# On a Stochastic Lotka-Volterra Competitive System with Distributed Delay and General Lévy Jumps

Authors: Lijie Zhang; Chun Lu; Hui Liu
Journal: Mathematical Problems in Engineering
 (2016)

---

## Abstract

This paper considers a stochastic competitive system with distributed delay and general Lévy jumps. Almost sufficient and necessary conditions for stability in time average and extinction of each population are established under some assumptions. And two facts are revealed: both stability in time average and extinction have closer relationships with the general Lévy jumps, firstly; and secondly, the distributed delay has no effect on the stability in time average and extinction of the stochastic system. Some simulation figures, which are obtained by the split-stepθ-method to discretize the stochastic model, are introduced to support the analytical findings.


---

## Body

## 1. Introduction

In recent years, delay differential equations has been used in the study of population dynamics. A famous competitive system with distributed delay can be expressed by(1)dy1tdt=y1tb1-a11y1t-a12∫-τ20y2t+θdμ2θ,dy2tdt=y2tb2-a21∫-τ10y1t+θdμ1θ-a22y2t,where yi(t) denotes the size of the ith population, bi, aij, and τi are all positive constants, and μi is a probability measure on [-τi,0]. There is an extensive literature concerned with the dynamics of (1) and we here only mention Kuang and Smith [1], Faria [2], Freedman and Wu [3], Bereketoglu and Győri [4], and Gopalsamy [5] among many others. In particular, Kuang (see [6, p. 231]) claimed that if Ψ1>0 and Ψ2>0, then model (1) has a positive equilibrium x∗=(x1∗,x2∗)=(Ψ1/Ψ,Ψ2/Ψ) which is globally asymptotically stable, where Ψ=a11a22-a12a21, Ψ1=b1a22-b2a12, and Ψ2=b2a11-b1a21. It is important to point out that if Ψ1>0 and Ψ2>0, then Ψ>0.In the real world, the intrinsic growth rates of many species are always disturbed by environmental noises (see, e.g., [7–10]), which was recognized by many scholars in recent years (see, e.g., [11–14]). In particular, May [7] has pointed out that, due to environmental noises, the birth rates, carrying capacity, and other parameters involved in the system should be stochastic. In this paper, we assume that the parameters b1 and b2 are stochastic; then by the central limit theorem, we can replace b1 and b2 by (2)b1⟶b1+σ1B˙1t,b2⟶b2+σ2B˙2t,where, for i=1,2, Bi(t) represents a standard Brownian motion defined on a complete probability space (Ω,F,P) and σi2 is the intensity of the noise.On the other hand, the population systems may suffer sudden environmental perturbations, that is, some jump type stochastic perturbations, for example, earthquakes, hurricanes, and epidemics. Some scholars have concentrated on the population systems with compensator jumps, and some significant and interesting results have been obtained (see, e.g., [15–19]). Bao et al. [15, 16] did pioneering work in this field. In addition, Zou et al. [20–22] introduce a general Lévy jumps, which is more reasonable and complicated than the compensator jumps from the viewpoint of biomathematics (see [20]), into population models for the first time. However, there are no articles introducing the general Lévy jumps into population models with distributed delay, to the best of our knowledge. Motivated by these, we consider the famous stochastic competitive system with distributed delay and general Lévy jumps:(3)dy1t=y1t-b1-a11y1t--a12∫-τ20y2t-+θdμ2θdt+σ1y1t-dB1t+y1t-∫Yγ1uNdt,du,dy2t=y2t-b2-a21∫-τ10y1t-+θdμ1θ-a22y2t-dt+σ2y2t-dB2t+y2t-∫Yγ2uNdt,du,where yi(t-)=lims↑t⁡yi(s), N(dt,du) is a real-valued Poisson counting measure with characteristic measure λ on a measurable subset Y of R¯+ with λ(Y)<+∞, N~(dt,du)=N(dt,du)-λ(du)dt, γ(u) is bounded function, and γ(u)>-1, u∈Y; furthermore, we assume that Bi(t) is independent of N. Let the initial data ξ(t)=ξ1t,ξ2t∈C-τ,0,R+2, where C-τ,0;R+2 represents the family of continuous functions from [-τ,0] to R+2 with the norm ξi=sup-τ≤t≤0⁡ξit, i=1,2, τ=max⁡{τ1,τ2}. Other parameters are defined and required as before.For convenience, we introduce the following notations:(4)R+2=g=g1,g2∈R2∣gi>0,i=1,2,ft=t-1∫0tfsds,ηi=∫Yln⁡11+γiuλdu,i=1,2,Ψ~1=0.5a22σ12-0.5a12σ22+a22η1-a12η2,Ψ~2=0.5a11σ22-0.5a21σ12+a11η2-a21η1.Moreover, we impose the following assumptions in this paper.Assumption 1.
There exists a positive constantc such that ln⁡1+γu≤c for γu>-1.In this paper, we consider a stochastic competitive system with distributed delay and general Lévy jumps. Unlike the deterministic system, the stochastic system does not have an interior equilibrium. Therefore, we cannot investigate the stability of the stochastic system. In Section2, we show that the solution to system (3) will tend to a point in time average. Furthermore, we establish almost sufficient and necessary conditions for stability in time average and extinction of each population. In Section 3, we present an example to illustrate our mathematical findings. Section 4 gives the conclusions and future directions of the research.


