# Stochastic Periodic Solution and Persistence of a Nonautonomous Impulsive System with Nonlinear Self-Interaction

Authors: Weili Kong; Yuanfu Shao
Journal: Discrete Dynamics in Nature and Society
 (2020)

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## Abstract

Considering periodic environmental changes and random disturbance, we explore the dynamical behaviors of a stochastic competitive system with impulsive and periodic parameters in this paper. Firstly, by use of extreme-value theory of quadratic function and constructing suitable functional, we study the existence of periodic Markovian process. Secondly, by comparison theory of the stochastic differential equation, we study the extinction and permanence in the mean of all species. Thirdly, applying an important lemma, we investigate the stochastic persistence of this system. Finally, some numerical simulations are given to illustrate the main results.


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## Body

## 1. Introduction

In real world, the habitat space and food for species are relatively scare; hence, the interspecific competition phenomenon exists extensively. Usually, the competitive interaction is assumed to be linear, see [1, 2]. However, some experimental tests showed that the term of self-interaction may be nonlinear, so Ayala et al. [3] proposed the following model:(1)dx1t=x1tr1t−a11tx1t−a12tx2t−b1tx12t,dx2t=x2tr2t−a21tx1t−a22tx2t−b2tx22t.In practice, the environmental white noise is almost everywhere and inevitably affects the growth of species. Thus, in the progress of mathematical modeling, random disturbance is introduced to reveal the effect of white noise [4–9]. For ecological system, some discrete effects often appear at some short time interval, such as periodic spraying pesticides, harvesting, and stocking, which affect the growth of species and are often modeled by impulsive parameters. In last decades, many impulsive systems have been proposed and many good results have been reported, see, e.g., [10–15] and references cited therein. For example, Tan et al. [15] investigated the existence of solution, stochastic permanence of the following impulsive model:(2)dx1t=x1tr1t−a11tx1t−a12tx2t−b1tx12t+σ1tx1tdω1t,t≠tk,dx2t=x2tr2t−a21tx1t−a22tx2t−b2tx22t+σ2tx2tdω2t,t≠tk,x1tk+=1+λ1kx1tk,t=tk,x2tk+=1+λ2kx2tk,t=tk,where σi2i=1,2 represents the density of white noise and ω1t and ω2t are independent standard Brownian motions defined on the probability space Ω,F,Ftt≥0,P, where Ftt≥0 denotes a filtration, which is right continuous, and all p-null set is contained in F0. For biological meanings of other parameters, refer to [15]. By constructing suitable functional and using inequality techniques, the stochastic permanence of (2) and extinction of x1t were studied.In natural world, due to individual life cycle and seasonal variation, the carrying capacity of species, birth rate, and other parameters always present periodic changes for population systems [16–18]. For the determinate biological system, the existence of periodic solution is a very important dynamical behavior [10, 19–22]. Similarly, for stochastic system, it is very interesting to study the existence of stochastic periodic solution (periodic Markovian process). On the other hand, the extinction and permanence in the mean and stochastic persistence in probability are all very important dynamical behaviors (see [12, 13, 23, 24]), but all these are not investigated in [15]. Hence, it is necessary for us to further explore these dynamical behaviors of (2). For this purpose, we give the following assumptions.Assumption 1.
All coefficientsa1t,a2t,b1t,b2t,at,σ1t,andσ2t are bounded, continuous, and periodic functions with period T.Assumption 2.
The impulsive points satisfy0<t1<t2<⋯, limi⟶∞ti=+∞, and there exists an integer q such that tk+q=tk+Tandλi,k+q=λik,i=1,2,k∈N.Assumption 3.
By the biological meanings, we assumeλik>−1 for i=1,2,k∈N.
For the following stochastic differential equation (see [21]),(3)dxt=fxt,tdt+gxt,tdωt,t≥t0,with initial data xt0=x0∈Rn, we define the following differential operator:(4)L=∂∂t+∑k=1nfkx,t∂∂xk+12∑k,j=1ngTx,tgx,tkj∂2∂xk∂xj.
For any bounded and continuous functionft, we use the following notations:(5)fu=supft,t≥0,fl=infft,t≥0,f∗=limsupt⟶∞ft,f∗=liminft⟶∞ft,and f=1/T∫0tfsdsandfT=1/T∫0tfsds if ft is integrable.
The rest of this paper is organized as follows. Section2 begins with some definitions and lemmas. Section 3 focuses on the existence and uniqueness of the periodic Markovian solution. Section 4 is devoted to the extinction and permanence in the mean of species. The stochastic persistence of (2) is studied in Section 5. Some numerical examples are showed in Section 6 to validate the main results. Finally, a brief conclusion and discussion are given to conclude the paper in Section 7.


