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Dynamical behavior for a stochastic two-species competitive model

  • Changjin Xu EMAIL logo and Maoxin Liao
Published/Copyright: October 9, 2017
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Open Mathematics
From the journal Open Mathematics Volume 15 Issue 1

Abstract

This paper deals with a stochastic two-species competitive model. Some very verifiable criteria on the global stability of the positive equilibrium of the deterministic system are established. An example with its computer simulations is given to illustrate our main theoretical findings.

Keywords: Competitive model; Global stability; Stochastic perturbation
MSC 2010: 60H30; 60H10; 92D25

1 Introduction

It is well known that Schoener’s models play an important role in ecology and mathematical biology. In recent years, dynamics of Schoener’s models has attracted much attention due to its theoretical and practical significance. Many results on various Schoener’s models are available, for example, Lu and Chen [1] studied the asymptotic behaviors of a periodic Schoener’s model, Liu et al. [2] analyzed the persistence and global stability of a Schoener’s model with time delay and feedback control, Li and Wang [3] focused on permanence, periodic solution and globally asymptotic stability of a Schoener’s model with two populations, Wu et al. [4] addressed the permanence and global attractivity for a discrete Schoener’s model with delays, Lv et al. [5] discussed the dynamical properties of a stochastic two-species Schoener’s competitive model. For more details, we refer the reader to [6–10]. In 1991, Chen [11] investigated the following competitive system

dx1(t)dt=r1x1(t)[−a1−a11x1(t)−a12x2(t)+c1x1(t)+d1],dx2(t)dt=r2x2(t)[−a2−a21x1(t)−a22x2(t)+c2x2(t)+d2], (1)

where x1(t) and x2(t) represents the population densities of each species and riai(i = 1, 2) stands for its death rate, respectively, r1a11 and r2a22 denote intra-specific competition rates, r1a12 and r2a21 denote inter-specific competition rates. All the parameters ri, aij, ci, di(i, j = 1, 2) are positive constants.

Many authors argue [12–23] that in the real word the populations usually live in changing circumstances which have important effect on the growth rates. So we can think that the growth rate can be expressed as an average rate plus an error term. In view of the well known central limit theorem, we know that the error term follows a normal distribution [24–27]. Therefore the error term can be denoted by a white noise σi(t) B˙i (t), where σi2 (t) denotes the intensity of the noise and B˙i (t) denotes a standard white noise, where Bi(t) represents a Brownian motion defined on a complete probability space (Ω, F, P). Motivated by the discussed above, we can let the growth rates r1 and r2 become r1 + σ1 B˙1 (t) and r2 + σ2 B˙2 (t), respectively. Then we obtain the following stochastic system:

dx1=x1(t)[−a1−a11x1(t)−a12x2(t)+c1x1(t)+d1][r1dt+σ1dB1(t)],dx2=x2(t)[−a2−a21x1(t)−a22x2(t)+c2x2(t)+d2][r2dt+σ2dB2(t)], (2)

The initial conditions of system (2) are given by

x10=φ1(0)>0,x20=φ2(0)>0. (3)

The remainder of the paper is organized as follows: in Section 2, some sufficient conditions for the existence of global positive solution of system (2) with the initial conditions (3) are established, moreover, the sufficient condition is obtained to ensure that the equilibrium of system (2) with the initial conditions (3) is globally asymptotically stable. In Section 3, an example with its computer simulations is given to illustrate the feasibility and effectiveness of our results derived in this section. A brief conclusion is drawn in Section 4.

2 Main results

In this section, we shall present our main results.

Theorem 2.1

For any given initial value (x10, x20)∈ R+2 , system (2) has a unique global solution (x1(t), x2(t)) almost sure (a.s.), where R+2 = {x1 > 0, x2 > 0}.