## 2. Main Content

Lemma 2 (see Liu et al. [23]).
Suppose thatz(t)∈C(Ω×[0,+∞),R+).(i)
If there exist two positive constantsT and ρ0 such that ln⁡z(t)≤ρt-ρ0∫0tz(s)ds+∑i=12αiBi(t) for all t≥T, where αi, i=1,2, are constants, then(5)limsupt→+∞⁡zt≤ρρ0a.s.,ifρ≥0;limt→+∞⁡zt=0a.s.,ifρ<0.(ii)
If there exist three positive constantsT, ρ, and ρ0 such that ln⁡z(t)≥λt-ρ0∫0tz(s)ds+∑i=12αiBi(t) for all t≥T, then liminft→+∞⁡〈z(t)〉≥ρ/ρ0 a.s.In order for the model to be significant, we shall show that the solution is global and nonnegative. However, theorem of existence and uniqueness ([24–28]) is not satisfied in system (3). By using method established by Mao et al. [8], we will show existence and uniqueness of the global positive solution of system (3).Lemma 3.
Let Assumption1 hold. For any given initial value ξ(t)=ξ1t,ξ2t∈C-τ,0,R+2; then system (3) has a unique positive solution x(t)=y1t,y2t on t≥-τ a.s. and the solution satisfies(6)limsupt→+∞⁡ln⁡yitln⁡t≤1a.s.,i=1,2.Proof.
The proof is similar to Han et al. [29] by defining(7)Vx=V1x+V2x,where(8)V1x=x1-1-0.5ln⁡x1,V2x=x2-1-0.5ln⁡x2.In addition, applying the inequality, for i=1,2,(9)∫0t∫-τ0xi2s+θ-dμθds=∫-τ0dμθ∫θtxi2s-ds≤∫-τ0dμθ∫-τtxi2s-ds≤∫-τ0ξi2s-ds+∫0txi2s-ds.So we omit it here. Now let us prove inequality (6).
Case  1(i=1). For any t≥0, applying the generalized Itô’s formula [30] to (3) results in (10)
d
e
t
ln
⁡
y
1
=
e
t
ln
⁡
y
1
d
t
+
e
t
d
ln
⁡
y
1
=
e
t
ln
⁡
y
1
+
b
1
-
0.5
σ
1
2
+
∫
Y
ln
⁡
1
+
γ
1
u
λ
d
u
-
a
11
y
1
-
a
12
∫
-
τ
2
0
y
2
t
-
+
θ
d
μ
2
θ
d
t
+
σ
1
d
B
1
t
+
∫
Y
ln
⁡
1
+
γ
1
u
N
~
d
t
,
d
u
≤
e
t
ln
⁡
y
1
+
b
1
-
0.5
σ
1
2
+
∫
Y
ln
⁡
1
+
γ
1
u
λ
d
u
-
a
11
y
1
d
t
+
e
t
σ
1
d
B
1
t
+
e
t
∫
Y
ln
⁡
1
+
γ
1
u
N
~
d
t
,
d
u
.Thus(11)etln⁡y1t≤ln⁡y10+∫0tesln⁡y1s+b1-0.5σ12+∫Yln⁡1+γ1uλdu-a11y1sds+∫0tesσ1dB1s+∫0tes∫Yln⁡1+γ1uN~ds,du.The rest of proof is analogous with Lemma  4.4 in [15]; we omitted it here.
Case  2 (i=2). The proof is similar to Case  1; we left out it here. The proof is complete.Theorem 4.
For system (3), we suppose that Assumption 1,Ψ1>0 and Ψ2>0, holds. (I)
Ifb1<0.5σ12+η1 and b2<0.5σ22+η2, then both y1 and y2 are extinctive almost surely (a.s.); that is, limt→+∞⁡yi(t)=0 a.s., i=1,2.(II)
