# Effects of Dispersal for a Logistic Growth Population in Random Environments

Authors: Xiaoling Zou; Dejun Fan; Ke Wang
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

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.


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## Body

## 1. 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 [2–9] 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 [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 single-population 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 [17–19]. 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:(1)dx(t)=x(t)(r-kx(t))dt.
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
(2)dx1=[x1(r1-k1x1)+ε12(x2-x1)]dt,dx2=[x2(r2-k2x2)+ε21(x1-x2)]dt,
where xi represents the population density of the species in ith patch. ri and ki are the growth rate and self-competition coefficient of the population in the ith patch. εij is a nonnegative diffusion coefficient for the species from jth patch to ith patch (i≠j). It is supposed that the net exchange from ith patch to jth patch is proportional to the difference of population densities xi-xj 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(3)r1⟶r1+σ1B˙1(t),r2⟶r2+σ2B˙2(t),
where B1(t), B2(t) are mutually independent Brownian motions and σ1 and σ2 reflect the intensities of the white noises. Then, the corresponding Itô-type stochastic system which takes the dispersal phenomenon into consideration becomes
(4)dx1=[x1(r1-k1x1)+ε12(x2-x1)]dt+σ1x1dB1(t),dx2=[x2(r2-k2x2)+ε21(x1-x2)]dt+σ2x2dB2(t).
Throughout this paper, unless otherwise specified, we let (Ω,F,{ℱ}t≥0,P) be a complete probability space. {ℱ}t≥0 is a filtration defined on this space satisfying the usual conditions (It is right continuous, and ℱ0 contains all P-null sets.).