## 2. Preliminaries

In this section, we introduce the definitions of the periodic Markovian process, the solution of the impulsive stochastic differential equation, and some auxiliary results of the existence of the periodic Markovian process.Definition 1.
(see [21]). Let ξt=ξt,ω−∞<t<∞ be a stochastic process and t1,t2,…,tn be a finite sequence of numbers. If the joint distribution of random variables ξt1+h,…,ξtn+h is independent of h, where h=kT,k=±1,±2,…, then ξt is said to be T-periodic stochastic process.Definition 2.
(see [13, 15]). The impulsive stochastic equation is given as follows:(6)dxt=ft,xtdt+gt,xtdωt,t≠tk,xtk+=1+λkxtk,k∈N,with x0=x0. A stochastic process xt=x1t,x2t,…,xntT,t∈R+=0,+∞, is said to be a solution of the above system, if(i)
xt is continuous on 0,t1 and tk,tk+1⊂R+k∈N and Ft-adapted, ft,xt∈LR+,Rn and gt,xt∈L2R+,Rn, and ∫0tftpdt<∞ almost surely for all T>0,p=1,2.(ii)
xtk+=limt⟶tk+xt and xtk−=limt⟶tk−xt exist, and xtk=xtk− with probability one, k∈N.(iii)
xt satisfies the following integral equation:(7)xt=x0+∫0tfs,xsds+∫0tgs,xsdωs,for t∈0,t1 and satisfies(8)xt=xtk++∫tktfs,xsds+∫tktgs,xsdωs,for t∈tk,tk+1,k∈N.Lemma 1 (see [13, 21]).
For the following Itô’s differential equation,(9)dxt=bt,xtdt+σt,xtdωt,all coefficients of (9) satisfy linear growing condition and the Lipschitz condition in every cylinder Ul×R+l>0,Ul=x:x≤l and are T-periodic in t. Furthermore, there exists a T-periodic and once continuously differentiable function v=vt,x in t, which is twice continuously differentiable with respect to x and satisfies the following conditions:(10)infx>lvt,x⟶∞,asl⟶∞,Lvt,x≤−1,outside some compact set,and then there exists a solution of (9) which is T-periodic Markov process.Lemma 2 (see [14]).
Suppose thatZt∈CΩ×0,+∞,R+ and limt⟶∞Ft/t=0,a.s.(a)
If there existt0>0andλ0>0 such that, for all t>t0,(11)lnZt≤λt−λ0∫0tzsds+Ft,a.s,then(12)Z∗≤λλ0,a.s.,ifλ≥0,limt⟶+∞Zt=0,a.s.,ifλ<0.(b)
If there existt0>0,λ0>0,andλ such that, for all t>t0,(13)lnZt≥λt−λ0∫0tzsds+Ft,a.s,then(14)Z∗≥λλ0,a.s.
To investigate (2), we consider the following nonimpulsive system:(15)du1t=u1tr1t+1T∑0≤tk<tln1+λ1k−a11tA1tu1t−a12tA2tu2t−b1tA12tu12tdt+σ1tu1tdω1t,du2t=u2tr2t+1T∑0≤tk<tln1+λ2k−a21tA1tu1t−a22tA2tu2t−b2tA22tu22tdt+σ2tu2tdω2t,where(16)A1t=∏j=1q1+λ1j−t/T∏0≤tk<t1+λ1k,A2t=∏j=1q1+λ2j−t/T∏0≤tk<t1+λ2k.Assume that the product equals unity if the number of factors is zero. Obviously,A1tandA2t are both T-periodic (for details, see [20]).For (2) and (15), we have the following.Lemma 3.
Letu1t=A1tx1tandu2t=A2tx2t:(1)
Ifx1t,x2t is solution of (2), then u1t,u2t is solution of (15)(2)
Ifu1t,u2t is solution of (15), then x1t,x2t is solution of (2).Remark 1.
The proof is similar to that of Theorem 3.1 in [20] and is omitted. Lemma 3 reveals the equivalence of (2) and (15), and hence, we will consider (15) later.Lemma 4.
For any given initial valuex10,x20∈R+2, if a11l>b1uA1uanda22l>b2uA2u, then system (2) has a unique solution x1t,x2t on t≥0 and the solution remains in R+ with probability one.Remark 2.
By reference [6], the existence of solutions of (15) can be derived; then, by Lemma 3, the required assertion is obtained.