Ifb1>0.5σ12+η1 and b2<0.5σ22+η2, then y2 is extinctive a.s. and y1 is stable in time average a.s.; that is, (12)limt→+∞⁡t-1∫0ty1sds=b1-0.5σ12-η1a11,a.s..(III)
Ifb1<0.5σ12+η1 and b2>0.5σ22+η2, then y1 is extinctive a.s. and y2 is stable in time average a.s.; that is, (13)limt→+∞⁡t-1∫0ty2sds=b2-0.5σ22-η2a22,a.s..(IV)
Ifb1>0.5σ12+η1, b2>0.5σ22+η2,
(A)
IfΨ1>Ψ~1 and Ψ2<Ψ~2, then y2 is extinctive a.s. and y1 is stable in time average a.s.: (14)limt→+∞⁡t-1∫0ty1sds=b1-0.5σ12-η1a11,a.s.;(B)
IfΨ1<Ψ~1 and Ψ2>Ψ~2, then y1 is extinctive a.s. and y2 is stable in time average a.s.: (15)limt→+∞⁡t-1∫0ty2sds=b2-0.5σ22-η2a22,a.s.;(C)
IfΨ1>Ψ~1 and Ψ2>Ψ~2, then both y1 and y2 are stable in time average a.s.: (16)limt→+∞⁡t-1∫0ty1sds=Ψ1-Ψ~1Ψ,limt→+∞⁡t-1∫0ty2sds=Ψ2-Ψ~2Ψ,a.s.Proof.
Applying Itô’s formula [30] to the first equality in (3), we get (17)ln⁡y1t-ln⁡y10=b1-0.5σ12-η1t-a11∫0ty1sds-a12∫0t∫-τ20y2s+θdμ2θds+σ1B1t+∫0t∫Yln⁡1+γ1uN~ds,du.Making use of the Fubini theorem and a substitution technique, we have (18)∫0t∫-τ20y2s+θdμ2θds=∫-τ20dμ2θ∫0ty2s+θds=∫-τ20dμ2θ∫θt+θy2sds=∫-τ20dμ2θ∫θ0y2sds+∫0ty2sds+∫tt+θy2sds=∫-τ20dμ2θ∫θ0y2sds+∫0ty2sds+∫-τ20dμ2θ∫tt+θy2sds.Therefore, we derive(19)ln⁡y1t-ln⁡y10=b1-0.5σ12-η1t-a11∫0ty1sds-a12∫0ty2sds-a12∫-τ20dμ2θ∫θ0y2sds-a12∫-τ20dμ2θ∫tt+θy2sds+σ1B1t+∫0t∫Yln⁡1+γ1uN~ds,du.Similarly,(20)ln⁡y2t-ln⁡y20=b2-0.5σ22-η2t-a22∫0ty2sds-a21∫0ty1sds-a21∫-τ10dμ1θ∫θ0y1sds-a21∫-τ10dμ1θ∫tt+θy1sds+σ2B2t+∫0t∫Yln⁡1+γ2uN~ds,du.(I) Assume that b1<0.5σ12+η1 and b2<0.5σ22+η2. From (9), (21)t-1ln⁡y1ty10≤b1-0.5σ12-η1+t-1σ1B1t+t-1∫0t∫Yln⁡1+γ1uN~ds,du.By the strong law of large numbers for martingales, we therefore have limt→+∞⁡Bi(t)/t=0 a.s., i=1,2; meanwhile, we define, for t≥0, Mi(t)=∫0t∫Yln⁡1+γi(u)N~(ds,du), i=1,2. Under Assumption 1,〈Mi〉(t)=∫0t∫Yln⁡(1+γi(u)2λ(du)ds≤c2tλ(Y), making use of the strong law of large numbers for local martingales (see, e.g., [31]), we then derive (22)limt→+∞⁡1t∫0t∫Yln⁡1+γiuN~ds,du=0a.s.,and b1<0.5σ12+η1 that (23)limsupt→+∞⁡t-1ln⁡y1t≤b1-0.5σ12-η1<0.Consequently, limt→+∞⁡y1(t)=0, a.s. Similarly, by (19), we can show that if b2<0.5σ22+η2, then limt→+∞⁡y2(t)=0, a.s.