Proof

Consider the following system

df1={r1[−a1−a11ef1−a12ef2+c1ef1+d1]−σ122[−a1−a11ef1−a12ef2+c1ef1+d1]2}dt+σ1[−a1−a11ef1−a12ef2+c1ef1+d1]dB1(t),df2={r2[−a2−a21ef1−a22ef2+c2ef2+d2]−σ222[−a2−a21ef1−a22ef2+c2ef2+d2]2}dt+σ2[−a2−a21ef1−a22ef2+c2ef2+d2]dB2(t) (4)

with initial value f10 = ln x10, f20 = ln x20. Obviously, the coefficients of (4) satisfy the local Lipschitz condition, then there is unique local solution (f1(t), f2(t)) on [0, τe), where τe is the explosion time. Thus, in view of Itô’s formula, (x1(t), x2(t),) = (ef1(t), ef2(t)) is the unique positive local solution to system (2) with the initial value x10 > 0, x20 > 0.

Assume that k0 > 0 is sufficiently large such that x10 and x20 lying within [1k0,k0]. For integer k > k0, define the stopping times

τk=inf{t∈[0,τe):x1(t)∉(1k,k)orx2(t)∉(1k,k)}. (5)

Let τ∞ = limk→∞τk. Then τ∞ ≤ ∞ a.s.. To complete the proof of Lemma 2.1 we need only to prove τ∞ = ∞. Assume that the statement does not hold, then there exists a constant T ≥ 0 and ϵ ∈ (0, 1) such that P{τ∞ ≤ T} > ϵ. Thus there is an integer k1 ≥ k0 such that P{τk ≤ T} ≥ ϵ. We define

V(x1,x2)=(x1−1−0.5ln⁡x1)+(x2−1−0.5ln⁡x2). (6)

If (x1(t), x2(t)) ∈ R+2 , then

dV(x1,x2)=0.5r1(x1−1)[−a1−a11x1−a12x2+c1x1+d1]dt+σ12(2−x1)8[−a1−a11x1−a12x2+c1x1+d1]2dt+0.5σ1(x1−1)[−a1−a11x1−a12x2+c1x1+d1]dB1(t)+0.5r2(x2−1)[−a2−a21x1−a22x2+c2x2+d2]dt+σ22(2−x2)8[−a2−a21x1−a22x2+c2x2+d2]2dt+0.5σ2(x2−1)[−a2−a21x1−a22x2+c2x2+d2]dB2(t)≤0.5r1[−a1x1−a11x132−a12x2x1+c1x1x1+d1+a1+a11x1+a12x2−c1x1+d1]dt+0.5r2[−a2x2−a22x232−a21x1x2+c1x2x2+d1+a2+a22x2+a21x1−c2x2+d2]dt+σ128[2a12+2a112x12+2a122x22+2(c1x1+d1)2+4a1a11x1+4a1a12x2−4a1c1x1+d1+4a11a12x1x2−4a11c1x1x1+d1−4a12c1x2x1+d1−a12x1−a112x152−a122x22x1−(c1x1+d1)2x1−2a1a11x132−2a1a12x2x1+2a1c1x1x1+d1−2a11a12x132x2+2a11c1x132x1+d1+2a12c1x2x132x1+d1]dt+σ228[2a22+2a222x22+2a212x12+2(c2x2+d2)2+4a2a22x2+4a2a21x1−4a2c2x2+d2+4a22a21x2x1−4a22c2x2x2+d2−4a21c2x1x2+d2−a22x2−a222x252−a212x12x2−(c2x2+d2)2x2−2a2a22x232−2a2a21x1x2+2a1c2x2x2+d2−2a22a12x232x1+2a22c2x232x2+d2+2a21c2x1x232x2+d2]dt+0.5σ1(x1−1)[−a1−a11x1−a12x2+c1x1+d1]dB1(t)+0.5σ2(x2−1)[−a2−a21x1−a22x2+c2x2+d2]dB2(t)≤0.5r1[c1x1x1+d1+a1+a11x1+a12x2]dt+0.5r2[c1x2x2+d1+a2+a22x2+a21x1]dt+σ128[2a12+2a112x12+2a122x22+2(c1x1+d1)2+4a1a11x1+4a1a12x2+4a11a12x1x2+2a1c1x1x1+d1+2a11c1x132x1+d1+2a12c1x2x132x1+d1]dt+σ228[2a22+2a222x22+2a212x12+2(c2x2+d2)2+4a2a22x2+4a2a21x1+4a22a21x2x1+2a1c2x2x2+d2+2a22c2x232x2+d2+2a21c2x1x232x2+d2]dt+0.5σ1(x1−1)[−a1−a11x1−a12x2+c1x1+d1]dB1(t)+0.5σ2(x2−1)[−a2−a21x1−a22x2+c2x2+d2]dB2(t)≤M1dt+M2dt+0.5σ1(x1−1)[−a1−a11x1−a12x2+c1x1+d1]dB1(t)+0.5σ2(x2−1)[−a2−a21x1−a22x2+c2x2+d2]dB2(t), (7)