## 3. Existence and Uniqueness of the Positive Solution for System (4)

Population densitiesx1(t) and x2(t) should be nonnegative by their biological significance. For this reason, we want to study system (4) in the region
(5)R+2={(x1,x2)∈R2∣x1>0,x2>0}.
Now, we will show that R+2 is a positive invariant set.Theorem 1.
For any initial value(x1(0),x2(0))∈R+2, there is a unique solution (x1(t),x2(t)) to system (4) on t≥0, and the solution will remain in R+2 with probability 1.Proof.
Our proof is motivated by the works of Mao et al.  [22]. All the coefficients in system (4) are locally Lipschitz continuous; then for any given initial value (x1(0),x2(0))∈R+2, there is a unique maximal local solution (x1(t),x2(t)) on t∈[0,τe], where τe is an explosion time (see e.g. [23, 24]). In order to show this solution is global, we only need to prove τe=∞. Let k0>0 be so large that xi(0), i=1,2 lying within the interval [1/k0,k0]. For each integer k>k0, define stopping times as follows:
(6)τk=inf{t∈[0,τe]:xi(t)∉(1k,k)forsomei=1,2}.
It is easy to see τk is increasing as k→∞. Set τ∞=limk→∞τk; hence τ∞≤τe a.s. If we can prove τ∞=∞ a.s., then τe=∞ a.s. and (x1(t),x2(t))∈R+2 a.s. for all t≥0. In other words, we only need to prove τ∞=∞ a.s.. For if this statement is false, then there are two constants T>0 and ε∈(0,1) such that
(7)P{τ∞≤T}>ε.
Consequently, there is an integer k1≥k0 satisfying
(8)P{τk≤T}≥ε
for all k≥k1. Define a C2-function V:R+2→R+ by
(9)V(x1,x2)=(x1-1-0.5lnx1)+(x2-1-0.5lnx2).
The nonnegativity of this function can be seen from
(10)y-1-0.5lny≥0,ony>0.
If (x1,x2)∈R+2, Itô's formula shows that
(11)dV=∂V∂x1dx1+∂V∂x2dx2+12∂2V∂2x1σ12x12+12∂2V∂2x2σ22x22=0.5(x1-0.5-x1-1)[x1(r1-k1x1)+ε12(x2-x1)]dtnn+0.5(x2-0.5-x2-1)nn×[x2(r2-k2x2)+ε21(x1-x2)]dtnn+0.5(-0.25x1-1.5+0.5x1-2)σ12x12dtnn+0.5(-0.25x2-1.5+0.5x2-2)σ22x22dt+0.5(x1-0.5-x1-1)nn×σ1x1dB1(t)+0.5(x2-0.5-x2-1)σ2x2dB2(t)=[-0.5k1x11.5+0.5k1x1+0.5(r1-ε12-0.25σ12)x10.5nnn-0.5(r1-ε12-0.5σ12)]dtnn+[-0.5k2x21.5+0.5k2x2+0.5(r2-ε21-0.25σ22)x20.5nnnnn-0.5(r2-ε21-0.5σ22)]dtnn+[0.5(x1-0.5-x1-1)ε12x2+0.5(x2-0.5-x2-1)ε21x1]dtnn+0.5(x10.5-1)σ1dB1(t)+0.5(x20.5-1)σ2dB2(t).
There exists a constant N such that f(x)=x-0.5-x-1<N on t>0; so we can obtain
(12)dV≤[-0.5k1x11.5+0.5(k1+Nε12)x1mk+0.5(r1-ε12-0.25σ12)x10.5mk-0.5(r1-ε12-0.5σ12)x11.5]dtm+[-0.5k2x21.5+0.5(k2+Nε21)x2mklll+0.5(r2-ε21-0.25σ22)x20.5mklll-0.5(r2-ε21-0.5σ22)x21.5]dtm+0.5(x10.5-1)σ1dB1(t)+0.5(x20.5-1)σ2dB2(t)≤Mdt+0.5(x10.5-1)σ1dB1(t)m+0.5(x20.5-1)σ2dB2(t)
as long as (x1,x2)∈R+2. Integrating both sides from 0 to τk∧T and then taking expectations yield
(13)EV(x1(τk∧T),x2(τk∧T))≤V(x1(0),x2(0))+ME(τk∧T)≤V(x1(0),x2(0))+MT.
Denote Ωk={τk≤T} for k≥k1, by (8), P(Ωk)≥ε. Note that, for every ω∈Ωk, there is some i such that xi(τk,ω) equals either k or 1/k, and V(x1(τk,ω),x2(τk,ω)) is no less than either k-1-0.5ln(k) or 1/k-1-0.5ln(1/k). Consequently,
(14)V(x1(τk,ω),x2(τk,ω))≥[k-1-0.5ln(k)]∧[1k-1-0.5ln(1k)].
It is follows from (13) that
(15)V(x1(0),x2(0))+MT≥E[IΩkV(x1(τk,ω),x2(τk,ω))]≥ε([k-1-0.5ln(k)]∧[1k-1-0.5ln(1k)]).
Letting k→∞ leads to the contradiction
(16)∞>V(x1(0),x2(0))+MT=∞.
So we must have τ∞=∞ a.s.Theorem1 shows that the solution of system (4) will remain in the positive cone R+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 R+2 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
(17)d(x1(t)x2(t))=(x1(r1-k1x1)+ε12(x2-x1)x2(r2-k2x2)+ε21(x1-x2))dt+(σ1x10)dB1(t)+(0σ2x2)dB2(t).
Its diffusion matrix can be presented as
(18)A(x1,x2)=(σ12x1200σ22x22).Assumption B.
There exists a bounded domainR+2 with regular boundary, having the following properties.(B1)