## 3. Existence of Stochastic Periodic Solution

Theorem 1.
Suppose the following condition holds:(17)H1η1a11lA1l2b1uA1u2+η2a22lA2l2b2uA2u2>maxϑ,χ,where ηi≔rit−σi2t/2T+1/T∑0≤tk<tln1+λik,i=1,2, and ϑ,χ are defined later; then, there exists a periodic Markovian process for system (2).Proof.
Obviously, Lemma4 implies the existence of positive solutions of (2); then, according to Lemma 3, it is only needed to prove the solution of (15) is a periodic Markovian process. By Lemma 1, it suffices to find a C2-function Vt,u1,u2 and a closed set U∈R+2 such that all conditions of Lemma 1 hold for (15). Define(18)V1t,u1=u1−a11lA1l2b1uA1u2lnu1+a11lA1l2b1uA1u2w1t,V2t,u2=u2−a22lA2l2b2uA2u2lnu2+a22lA2l2b2uA2u2w2t,where wit is a continuous differentiable function satisfying(19)wi′t=rit+1T∑0≤tk<tln1+λik−σi2t2−ηi,i=1,2.
LetVt,u1,u2=V1t,u1+V2t,u2, then it is not difficult to verify that Vt,u1,u2 is T-periodic function on 0,∞ and(20)liminfu1,u2∈R+2/UkVt,u1,u2⟶∞,ask⟶∞,where Uk=u1,u2:u1,u2∈1/k,k×1/k,k. Hence, the first condition of Lemma 1 is satisfied. Now, we are in the progress of proving the second condition of Lemma 1. Using Itô’s formula on V1t,u1 and V2t,u2, respectively, we obtain(21)LV1=a11lA1l2b1uA1u2w1′t+1−a11lA1l2b1uA1u21u1u1r1t+1T∑0≤tk<tln1+λ1k−a11tA1tu1−a12tA2tu2−b1tA12tu12+12a11lA1l2b1uA1u2σ12t=a11lA1l2b1uA1u2w1′t−r1t−1T∑0≤tK<tln1+λ1k+12σ12t+r1tu1+1T∑0≤tk<tln1+λ1ku1−a11tA1tu12−a12tA2tu1u2−b1tA12tu13+a11lA1l2b1uA1u2a11tA1tu1+a12tA2tu2+b1tA12tu12≤−η1a11lA1l2b1uA1u2−a11lA1l2u12+r1u+1T∑0≤tk<tln1+λ1k+a11lA1la11u2b1uA1uu1+a11lA1la12uA2u2b1uA1u2u2,LV2=a22lA2l2b2uA2u2w2′t+1−a22lA2l2b2uA2u21u2u2r2t+1T∑0≤tk<tln1+λ2k−a21tA1tu1−a22tA2tu2−b2tA22tu22+12a22lA2l2b2uA2u2σ22t=a22lA2l2b2uA2u2w2′t−r2t−1T∑0≤tk<tln1+λ2k+12σ22t+r2tu2+1T∑0≤tk<tln1+λ2ku2−a21tA1tu1u2−a22tA2tu22−b2tA22tu23+a22lA2l2b2uA2u2a21tA1tu1+a22tA2tu2+b2tA22tu22≤−η2a22lA2l2b2uA2u2−a22lA2l2u22+r2u+1T∑0≤tk<tln1+λ2k+a22lA2la22u2b2uA2uu2+a22lA2lA1ua21u2b2uA2u2u1.
Therefore,(22)LVt,u1,u2≤−η1a11lA1l2b1uA1u2−η2a22lA2l2b2uA2u2−a11lA1l2u12−a22lA2l2u22+r1u+1T∑0≤tk<tln1+λ1k+a11lA1la11u2b1uA1u+a22lA2lA1ua21u2b2uA2u2u1+r2u+1T∑0≤tk<tln1+λ2k+a22lA2la22u2b2uA2u+a11lA1lA2ua12u2b1uA1u2u2.
Denote(23)fu1=−a11lA1l2u12+r1u+1T∑0≤tk<tln1+λ1k+a11lA1la11u2b1uA1u+a22lA2lA1ua21u2b2uA2u2u1,fu2=−a22lA2l2u22+r2u+1T∑0≤tk<tln1+λ2k+a22lA2la22u2b2uA2u+a11lA1lA2ua12u2b1uA1u2u2.
Thus,(24)fuu1=r1u+1/T∑0≤tk<tln1+λ1k+a11lA1la11u/2b1uA1u+a22lA2lA1ua21u/2b2uA2u222a11lA1l≔ϑ,fuu2=r2u+1/T∑0≤tk<tln1+λ2k+a22lA2la22u/2b2uA2u+a11lA1lA2ua12u/2b1uA1u222a22lA2l≔χ.
For any small positiveε<1, define a closed set(25)Dε=u1,u2:u1,u2∈ε,1ε×ε,1ε,where Dε is compact and its component DεC=R+2/Dε=∪i=14Dεi, in which(26)Dε1=u1,u20<u1<ε,Dε2=u1,u20<u2<ε,Dε3=u1,u2u1>1ε,Dε3=u1,u2u2>1ε.
DiscussLV as follows:(i)
Ifu1,u2∈Dε1, by H1, we choose ε>0 small enough such that(27)r1u+1T∑0≤tk<tln1+λ1k+a11lA1la11u2b1uA1u+a22lA2lA1ua21u2b2uA2u2ε+χ≤η1a11lA1l2b1uA1u2+η2a22lA2l2b2uA2u2,and hence,LV<−1.(ii)
Ifu1,u2∈Dε2, by H1 again, for ε>0 small enough, we have(28)r2u+1T∑0≤tk<tln1+λ2k+a22lA2la22u2b2uA2u+a11lA1lA2ua12u2b1uA1u2ε+ϑ≤η1a11lA1l2b1uA1u2+η2a22lA2l2b2uA2u2.Therefore,LV<−1.(iii)
Ifu1,u2∈Dε3 or u1,u2∈Dε4, by the monotonicity, obviously, LV⟶−∞.
To summarize, the conditions of Lemma 1 are all satisfied and the required assertion is directly derived. This completes the proof.Remark 3.
Based on the existence theorem of periodic Markovian process and extreme-value theory of quadratic function, the sufficient conditions assuring the existence of stochastic periodic solution are established, which is not discussed in [15]. Theorem 1 shows that impulsive disturbance and stochastic disturbance affect the periodic behavior of (2), which is shown in Figures 1(g) and 1(h) in Section 6.Figure 1
Periodic Markovian process for (2) with initial data x10=x20=2. (a) The time series graph of x1t, (b) the time series graph of x2t, (c) the phase graph of deterministic system (ξik=0;σit=0;i=1,2), (d) the phase graph of impulsive system with σit=0;λ1k=0.2+0.1sintk;λ2k=0.2+0.1sintk;tk=0.1k, (e) the phase graph of stochastic system with σ1t=0.1+0.02sint;σ2t=0.01+0.01sint;λik=0, (f) the phase graph of stochastic impulsive system with σ1t=0.2+0.02sint;σ2t=0.1+0.01sint;λ1k=0.1+0.1sintk;λ2k=0.1+0.1sintk;tk=0.1k;k∈N, (g) the phase graph of stochastic impulsive system with large stochastic disturbance σ1t=3+0.02sint;σ2t=3+0.01sint, and (h) the phase graph of stochastic impulsive system with negative impulsive λ1k=−0.4+0.1sintk;λ2k=−0.4+0.1sintk;tk=0.1k,k∈N.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)