where M1 and M2 are positive numbers. Integrating both sides of (7) from 0 to tk ∧ T, and taking the expectations, we have

EV(x1(τk∧T),x2(τk∧T))≤V(x10,x20)+(M1+M2)T. (8)

Set Φk = {τk ≤ T}, then it follows that P(Φk) ≥ ϵ. Since for every ϖ ∈ Φk, there exists some i such that xi(τk, ϖ) is equal to either k or 1k for i = 1, 2. Thus V(x1(τk, ϖ), x2(τk, ϖ) is no less than

min{(k−1−0.5ln⁡k),(1k−1−0.5ln⁡1k)}. (9)

In view of (8), we have

V(x10,x20)+(M1+M2)T≥E[1Φk(ϖ)V(x1(τk),x2(τn))]≥ϵmin{(k−1−0.5ln⁡k),(1k−1−0.5ln⁡1k)}, (10)

where 1Φk denotes the indicator function of Φk. Letting k → ∞, then

∞>V(x10,x20)+(M1+M2)T=∞, (11)

which is a contradiction. The proof of Theorem 2.1 is completed.□

Let E (x1∗,x2∗) be the positive equilibrium of system (2), then x1∗ and x2∗ satisfy the following equations

−a1−a11x1∗−a12x2∗+c1x1∗+d1=0,−a2−a21x1∗−a22x2∗+c2x2∗+d2=0. (12)

Now we will state our second main result of the article.

Theorem 2.2

Let

ϱ1=−r1a11+r1c1(x1∗+d1)d1+σ12x1∗a1122+c12(x1∗+d1)d1+2a11c12(x1∗+d1)d1+σ22x2∗2a222,ϱ2=−r1a12+2a11a12+2a12c12(x1∗+d1)d1−r2a21+2a22a21+2a21c22(x2∗+d2)d2,ϱ3=−r2a22+r2c2(x2∗+d2)d2+σ22x2∗a2122+c22(x2∗+d2)d2+2a22c22(x2∗+d2)d2+σ12x1∗2a122.

If ϱ1 < 0, 4ϱ1ϱ3 > ϱ22, then the equilibrium (x1∗,x2∗) of system (2) is globally asymptotically stable, that is, for any initial value x10(0) > 0, x20(0) > 0, the solution of system (2) satisfies

limt→+∞x1(t)=x1∗,limt→+∞x2(t)=x2∗,a.s.

Proof

Define the following functions

V1(x1)=∫0x1−x1∗θθ+x1∗dθ, (13)

V2(x2)=∫0x2−x2∗θθ+x2∗dθ. (14)