(II) Suppose thatb1>0.5σ12+η1 and b2<0.5σ22+η2. Since b2<0.5σ22+η2, then, by (I), limt→+∞⁡y2(t)=0, a.s. Hence, making use of (9), for arbitrary ε>0, there is T>0 such that, for t≥T, (24)-ε2≤a12t-1∫0t∫-τ20y2s+θdμ2θds≤ε2,-ε2≤t-1ln⁡y10≤ε2.Substituting the above inequalities into (19), we can see that, for t≥T,(25)ln⁡y1t≤b1-0.5σ12-η1+εt-a11∫0ty1sds+σ1B1t+∫0t∫Yln⁡1+γ1uN~ds,du,(26)ln⁡y1t≥b1-0.5σ12-η1-εt-a11∫0ty1sds+σ1B1t+∫0t∫Yln⁡1+γ1uN~ds,du.Since b1>0.5σ12+η1, we can choose ε sufficiently small such that b1-0.5σ12-η1-ε>0. Applying (i) and (ii) in Lemma 2 to (25) and (26), respectively, we derive(27)b1-0.5σ12-η1-εa11≤liminft→+∞⁡y1t≤limsupt→+∞⁡y1t≤b1-0.5σ12-η1+εa11,a.s.Let ε→0. Then, we have limt→+∞⁡〈y1(t)〉=b1-0.5σ12-η1/a11, a.s.
The proof of (III) is homogeneous with (II) by symmetry; hence it is omitted.
Now let us prove (IV). Fori=1,2, consider the following equation:(28)dzit=zitbi-aiizitdt+σizitdBit+z1t-∫YγiuNdt,du,ziθ=yiθ,θ∈-τ,0.In virtue of the classic stochastic comparison theorem [32], we can find that(29)y1t≤z1t,y2t≤z2t.Since bi>0.5σi2+ηi, i=1,2, similar to the proof of (II), we can show that (30)limt→+∞⁡zit=limt→+∞⁡t-1∫0tzisds=bi-0.5σi2-η2aii,a.s.,i=1,2,limt→+∞⁡zit=limt→+∞⁡t-1∫0t+θzisds=bi-0.5σi2-η2aii,a.s.,-τ≤θ≤0,i=1,2.Thus(31)⁡limt→+∞⁡t-1∫t+θtz1sds=limt→+∞⁡t-1∫0tz1sds-t-1∫0t+θz1sds=0,limt→+∞⁡t-1∫t+θtz2sds=0,a.s.,which, together with (29), implies that(32)limt→+∞⁡t-1∫t+θty1sds=0,limt→+∞⁡t-1∫t+θty2sds=0,a.s.On the other hand, calculating (2.5)×a11-(2.4)×a21 deduces(33)a11ln⁡y2ty20=-a11a21∫-τ10dμ1θ∫θ0y1sds-a11a21∫-τ10dμ1θ∫tt+θy1sds+a21a12∫-τ20dμ2θ∫θ0y2sds+a21a12∫-τ20dμ2θ∫tt+θy2sds+a21ln⁡y1ty10+Ψ2-Ψ~2t-Ψ∫0ty2sds-a21σ1B1t-a21∫0t∫Yln⁡1+γ1uN~ds,du+a11σ2B2t+a11∫0t∫Yln⁡1+γ2uN~ds,du.By virtue of (6) and (32), for arbitrary ε>0, there is T>0 such that, for t≥T, (34)t-1a21ln⁡y1ty10<ε3,t-1a11ln⁡y20<ε3,-a11a21t-1∫-τ10dμ1θ∫θ0y1sds-a11a21t-1∫-τ10dμ1θ∫tt+θy1sds+a21a12t-1∫-τ20dμ2θ∫θ0y2sds+a21a12t-1∫-τ20dμ2θ∫tt+θy2sds≤ε3.Substituting the above inequalities into (33) results in(35)a11ln⁡y2t≤Ψ2-Ψ~2+εt-Ψ∫0ty2sds-a21σ1B1t+a11σ2B2+a11∫0t∫Yln⁡1+γ2uN~ds,du,for t>T. Meanwhile, calculating (2.4)×a22-(2.5)×a12 yields(36)a22ln⁡y1ty10=-a11a21∫-τ10dμ1θ∫θ0y1sds-a11a21∫-τ10dμ1θ∫tt+θy1sds+a12a21∫-τ10dμ1θ∫θ0y1sds-a12a21∫-τ10dμ1θ∫tt+θy1sds+a12ln⁡y2ty20+Ψ1-Ψ~1t-Ψ∫0ty1sds+a22σ1B1t+a22∫0t∫Yln⁡1+γ1uN~ds,du-a12σ2B2t-a12∫0t∫Yln⁡1+γ2uN~ds,du.Using the same way, by (36) we can have that, for t>T,(37)a22ln⁡y1t≤Ψ1-Ψ~1+εt-Ψ∫0ty1sds+a22σ1B1t+a22∫0t∫Yln⁡1+γ1uN~ds,du-a12σ2B2t-a12∫0t∫Yln⁡1+γ2uN~ds,du.