## 4. Extinction and Permanence in the Mean

Lemma 5.
The solutionxit of (2) with initial value x10,x20T∈R+2 satisfies(29)limsupt⟶∞Exipt<Hip,for any p>0, where Hipi=1,2 is a positive constant. Furthermore,(30)limsupt⟶∞Pxit>ζi<ε,i=1,2,that is, the solution of (2) is stochastically ultimately bounded.Proof.
By Lemma 3, it is only needed to study the equivalent system (15). Define Vu1,u2=etu1p+u2p for u1,u2∈R+2 and p≥1. Applying Itô’s formula to Vu1,u2, we obtain(31)LVu1,u2=etu1p+u2p+pu1pr1t+1T∑0≤tk<tln1+λ1k+p−1σ12t2−a11tA1tu1t−a12tA2tu2t−b1tA12tu12t+pu2pr2t+1T∑0≤tk<tln1+λ2k−a21tA1tu1t−a22tA2tu2t−b2tA22tu22t+p−1σ22t2≤etu1p+u2p+pu1pr1u+1T∑0≤tk<tln1+λ1k+p−1σ1u22+pu2pr2u+1T∑0≤tk<tln1+λ2k+p−1σ2u22≔etH⌣p.
Integrating both sides of the above inequality from 0 tot and taking expectation yields,(32)etEu1p+u2p≤u1p0+u2p0+E∫0tesHpds=u1p0+u2p0+H⌣pet−1.
By comparison theorem,(33)limsupt⟶∞Eu1p+u2p≤H⌣p<∞,which means the existence of constants H⌣1p and H⌣2p such that(34)limsupt⟶∞Eu1pt≤H⌣1p,limsupt⟶∞Eu2pt≤H⌣2p.
Hence, there existst1>0; for any t≥t1, we have Eu1pt≤H⌣1pandEu2pt≤H⌣2p. Furthermore, by the continuity of Eu1pt and Eu2pt, for any t≤t1, there exist H˜1p>0 and H˜2p>0 such that Eu1pt≤H˜1p and Eu2pt≤H˜2p. Let H1p=maxH⌣1p,H˜1p and H2p=maxH⌣2p,H˜2p, then(35)Eu1pt≤H1p,Eu2pt≤H2p,hold for all t≥0. Applying Chebyshev inequality, we obtain(36)limsupt⟶∞Puit>ζi<ε,i=1,2,which means (15) is stochastically ultimately bounded. This completes the proof.Theorem 2.
For system (2), let κ1=a12uA2uζ2andκ2=a21uA1uζ1, then the following results hold:(i)
Ifη1<0andη2<0, then all species are extinct, i.e., limt⟶∞x1t=0andlimt⟶∞x2t=0(ii)
Ifη1>0andη2<0, then x1t is permanent in the mean and x2t is extinct, i.e., α˜1≤x1t≤β1andlimt⟶∞x2t=0(iii)
Ifη1<0andη2>0, then x1t is extinct and x2t is permanent in the mean, i.e., limt⟶∞x1t=0 and α˜2≤x2≤β2(iv)
Ifη1>κ1andη2>κ2, then x1tandx2t are both permanent in the mean, i.e., αi≤x1≤βi,i=1,2, where ζi and ηi are defined as before and α˜i,αi, and βi are all constants defined later in the proof, i=1,2.Proof.
By Lemma 3, it suffices to prove that there exist some positive constantsα˜i′,αi′,andβi′i=1,2 such that these conclusions hold for system (15). Above all, applying Itô’s formula to lnu1t, we have(37)dlnu1=r1t+1T∑0≤tk<tln1+λ1k−σ1t22−a11tA1tu1t−a12tA2tu2t−b1tA12tu12tdt+σ1tdω1t.