It follows from Itô’s formula that

LV1(x1)=r1(x1−x1∗)[−a1−a11x1−a12x2+c1x1+d1]+σ12x1∗2[−a1−a11x1−a12x2+c1x1+d1]2=r1(x1−x1∗)[a11(x1∗−x1)+a12(x2∗−x2)−c1x1∗+d1+c1x1+d1]+σ12x1∗2[a11(x1∗−x1)+a12(x2∗−x2)−c1x1∗+d1+c1x1+d1]2=−r1a11(x1−x1∗)2−r1a12(x1−x1∗)(x2−x2∗)−r1c1(x1−x1∗)2(x1∗+d1)(x1+d1)+σ12x1∗2[a112(x1−x1∗)2+a122(x2−x2∗)2+c12(x1−x1∗)2(x1∗+d1)(x1+d1)+2a11a12(x1−x1∗)(x2−x2∗)+2a11c12(x1−x1∗)2(x1∗+d1)(x1+d1)+2a12c12(x1−x1∗)(x2−x2∗)(x1∗+d1)(x1+d1)]≤[−r1a11+r1c1(x1∗+d1)d1+σ12x1∗a1122+c12(x1∗+d1)d1+2a11c12(x1∗+d1)d1](x1−x1∗)2+σ12x1∗2a122(x2−x2∗)2+[−r1a12+2a11a12+2a12c12(x1∗+d1)d1]|x1−x1∗||x2−x2∗| (15)

and

LV2(x2)=r2(x2−x2∗)[−a2−a21x1−a22x2+c2x2+d2]+σ22x2∗2[−a2−a22x1−a21x2+c2x2+d2]2=r2(x2−x2∗)[a21(x1∗−x1)+a22(x2∗−x2)−c2x2∗+d2+c2x2+d2]+σ22x2∗2[a22(x1∗−x1)+a21(x2∗−x2)−c2x2∗+d2+c2x2+d2]2=−r2a22(x2−x2∗)2−r2a21(x1−x1∗)(x2−x2∗)−r2c2(x2−x2∗)2(x2∗+d2)(x2+d2)+σ22x2∗2[a222(x1−x1∗)2+a212(x2−x2∗)2+c22(x2−x2∗)2(x2∗+d2)(x2+d2)+2a22a21(x2−x2∗)(x1−x1∗)+2a22c22(x2−x2∗)2(x2∗+d2)(x2+d2)+2a21c22(x2−x2∗)(x1−x1∗)(x2∗+d2)(x2+d2)]≤[−r2a22+r2c2(x2∗+d2)d2+σ22x2∗a2122+c22(x2∗+d2)d2+2a22c22(x2∗+d2)d2](x2−x2∗)2+σ22x2∗2a222(x1−x1∗)2+[−r2a21+2a22a21+2a21c22(x2∗+d2)d2]|x1−x1∗||x2−x2∗|. (16)

Now we define

V(t)=V1(x1)+V2(x2). (17)

Then we have

LV(t)=LV1(x1)+LV2(x2)≤[−r1a11+γ1C1(x1∗+d1)d1+σ12x1∗a1122+c12(x1∗+d1)d1+2a11c12(x1∗+d1)d1](x1−x1∗)2+σ12x1∗2a122(x2−x2∗)2+[−r1a12+2a11a12+2a12c12(x1∗+d1)d1]|x1−x1∗||x2−x2∗|+[−r2a22+r2c2(x2∗+d2)d2+σ22x2∗a2122+c22(x2∗+d2)d2+2a22c22(x2∗+d2)d2](x2−x2∗)2+σ22x2∗2a222(x1−x1∗)2+[−r2a21+2a22a21+2a21c22(x2∗+d2)d2]|x1−x1∗||x2−x2∗|=ϱ1(x1−x1∗)2+ϱ2|x1−x1∗||x2−x2∗|+ϱ3(x2−x2∗)2. (18)

Let |Z−Z∗|=(|x1−x1∗|,|x2−x2∗|)T, then it follows from (18) that

LV(t)≤12|Z−Z∗|T2ϱ1ϱ2ϱ22ϱ3|Z−Z∗|. (19)

In view of the conditions in Theorem 2.2, we can conclude that LV(t) < 0 along all trajectories in the first quadrant except x1∗,x2∗ ). Then

limt→+∞x1(t)=x1∗,limt→+∞x2(t)=x2∗.