In the domainU and some neighborhood thereof, the smallest eigenvalue of the diffusion matrix A(x) is bounded away from zero.(B2)
supKnExτ<∞ for all n, where Kn is a family of countable compact subsets such that R+2=⋃n=1∞‍Kn; Exτ is the mean time τ at which a path issuing from x reaches the set Kn.Lemma 2 (see [25]).
If (B) holds, then the Markov processX(t)=(x1,x2) has a stable stationary distribution μ(·) confined on R+2.To validate (B1), it suffices to proveF is uniformly elliptical in U, where Fu=b(x)·ux + [tr(A(x)uxx)]/2; that is, there is a positive number M such that
(19)∑i,j=1kaij(x)ξiξj≥M|ξ|2,x∈U,ξ∈Rk
(see Chapter 3 of [29] and Rayleigh's principle in [30]). To verify (B2), it suffices to show that there exists some neighborhood U and a nonnegative C2-function such that, for any x∈El∖U, LV is negative (see[31]).The deterministic system (2) has two equilibrium points, namely, E0(0,0) and E*(x1*,x2*). Takeuchi [9] has proved that this single species diffusion model has a positive and globally stable equilibrium point E*(x1*,x2*) for any diffusion rate; the results obtained in his paper show that no diffusion rate; can change the global stability of the deterministic model.SupposeE*(x1*,x2*) is the equilibrium points of system (2). Then, they meet the following equations:
(20)r1=k1x1*-ε12x2*x1*+ε12,r2=k2x2*-ε21x1*x2*+ε21.
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.Theorem 3.
Letσ1>0, σ2>0 such that
(21)4k1k2>ε12ε21β2,(22)r1ε21x1*+r2ε12x2*>σ122ε21x1*+σ222ε12x2*.
Then there is a stationary distribution μ(·) with respect to R+2 for system (4) with any initial value (x1(0),x2(0))∈R+2, where
(23)β=(1x1*-1x2*)2.
In addition, condition (22) can be satisfied when
(24)r1>σ122,r2>σ222.Proof.
DefineV:R+2→R+,
(25)V(x1,x2)=(x1-x1*-x1*lnx1x1*)+k(x2-x2*-x2*lnx2x2*)=V1+kV2,
where k is a positive constant to be determined later. V(x1,x2) is a positive definite function for all (x1,x2)≠(x1*,x2*). By Itô's formula, we can calculate
(26)dV1(x1,x2)=(1-x1*x1)dx1+12x1*x12σ12x12dt=(x1-x1*)[(r1-k1x1+ε12x2x1-ε12)dt=(x1-x1*)m+σ1dB1(t)(r1-k1x1+ε12x2x1-ε12)]+12x1*σ12dt=(x1-x1*)[k1x1*-ε12x2*x1*+ε12(x1-x1*)mm-k1x1+ε12x2x1-ε12]dt=+12x1*σ12dt+σ1(x1-x1*)dB1(t)=(x1-x1*)[k1(x1*-x1)+ε12(x2x1-x2*x1*)]dt=+12x1*σ12dt+σ1(x1-x1*)dB1(t)=[(x2x1-x2*x1*)-k1(x1-x1*)2+ε12(x1-x1*)(x2x1-x2*x1*)+12x1*σ12(x2x1-x2*x1*)]dt+σ1(x1-x1*)dB1(t)=LV1(x1,x2)dt+σ1(x1-x1*)dB1(t),(27)dV2(x1,x2)=(1-x2*x2)dx2+12x2*x22σ22x22dt=(x2-x2*)[(r2-k2x2+ε21x1x2-ε21)dt(x2-x2*)mm+σ2dB2(t)(r2-k2x2+ε21x1x2-ε21)]+12σ22x2*dt=(x2-x2*)[k2x2*-ε21x1*x2*+ε21-k2x2(x2-x2*)mm+ε21x1x2-ε21]dt=+12σ22x2*dt+σ2(x2-x2*)dB2(t)=(x2-x2*)[k2(x2*-x2)+ε21(x1x2-x1*x2*)]dt=+12σ22x2*dt+σ2(x2-x2*)dB2(t)=[12σ22x2*-k2(x2-x2*)2+ε21(x2-x2*)(x1x2-x1*x2*)mm+12σ22x2*]dt+σ2(x2-x2*)dB2(t)=LV2(x1,x2)dt+σ2(x2-x2*)dB2(t).
Then we have
(28)dV(x1,x2)=dV1(x1,x2)+kdV2(x1,x2)=LV(x1,x2)dt+σ1(x1-x1*)dB1(t)+σ2(x2-x2*)dB2(t).
Choosing k=ε12/ε21, we can obtain
(29)LV(x1,x2)=-k1(x1-x1*)2-k2ε12ε21(x2-x2*)2+12x1*σ12+12ε12ε21σ22x2*+ε12×[(x1-x1*)(x2x1-x2*x1*)+(x2-x2*)(x1x2-x1*x2*)]-k1(x1-x1*)2-k2ε12ε21(x2-x2*)2+σ-ε12Γ(x1,x2),
where
(30)σ=12x1*σ12+12ε12ε21σ22x2*,Γ(x1,x2)=-[(x1-x1*)(x2x1-x2*x1*)+(x2-x2*)(x1x2-x1*x2*)]=-(x1-x1*)(x1*x2-x1x2*)x1x1*-(x2-x2*)(x1x2*-x2x1*)x2x2*=-(x1-x1*)(x2(x1*-x1)+x1(x2-x2*))x1x1*m-(x2-x2*)(x1(x2*-x2)+x2(x1-x1*))x2x2*=x2(x1-x1*)2x1x1*+x1(x2-x2*)2x2x2*m-(1x1*+1x2*)(x1-x1*)(x2-x2*)≥(21x1*x2*-1x1*-1x2*)(x1-x1*)(x2-x2*)=-(1x1*-1x2*)2(x1-x1*)(x2-x2*).
Then we have
(31)LV≤-k1(x1-x1*)2-k2ε12ε21(x2-x2*)2+σ+ε12(1x1*-1x2*)2(x1-x1*)×(x2-x2*)≜R(x1,x2).
If we denote x¯1=x1-x1*,  x¯2=x2-x2*, then
(32)LV≤-k1x¯12-k2ε12ε21x¯22+σ+ε12(1x1*-1x2*)2x¯1x¯2≜G(x¯1,x¯2).
It is well known that G(x¯1,x¯2)=0 is an elliptic-curve when