Integrating both sides of (37) from 0 to t, we obtain(38)lnu1t=lnu10+∫0tr1s+1T∑0≤tk<sln1+λ1k−σ1s22ds−∫0ta11sA1su1sds−∫0ta12sA2su2sds−∫0tb1sA12su12sds+∫0tσ1sdω1s,≤lnu10+∫0tr1s+1T∑0≤tk<sln1+λ1k−σ1s22ds−a11uA1u∫0tu1sds+∫0tσ1sdω1s.
Sincelimt⟶∞1/T∫0tσ1sdω1s=0, under η1>0, using comparison theorem of stochastic differential equation, we obtain(39)u1∗≤r1t−σ1t2/2T+1/T∑0≤tk<tln1+λ1ka11uA1u≔β1′.
On the other hand, by (15), we have(40)dlnu2t=u2tr2t+1T∑0≤tk<tln1+λ2k−σ22t2−a21tA1u1t−a22tA2u2t−b2tA22tu22tdt+σ2tdω2t.
In the same manner, under conditionη2>0, we have(41)u2∗≤r2t−σ2t2/2T+1/T∑0≤tk<tln1+λ2ka22uA2u≔β2′.(i)
Ifη1<0andη2<0, then it directly follows from (39) and (41) that limt⟶∞uit=0,i=1,2. That is, all species of (15) are extinct.(ii)
Ifη2<0, then limt⟶∞u2t=0. Using (37) again, we have(42)lnu1t=lnu10+∫0tr1s+1T∑0≤tk<sln1+λ1k−σ1s22ds−∫0ta11sA1su1sds−∫0ta12sA2su2sds−∫0tb1sA12su12sds+∫0tσ1sdω1s,≥lnu10+∫0tr1s+1T∑0≤tk<sln1+λ1k−σ1s22ds−a11uA1u+b1uA1u2ζ1∫0tu1sds+∫0tσ1sdω1s.Applying Lemma2 yields(43)u1∗≥r1t−σ1t2/2T+1/T∑0≤tk<tln1+λ1ka11uA1u+b1uA1u2ζ1≔α˜1′.Therefore,α˜1′<u1<β1′ holds, that is, species u1t is permanent in the mean.(iii)
Ifη1<0, then limt⟶∞u1t=0. From (40), we have(44)lnu2t=lnu20+∫0tr2s+1T∑0≤tk<sln1+λ2k−σ2s22ds−∫0ta21sA1su1sds−∫0ta22sA2su2sds−∫0tb2sA22su22sds+∫0tσ2sdω1s,≥lnu20+∫0tr2s+1T∑0≤tk<sln1+λ2k−σ2s22ds−a22uA2u+b2uA2u2ζ2∫0tu2sds+∫0tσ2sdω1s.By assumptionη2>0, we obtain(45)u2∗≥r2t−σ2t2/2T+1/T∑0≤tk<tln1+λ2ka22uA2u+b2uA2u2ζ2≔α˜2′.(iv)
Ifη1>κ1>0andη2>κ2>0, then using (37) again, we have(46)lnu1t≥lnu10+∫0tr1s+1T∑0≤tk<sln1+λ1k−σ1s22ds−a12uA2uζ2−a11uA1u+b1uA1u2ζ1∫0tu1sds+∫0tσ1sdω1s.
Lemma 2 implies(47)u1∗≥r1t−σ1t2/2T+1/T∑0≤tk<tln1+λ1k−a12uA2uζ2a11uA1u+b1uA1u2ζ1≔α1′.
Similarly, we derive from (41) that(48)lnu2≥lnu20+∫0tr2s+1T∑0≤tk<sln1+λ2k−σ2s22ds−a21uA1uζ1−a22uA2u+b2uA2u2ζ2∫0tu2sds+∫0tσ2sdω1s.
Using Lemma 2 yields(49)u2∗≥r2t−σ2t2/2T+1/T∑0≤tk<tln1+λ2k−a21uA1uζ1a22uA2u+b2uA2u2ζ2≔α2′.
Combining (39) and (41), α1′≤uit≤β1′ holds for i=1,2.
To summarize the above discussion and combine Lemma 3, the required results are obtained. The proof is completed.Remark 4.
The result of extinction for all species (case (i)) is in accordance with Theorem 3.2 of [15], but other dynamics such as permanence in the mean for all species (case (ii)–case (iv)) is not studied in [15], which reveals richer dynamical behaviors of this system.