The proof of Theorem 2.2 is completed.□

3 Computer simulations

In this section, we give an example to illustrate our main results obtained in previous sections by the Milstein method [27]. Consider the following stochastic three-species clockwise chain predator-prey model

x1,k+1=x1,k+r1x1,k[−a1−a11x2,k−a12x2,k+c1x1,k+d1]Δt+σ1x1,k[−a1−a11x1,k−a12x2,k+c1x1,k+d1]Δtξk+σ122x1,k[−a1−a11x1,k−a12x2,k+c1x1,k+d1](ξk2−1)Δt,x2,k+1=x2,k+r2x2,k[−a2−a21x1,k−a22x2,k+c2x2,k+d2]Δt+σ2x2,k[−a2−a21x1,k−a22x2,k+c2x2,k+d2]Δtηk+σ222x2,k[−a2−a21x1,k−a22x2,k+c2x2,k+d2](ηk2−1)Δt, (20)

where ξk and ηk are Gaussian random variables that follow N(0, 1). We choose r1 = 0.5, r2 = 0.2, a1 = 0.1, a2 = 0.2, a11 = 0.4, a22 = 0.3, a21 = 0.1, a22 = 0.4, c1 = 0.1, c2 = 0.2, d1 = 0.3, d2 = 0.2. Let σ12 = 0.2, σ22 = 0.15. We can easily check that all the conditions in Theorem 2.2 are fulfilled. Hence we can conclude that the equilibrium (x1∗,x2∗) is globally asymptotically stable which is shown in Figure 1. Let σ12 = 0.4, σ22 = 0.2 and also we can easily check that all the conditions in Theorem 2.2 are fulfilled. Hence we can also know that the equilibrium (x1∗,x2∗) is globally asymptotically stable which is shown in Figure 2.

Figure 1 The solutions of system (20) with σ12 $\sigma_{1}^{2} $ = 0.2 and σ22 $\sigma_{2}^{2} $ = 0.15, where the red line stands for x1 and the blue line stands for x2.
Figure 1

The solutions of system (20) with σ12 = 0.2 and σ22 = 0.15, where the red line stands for x1 and the blue line stands for x2.

Figure 2 The solutions of system (20) with σ12 $\sigma_{1}^{2} $ = 0.4 and σ22 $\sigma_{2}^{2} $ = 0.2, where the red line stands for x1 and the blue line stands for x2.
Figure 2

The solutions of system (20) with σ12 = 0.4 and σ22 = 0.2, where the red line stands for x1 and the blue line stands for x2.

4 Conclusions and further research

In this paper, we are concerned with the dynamical properties of a stochastic two-species competitive model. We have shown that under suitable conditions, the stochastic system has a unique global positive solutions for any initial values. Some sufficient conditions which ensure the global stability of the stochastic system are established. Form the viewpoint of biology, the results play an important role in practical applications. The fact that a positive equilibrium is globally asymptotically stable implies that all the species could exist. It is well known that two-species competitive models have attracted much attention due to their theoretical and practical significance. To the best of our knowledge, it is the first attempt to carry out such a study in a stochastic case. Since discontinuity is a common phenomenon in real natural world and fractal calculus is valid for discontinuous problems, then the fractal effect has received much attention and have been widely investigated. Motivated by the discussion, the fractal two-species competitive models will be our future work.

Acknowledgement

This work is supported by National Natural Science Foundation of China (No.61673008), Project of High-level Innovative Talents of Guizhou Province ([2016]5651) and Major Research Project of The Innovation Group of The Education Department of Guizhou Province (2017).

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Received: 2017-1-26
Accepted: 2017-8-24
Published Online: 2017-10-9

© 2017 Xu and Liao

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

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https://doi.org/10.1515/math-2017-0105
Keywords for this article
Competitive model; Global stability; Stochastic perturbation
Creative Commons
BY-NC-ND 4.0

Articles in the same Issue

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  2. Integrals of Frullani type and the method of brackets
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  4. Edge of chaos in reaction diffusion CNN model
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