(A) Suppose Ψ1>Ψ~1 and Ψ2<Ψ~2. Note that Ψ2<Ψ~2, and then let ε be sufficiently small such that Ψ2-Ψ~2+ε<0. Applying (i) in Lemma 2 to (35) gives limt→+∞⁡y2(t)=0, a.s. The proof of limt→+∞⁡〈y1(t)〉=(b1-0.5σ12-η1)/a11, a.s., is similar to (II) and hence is omitted.
The proof of (B) is similar to (A) by symmetry and hence is left out.
(C) Suppose thatΨ1>Ψ~1 and Ψ2>Ψ~2. Since Ψ2>Ψ~2, it then follows from (33) and Lemma 2 that(38)limsupt→+∞⁡y2t≤Ψ2-Ψ~2+εΨ,a.s.Making use of the arbitrariness of ε, we can see that(39)limsupt→+∞⁡y2t≤Ψ2-Ψ~2Ψ,a.s.It follows from (37), Lemma 2, and the arbitrariness of ε that(40)limsupt→+∞⁡y1t≤Ψ1-Ψ~1Ψ,a.s.likewise. Let ε be sufficiently small such that a11Ψ2-Ψ~2/Ψ-ε>0. When (32) and (39) are used in (19), we get(41)t-1ln⁡y1t=t-1ln⁡y10+b1-0.5σ12-η1-a11y1t-a12y2t+σ1B1tt+t-1∫0t∫Yln⁡1+γ1uN~ds,du-a12t-1∫-τ20dμ2θ∫θ0y2sds+∫-τ20dμ2θ∫tt+θy2sds≥b1-0.5σ12-η1-ε-a11y1t-a12limsupt→+∞⁡y2t+σ1B1tt+t-1∫0t∫Yln⁡1+γ1uN~ds,du≥b1-0.5σ12-η1-ε-a11y1t-a12Ψ2-Ψ~2Ψ+σ1B1tt+t-1∫0t∫Yln⁡1+γ1uN~ds,du=a11Ψ1-Ψ~1Ψ-ε-a11y1t+σ1B1tt+t-1∫0t∫Yln⁡1+γ1uN~ds,du,for sufficiently large t. In virtue of (ii) in Lemma 2 and the arbitrariness of ε, we get(42)liminft→+∞⁡y1t≥Ψ1-Ψ~1Ψ,a.s.Similarly, substituting (32) and (40) into (20) brings about liminft→+∞⁡y2t≥Ψ2-Ψ~2/Ψ, a.s. This, together with (39), (40), and (42), means limt→+∞⁡〈y1(t)〉=Ψ1-Ψ~1/Ψ and limt→+∞⁡〈y2(t)〉=Ψ2-Ψ~2/Ψ, a.s.Remark 5.
It is important to designate that ifb1>0.5σ12+η1, b2>0.5σ22+η2, and Ψ>0, then Ψ1<Ψ~1 and Ψ2<Ψ~2 cannot hold simultaneously.Remark 6.
Theorem4 implies an important fact that when -1<γi(u)<0, i=1,2, the jump process can result in extinction of the population yi(t), for example, earthquakes and hurricanes, and when γi(u)>0, i=1,2, the jump process is always advantage for the population yi(t), for example, ocean red tide.Remark 7.
From the perspective of the condition in Theorem4, the distributed delay does not influence some the properties including extinction and stability in time average.