## 5. Stochastic Persistence in Probability

Firstly, we give the following lemmas.Lemma 6.
The solutions of (2) are uniformly continuous.Proof.
Using the property of expectation and Lemma 5, we have(50)Eu1tr1t+1T∑0≤tk<tln1+λ1k−a11tA1tu1t−a12tA2tu2t−b1tA12tu12tdtp≤12Eu12p+12Er1t+1T∑0≤tk<tln1+λ1k−a11tA1tu1t−a12tA2tu2t−b1tA12tu12t2p≤12H12p+42p−1r1u+1T∑0≤tk<tln1+λ1k2p+a11uA1uH12p+a12uA2uH22p+b1uA1u2tH122p≔G1p.
By stochastic integral inequality, for0<t1<t2 and p>2, we obtain(51)E∫t1t2σ1su1sdω1sp≤σ1uppp−12p/2t2−t1p−2/2∫t1t2u1pds≤σ1uppp−12p/2t2−t1p/2H1p.
Hence, when0<t1<t2<∞,t2−t1≤1, and 1/p+1/q=1, we have(52)Eu1t2−u1t1p=Eu1tr1t+1T∑0≤tk<tln1+λ1k−a11tA1tu1t−a12tA2tu2t−b1tA12tu12tds+∫t1t2σ1su1sdω1sp≤2p−1Eu1tr1t+1T∑0≤tk<tln1+λ1k−a11tA1tu1t−a12tA2tu2t−b1tA12tu12tdsp+2p−1E∫t1t2σ1su1sdω1sp≤2p−1t2−t1p/qu1tr1t+1T∑0≤tk<tln1+λ1k−a11tA1tu1t−a12tA2tu2t−b1tA12tu12tpds+2p−1σ1uE∫t1t2y1sdω1sp≤2p−1t2−t1p/q+1G1P+2p−1σ1uE∫t1t2y1sdω1sp≤2p−1t2−t1p/21+pp−12p/2G2P,where G2p=maxG1p,σ1uH1p. Therefore, the solution u1t of (15) is uniformly continuous. Similarly, we can obtain the uniform continuity of u2t. Lemma 3 implies the required assertion. This completes the proof.Lemma 7 (see [25]).
Letf be a nonnegative function defined on R+ such that f is integrated and uniformly continuous, then limt⟶∞ft=0.Theorem 3.
Suppose(53)H2ι1=a11t−a21t>0,ι2=a22t−a12t>0,then the solution x1t,x2t of (2) is globally attractive. Furthermore, for any other solution x˜1t,x˜2t, we have(54)limt⟶∞Ex1−x˜1=0,limt⟶∞Ex2−x˜2=0.Proof.
Similarly, we only prove the conclusion for (15). Define Vu1,u2=lnu1−lnu˜1+lnu2−lnu˜2. Applying Itô’s formula and computing the right derivative of function Vu1,u2 along the solution of (15) yield(55)d+V=sgnu1−u˜1−a11tA1tu1−u˜1−a12tA2tu2−u˜2−b1tA12tu1−u˜1u1+u˜1tdt+sgnu2−u˜2−a21tA1tu1−u˜1−a22tA2tu2−u˜2−b2tA22tu2−u˜2u2+u˜2tdt≤−A1ta11t+2b1tA1tζ−a21tu1−u˜1−A2ta22t+2b2tA2tζ−a12tu2−u˜2≤−A1ta11t−a21tu1−u˜1−A2ta22t−a12tu2−u˜2,where ζi is the sup of u1 and u2, i.e., ui≤ζii=1,2. According to H2, there exist ρ>0 and t0>0 such that a11t−a21t>ρanda22t−a12t>ρ for t≥t0; hence, D+Vi≤−ρu1−u˜1+u2−u˜2 for t≥t0. Integrating both sides of it from t0 to t leads to(56)Vt−Vt0≤−ρ∫t0tu1s−u˜1s+u2s−u˜2sds.
Hence,(57)Vt+ρ∫t0tu1s−u˜1s+u2s−u˜2sds≤Vt0<+∞.
That is,(58)u1t−u˜1t∈L10,+∞,u2t−u˜2t∈L10,+∞.
Therefore, the global attractiveness of (15) is obtained by Lemmas 6 and 7.
On the other hand,(59)dEu1dt≤r1u+1T∑0≤tk<tln1+λ1kEu1−a11lA1lEu12−a12lA2lEu1u2−b1lA1l2Eu13≤r1u+1T∑0≤tk<tln1+λ1kEu1≤r1u+1T∑0≤tk<tln1+λ1kH1p,and hence, Eu1 is uniformly continuous. The uniform continuity of Eu2 can be similarly obtained. According to (58) and Barbalat’s conclusion [25],(60)limt⟶∞Eu1t−u˜1t=0,limt⟶∞Eu2t−u˜2t=0.
This completes the proof.
Finally, we discuss the stochastic persistence in probability of (2). For the system(61)dx1t=xitfixtdt+∑j=1ngijxtdBjt,i=1,2,…,m,where xt=x1t,x2t,…,xmt, B1t,B2t,…,Bmt is an m-dimensional standard Brownian motion. Suppose S⊂R¯+n is a set such that xt∈S for t≥0. Denote the set of the states where the size of at least one species is less than or equal to η by Sη=x∈S:xi≤ηfor somei. S0 stands for the states where the size of one or more populations is 0. Let ∏t=1/T∫0tδxsds be the occupation measure of xtt≥0, where δxs represents a Dirac measure at xs.Definition 3.