(33)I2=|k1-ε12β2-ε12β2k2ε12ε21|>0,
that is,
(34)4k1k2>ε12ε21β2.
Now we take U to be the intersection of (G(x¯1,x¯2)≥0) and R+2 with U⊆R+2. So, for (x1,x2)∈R+2/U, LV is negative, which implies condition (B2) is satisfied. Besides, If U is bounded away from (0,0), that is,
(35)limx1,x2→0LV(x1,x2)=-k1(x1*)2-k2ε12ε21(x2*)2+σ<0,
which can be satisfied when
(36)k1ε21(x1*)2+k2ε12(x2*)2=r1ε21x1*+r2ε12x2*>σ122ε21x1*+σ222ε12x2*,
then there is a constant M>0 such that
(37)∑i,j=12‍aij(x1,x2)ξiξj=σ12x12ξ12+σ22x22ξ22≥M|ξ|2∀(x1,x2)∈U,ξ∈R2,
which implies condition (B1) is also satisfied. Therefore, system (4) has a stable stationary distribution μ(·) confined on R+2. These together with the positive invariant property of R+2 complete our proof.Lemma 4 (see [32, Corollary 1]).
Equationdx(t)=x(t)(r-kx(t))dt+αx(t)dB(t) has a nontrivial stationary distribution if and only if α2<2r.Remark 5.
Suppose there is no diffusion; that is,ε12=ε21=0. Then condition (21) is always satisfied, and the corresponding equations,
(38)dx1=x1(r1-k1x1)dt+σ1x1dB1(t),dx2=x2(r2-k2x2)dt+σ2x2dB2(t),
have stationary distributions when r1>σ12/2, r2>σ22/2. 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 ε12, ε21 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 r1<σ12/2, r2>σ22/2; that is, only species in the 2nd patch have a nontrivial stationary distribution when there is no diffusion. But we can choose ε21 sufficiently small such that conditions in Theorem 3 are satisfied, and there exists a stationary distribution for (x1(t),x2(t)). This implies small diffusion rate has some stabilizing effects on stochastic system. However, if we choose ε21 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 [33] to numerically simulate (4):
(39)x1k+1=x1k+[x1k(r1-k1x1k)+ε12(x2k-x1k)]Δt+σ1x1kΔtξk+σ122x1k(Δtξk2-Δt),x2k+1=x2k+[x2k(r2-k2x2k)+ε21(x1k-x2k)]Δt+σ2x2kΔtηk+σ222x2k(Δtηk2-Δt),
where ξk and ηk are the Gaussian random variables N(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.Example 8.
For system (38), we let r1=0.9, k1=1.2, σ1=2, r2=1.1, k2=0.2, and σ2=0.8. Note that r1<σ12/2 and r2>σ22/2; 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 [32].) Numerical simulations of (38) are showed in Figures 1(a) and 1(b).Numerical simulations of Example8 (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.
(a)
(b)Example 9.
For system (4), we let r1=0.9, k1=1.2, ε12=0.1, σ1=2, r2=1.1, k2=0.2, ε21=0.4, and σ2=0.8. Its corresponding deterministic system (2) has a globally asymptotically stable equilibrium point E*(x*,y*)=(1,4). We also have
(40)σ1=2>0,σ2=0.8>0,β=(1x1*-1x2*)2=14,4k1k2=0.96>0.0025=ε12ε21β2,r1ε21x1*+r2ε12x2*=0.8>0.528=σ122ε21x1*+σ222ε12x2*.
So, from Theorem 3, we obtain that there is a stationary distribution μ(·) with respect to R+2 for system (4) with initial value (x1(0),x2(0))∈R+2 (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.Numerical simulations of Example9. In this example, we choose ε21 sufficiently small such that conditions in Theorem 3 are satisfied, and there is a stationary distribution for (x1(t),x2(t)); (c) and (d) are distributions of x1 and x2, respectively.
(a)
(b)
(c)
(d)Example 10.
For system (4), we let r1=0.9, k1=1.2, ε12=0.1, σ1=2, r2=1.1, k2=0.2, ε21=5, and σ2=0.8. 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.Numerical simulations of Example10. In this example, we choose ε21 sufficiently large such that conditions in Theorem 3 are destroyed which show the destabilizing effect of large diffusion rate.
(a)
(b)


## 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.



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