## 3. Numerical Simulations

In this section, we employ the split-stepθ-method, whose approximate solution is mean-square convergent with order p=0.5 (see [32, 33]), to discretize (3). Here, we choose the initial data ξ(t)=0.3et,0.4et, b1=0.59, b2=0.51, a11=0.8, a12=0.39, a21=0.51, a22=0.69, τ1=τ2=0.3, Y=(0,+∞), and λ(Y)=1. Then Ψ=0.36. The main difference between the conditions of the following Case  1–Case  4 is that the values of γ1(u) and γ2(u) are different. In Case  1, we let γ1(u)=γ2(u)=0 and σ12=σ22=0. Then by virtue of Kuangs work [6], we have that the positive equilibrium x∗=Ψ1/Ψ,Ψ2/Ψ=0.612,0.277 is globally asymptotically stable. Figure 1(a) verifies this. In Case  2, we set γ1(u)=-0.1, γ2(u)=-0.2, and σ12=0.2,σ22=0.4. Then Ψ1>Ψ~1=-0.03216 and Ψ2=0.1<Ψ~2=0.2386. In view of (A) in Theorem 4, y2 goes to extinction and(43)limt→+∞⁡t-1∫0ty1sds=b1-0.5σ12-η1a11=0.5950.8=0.74.Figure 1(b) confirms this. In Case  3, we choose γ1(u)=-0.8, γ2(u)=0.01, σ12=0.2, and σ22=0.4. Then Ψ1=0.22<Ψ~1=1.11 and Ψ2=0.1>Ψ~2=0.03. It follows from (B) in Theorem 4 that y1 goes to extinction and (44)limt→+∞⁡t-1∫0ty2sds=b2-0.5σ22-η2a22=0.410.7=0.59.See Figure 1(c). In Case  4, we let γ1(u)=-0.03, γ2(u)=0.01, σ12=0.2, and σ22=0.4, that is, Ψ1=0.22>Ψ~1=-0.22 and Ψ2=0.1>Ψ~2=0.095. According to (C) in Theorem 4, we obtain (45)limt→+∞⁡t-1∫0ty1sds=Ψ1-Ψ~1Ψ=1.22,limt→+∞⁡t-1∫0ty2sds=Ψ2-Ψ~2Ψ=0.0139.Figure 1(d) validates this.Figure 1
The horizontal axis and the vertical axis in this and following figures represent the timet and the populations size (step size Δt=0.001).
(a) (b) (c) (d)


## 4. Conclusions and Remarks

This paper investigates a stochastic competitive system with distributed delay and general Lévy jumps. Under the assumptionΨ>0, the almost complete parameter analysis is fulfilled in detail. Our results imply that the general Lévy jumps can significantly change the properties of population models.Some interesting and significant topics deserve our further engagement. One may put forward a more realistic and sophisticated model to integrate the colored noise into the model [10, 11, 34]. Another significant problem is devoted to stochastic model with infinite delays and general Lévy jumps. We will leave these for future investigation.It should also be mentioned that “stability in time average” is not a good definition of persistence for stochastic population models. Some papers have introduced more appropriate definitions of permanence for stochastic population models, that is, stochastically persistent in probability or stochastic permanence (see, e.g., [35–37]). We will research these kinds of permanence of model (3) in detail in our following study.



---

*Source: 3407463-2016-12-15.xml*