(see [26, 27]). For any ε>0, if there exists η>0 such that limsupt⟶∞∏tSη≤ε with X0∈S0 a.s., then we say (61) is stochastically persistent.Remark 5.
The definition ofSη shows one or more populations have a density less than η; then, stochastic persistence means that all populations spend an arbitrarily small fraction of time at arbitrarily low densities. It is more appropriate than permanence in the mean and Definition 2 of [15], see [27].Lemma 8 (see [26, 27]).
LetΔ0=a∈R+mai=0for somei,1≤i≤m. System (61) is said to be stochastically persistent in probability if there exists a unique invariant probability measure μ such that μΔ0=0 and the distribution of xt converges to μ as t⟶∞ whenever x0∈R+m.Theorem 4.
SupposeH2 and ηi>κi, then (2) is stochastically persistent in probability, where ηiandκii=1,2 are defined in Theorem 2.Proof.
Firstly, the proof of stochastic persistence in probability is motivated by [20]. Let u1t,u10,u2t,u20 be the solution of (15) with initial data u10,u20∈R+2, pt,u0,dy be the transition probability of ut,u0, and Pt,u0,Γ be the probability of events ut,u0∈Γ. The stochastic boundedness of ut,u0 implies there exists a compact subset K⊆R+2 such that Pt,ϕ,K≥1−ε for any given ε>0, that is, pt,u0,dy:t≥0 is tight. PR+2 represents the probability measure on R+2; then, we prove the series pt,u0,R+2 is Cauchy in equation.
For anyP1,P2∈PR+2, define the metric of P1 and P2 as follows:(62)dFP1,P2=supf∈F∫R+2fuP1du−∫R+2fuP2du,where(63)F=f:R+2⟶Rfu1−fu2≤u1−u2,f⋅≤1.
For anyf∈F and s,t>0,(64)Efut+s,u0−Eut,u0=EfEut+s,u0ℱs−Efut,u0=∫R+2Efut,u0ps,u0,du0−Efut,u0≤∫R+2Efut,u^0−Efut,u0ps,u0,du^0=∫U¯KEfut,u^0−Efut,u0ps,u0,du^0+∫R+2/U¯KEfut,u^0−Efut,u0ps,u0,du^0,where U¯K=u∈R+2:u≤K. Because of the tightness of pt,u0,dy, one can find sufficiently large K such that Ps,u0,R+2/U¯K<ε for any s>0. Then,(65)∫R+2/U¯KEfut,u^0−Efut,u0ps,u0,du^0≤2Ps,u0,R+2/U¯K≤2ε.
By the global attractivity (Theorem 3), for arbitrarily givenε>0 and sufficiently large t, we have(66)∫U¯KEfut,u^0−Efut,u0ps,u0,du^0≤∫U¯KEut,u^0−ut,u0ps,u0,du^0≤∫U¯Kεps,u0,du^0=εPs,u0,U¯K≤ε.
Consequently, for anys>0 and sufficiently large t,(67)dFpt+s,u0,⋅,pt,u0,⋅=supf∈FEfut+s,u0−Efut,u0≤3ε.
Hence, for any initial data,u0=u10,u20∈R+2,pt,u0,⋅:t≥0 is Cauchy in ℙR+2 with the metric dF and there exists a probability measure ν⋅ such that, for fixed initial data u^0∈R+2,(68)limt⟶∞dFpt,u^0,⋅,ν⋅=0.
Finally, by the triangle inequality, we have(69)dFpt,u0,⋅,ν⋅≤dFpt,u0,⋅,pt,u^0,⋅+dFpt,u^0,⋅,ν⋅.
In view of the global attractivity, we can deduce(70)dFpt,u0,⋅,pt,u^0,⋅=supf∈FEfut,u0−Efut,u^0≤supf∈FEfut,u0−fut,u^0≤Eut,u0−ut,u^0⟶0.
Therefore, for anyu0∈R+2, limt⟶∞dFpt,u0,⋅,ν⋅=0, that is, for any u0∈R+2, there exists unique probability measure ν⋅ such that the transition probability pt,u0,⋅ of ut converges weakly to ν⋅ as t⟶∞. Therefore, (15) is asymptotically stable in distribution.
Meanwhile, Theorem2 implies(71)1t∫0tu1sds>0,1t∫0tu2sds>0,a.s.,and hence, (15) is stochastically persistent in probability by Lemma 8. Applying Lemma 3 yields the required assertion. This completes the proof.


