) , x 2 (0)) + MT. (13) Denote Ω k = {τ k ≤ T} for k ≥ k 1 , by (8), P(Ω k ) ≥ ε. Note that, for every ω ∈ Ω k , there is some i such that x i (τ k , ω) equals either k or 1/k, and V(x 1 (τ k , ω), x 2 (τ k , ω)) is no less than either√k − 1 − 0.5 ln(k) or 1/√k − 1 − 0.5 ln(1/k). Consequently, V (x 1 (τ k , ω) , x 2 (τ k , ω)) ≥ [√k − 1 − 0.5 ln (k)] ∧ [ 1 √k − 1 − 0.5 ln(1 k )] . (14) It is follows from (13) that V (x 1 (0) , x 2 (0)) + MT ≥ E [I Ωk V (x 1 (τ k , ω) , x 2 (τ k , ω))] ≥ ε ([√k − 1 − 0.5 ln (k)] ∧ [ 1 √k − 1 − 0.5 ln(1 k )]) . (15) Letting k → ∞ leads to the contradiction ∞ > V(x 1 (0) , x 2 (0)) + MT = ∞. (16) So we must have τ ∞ = ∞ a.s. Theorem 1 shows that the solution of system (4) will remain in the positive cone R + . 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 R + in more detail. 4. Stationary Distribution for System (4) In order to prove our main results, we require some results in [25], and the technique we used here is motivated by [26–28]. System (4) can be rewritten as d(x1 (t) x 2 (t) ) = ( x 1 (r 1 − k 1 x 1 ) + ε 12 (x 2 − x 1 ) x 2 (r 2 − k 2 x 2 ) + ε 21 (x 1 − x 2 ) ) dt


Introduction
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 [1].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][3][4][5][6][7][8][9] and references cited therein).The others dealt with competition or predator-prey interactions in patchy environments (see [10][11][12][13][14][15][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 [10] showed that two unstable competitive patches can be stabilized by diffusion; Levin [11] also showed that diffusion can destabilize a stable system by using a prey-predator model; Allen [4] 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 singlepopulation dynamics [9].
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][18][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.

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  1 (),  2 () are mutually independent Brownian motions and  1 and  2 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 (Ω, , {F} ≥0 , ) be a complete probability space.{F} ≥0 is a filtration defined on this space satisfying the usual conditions (It is right continuous, and F 0 contains all -null sets.).

Existence and Uniqueness of the Positive Solution for System (4)
Population densities  1 () and  2 () should be nonnegative by their biological significance.For this reason, we want to study system (4) in the region Now, we will show that  2 + is a positive invariant set.
Theorem 1 shows that the solution of system (4) will remain in the positive cone  2 + .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  2 + in more detail.

Stationary Distribution for System (4)
In order to prove our main results, we require some results in [25], and the technique we used here is motivated by [26][27][28].System (4) can be rewritten as Its diffusion matrix can be presented as Assumption B. There exists a bounded domain  2 + 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) sup      < ∞ for all , where   is a family of countable compact subsets such that  2 + = ⋃ ∞ =1   ;    is the mean time  at which a path issuing from  reaches the set   .
To verify (B2), it suffices to show that there exists some neighborhood  and a nonnegative  2 -function such that, for any  ∈   \,  is negative (see [31]).
The deterministic system (2) has two equilibrium points, namely,  0 (0, 0) and  * ( * 1 ,  * 2 ).Takeuchi [9] has proved that this single species diffusion model has a positive and globally stable equilibrium point  * ( * 1 ,  * 2 ) 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  * ( * 1 ,  * 2 ) 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.
Then there is a stationary distribution (⋅) with respect to  2

+
for system (4) with any initial value ( 1 (0),  2 (0)) ∈  2 + , where In addition, condition (22) can be satisfied when Proof.Define  : where  is a positive constant to be determined later.( 1 ,  2 ) is a positive definite function for all ( 1 ,  2 ) ̸ = ( * 1 ,  * 2 ).By Itô's formula, we can calculate Then we have d Abstract and Applied Analysis 5 Choosing  =  12 / 21 , we can obtain where Then we have It is well known that ( 1 ,  2 ) = 0 is an elliptic-curve when that is, Now we take  to be the intersection of (( 1 ,  2 ) ≥ 0) and which can be satisfied when then there is a constant  > 0 such that which implies condition (B1) is also satisfied.Therefore, system (4) has a stable stationary distribution (⋅) confined on  2 + .These together with the positive invariant property of  2 + complete our proof.
Remark 5. Suppose there is no diffusion; that is,  12 =  21 = 0. Then condition ( 21) is always satisfied, and the corresponding equations, have stationary distributions when  1 >  2 1 /2,  2 >  2 2 /2.This is in agreement with the results in the literature [32] (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  12 ,  21 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  1 <  2  1 /2,  2 >  2 2 /2; that is, only species in the 2nd patch have a nontrivial stationary distribution when there is no diffusion.But we can choose  21 sufficiently small such that conditions in Theorem 3 are satisfied, and there exists a stationary distribution for ( 1 (),  2 ()).This implies small diffusion rate has some stabilizing effects on stochastic system.However, if we choose  21 sufficiently large, then the conditions of Theorem 3 are destroyed which implies large diffusion rate also has some destabilizing effects on stochastic models.

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 [33] to numerically simulate (4): where   and   are the Gaussian random variables (0, 1).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.

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.

Figure 1 :
Figure 1: Numerical simulations of Example 8 (there is no diffusion).Species in the 1st patch has a Dirac delta distribution with mass concentrated in 0, and species in the 2nd patch has a stationary distribution.

Figure 2 :
Figure 2: Numerical simulations of Example 9.In this example, we choose  21 sufficiently small such that conditions in Theorem 3 are satisfied, and there is a stationary distribution for ( 1 (),  2 ()); (c) and (d) are distributions of  1 and  2 , respectively.

Figure 3 :
Figure 3: Numerical simulations of Example 10.In this example, we choose  21 sufficiently large such that conditions in Theorem 3 are destroyed which show the destabilizing effect of large diffusion rate.