## 6. Examples and Simulations

In this section, by use of the numerical method [28], we give some simulations.For system (2), with no special mention, we always take r1t=5+2sint, a11t=4+2cost, a12t=1+0.2sint, b1t=1+0.3sint, r2t=5+3sint, a21t=0.4+0.2cost, a22t=2+0.1sint, b2t=1+0.1sint, σ1t=0.2+0.02sint, σ2t=0.1+0.01sint, λ1k=0.1+0.1sintk, λ2k=0.1+0.1sintk,andtk=0.1k,k∈N.An easy computation showsH1 holds. Theorem 1 implies that the solution of (2) is a periodic Markovian process. Using the Milstein method and writing MATLAB code, we get the simulation result of periodic solution of (2), see Figure 1. Under H1, Figures 1(a) and 1(b) are the time series graphs of x1t and x2t for impulsive and stochastic case, respectively. Figures 1(c)–1(f) are the phase graphs of the periodic solution of deterministic system with σit=0andλik=0, impulsive system with σit=0,λ1k=0.2+0.1sintk,λ2k=0.2+0.1sintk,andtk=0.1k, stochastic system with σ1t=0.1+0.02sint,σ2t=0.01+0.01sint,andλik=0, and stochastic impulsive system with σit and λiki=1,2,k∈N as before, respectively. If σ1t=3+0.02sintandσ2t=3+0.01sint, that is, the stochastic disturbance is large, then the periodic process does not appear, illustrated in Figure 1(g). If λ1k=−0.4+0.1sintk,λ2k=−0.4+0.1sintk,andtk=0.1k,k∈N, then there exists no periodic process, see Figure 1(h). Figures 1(g) and 1(h) show that impulse and stochastic perturbation affects the existence of stochastic periodic solution.Next, we simulate the extinction and permanence in the mean of solution of (2). Let σ1t=3.5+0.02sintandσ2t=4+0.01sint or λ1k=−0.4+0.1sintandλ2k=−0.6+0.1sint, then η1<0andη2<0. Hence, it follows from Theorem 2 that x1t and x2t are both extinct, illustrated in Figures 2(a) and 2(b). Let σ1t=4+0.02sintandσ2t=2+0.01sint, then η1<0andη2>0. By Theorem 2 again, x1t is extinct and x2t is permanent in the mean, illustrated in Figure 2(c). Let σ1t=1+0.02sintandσ2t=5+0.01sint, then η1>0andη2<0. Using Theorem 2 yields that x1t is permanent in the mean and x2t is extinct, illustrated in Figure 2(d). Let σ1t=1+0.02sintandσ2t=1+0.01sint, then η1>κ1andη2>κ2. Similarly, Theorem 2 implies x1t and x2t are both permanent in the mean, see Figures 2(e) and 2(f). Figure 2 shows that the impulsive disturbance and stochastic disturbance bring some important influence to the dynamics of all species.Figure 2
Extinction and permanence in the mean for system (2) with initial data x10=x20=2. (a) The extinction of x1t and x2t for (2) with only impulse effects (σit=0;λ1k=−0.4+0.1sintk;λ2k=−0.6+0.1sintk;i=1,2,k∈N), (b) the extinction of x1t and x2t for (2) with only stochastic effects (σ1t=3.5+0.02sint;σ2t=4+0.01sint;λik=0;i=1,2,k∈N), (c) the extinction of x1t and permanence in mean of x2t with σ1t=4+0.02sint;σ2t=2+0.01sint, (d) the permanence in mean of x1t and extinction of x2t with σ1t=1+0.02sint;σ2t=5+0.01sint, (e) the permanence in mean of x1t with σ1t=1+0.02sint; σ2t=1+0.01sint, and (f) the permanence in mean of x2t with σ1t=1+0.02sint;σ2t=1+0.01sint.
(a)
(b)
(c)
(d)
(e)
(f)Finally, we simulate the global attractivity and stochastic persistence in probability of solution of (2). By computation, all parameters meet the conditions of Theorem 4; therefore, the solution of (2) is globally attractive (see Figures 3(a) and 3(b)) and the distribution of all species are asymptotic stability (see Figure 4(a)). The stochastic persistence is simulated in Figures 4(b)–4(d), which are the time series graphs of x1t, x2t, and zt=x1t+x2t1/2, respectively.Figure 3
The attractivity ofx1t and x2t of system (2) with initial data x10=x20=2. (a) The attractivity of x1t and (b) the attractivity of x2t.
(a)
(b)Figure 4
The distribution and stochastic persistence forx1t and x2t of (2) with initial data x10=x20=2. (a) The density graph of x1t and x2t, (b) the time series graph of x1t, (c) the time series graph of x2t, and (d) the time series graph of zt=x12t+x22t1/2.
(a)
(b)
(c)
(d)


## 7. Conclusions and Discussions

In this paper, for the stochastic competitive system with impulsive and nonlinear term of self-interaction proposed in [15], we further study such dynamics as the existence of periodic solution, the extinction and permanence in the mean, the global attractivity of solutions, and stochastic persistence of this system. Theorem 1 gives the sufficient conditions of the existence of periodic Markovian process. Theorem 2 gives the conditions of extinction and permanence in the mean of both species. Theorem 3 establishes the condition assuring the global attractivity. Theorem 4 establishes the condition of stochastic persistence in probability of this system. Finally, simulations (Figures 1–4) are given to verify the obtained results. Our main results are new and different from [15], which is presented by giving Remarks 3, Remark 4, and Remark 5 in detail.Three or more species often coexist in the real world, and time delays often appear in biological system, then how to deal with the effects of time delays on the stochastic bahaviors of three-species biological system is very interesting to be further investigated. On the other hand, regime switching is another common random perturbation, e.g., stochastic hybrid phytoplankton-zooplankton model with toxin-producing phytoplankton, stochastic tumor-immune model with regime switching, and impulsive perturbations. All these are necessary and very interesting for us to study in the future.



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