# Dynamical Behavior of a Nonautonomous Stochastic Modified Bazykin Model

Authors: Zhangzhi Wei; Zheng Wu; Lianglong Wang
Journal: Journal of Chemistry
 (2019)

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## Abstract

In this article, a nonautonomous stochastic modified Bazykin model is introduced. The positive solution is proved to be unique and global for any initial data via stochastic comparison theorem and Itô formula. Stochastic ultimate boundedness and stochastic permanence are also considered. Finally, some simulations are performed to verify the validity of results.


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## Body

## 1. Introduction

The study on population systems is one of the hot topic in applied mathematics and ecology. Mathematical modelling and qualitative analysis have received much concern, such as the famous Lotka–Volterra model [1]. For more than a decade, predator-prey models with ration-dependent response function have been discussed extensively [2–4]. Today, most predator-prey models assume that the functional response function is a function of prey population density [5–7].Considering the classical assumptions is not always appropriate which assume that predators encounter prey at random, and the functional response only depends on prey abundance. In 1998, Alexeev and Bazykin first proposed the following Bazykin’s model [2].(1)N˙t=Nta−bNt−cPt1+ANt,P˙t=Pt−g−hPtfNt1+ANt,where Nt and Pt denote the number (or density) of prey species and predator species. The parameters a,b,c,g,f, and m are greater than zero with their usual ecological meanings (for details refer [2]). Many research studies about system (1) have been received, more about its generalization [8–11].Taking the ratio-dependent of predator into account (more predators are good for hunting in serious situations), M. Haque remodeled a modified Bazykin model according to system (1)(2)N˙t=Nta−bNt−cPtPt+ANt,P˙t=Pt−g−hPtfNtPt+ANt,in which the analytical behavior of the system near origin is observed [1].In the real environment, there is the interference of random factors everywhere, such as emergencies, weather changes, and disease spread, and then the actual observed data have some deviation from the results of the deterministic model. Therefore, the influence of the real random factors on the populations should be considered, and the stochastic model can describe the real situation better [12–16].Considering that the parametersa and g are affected by white noise, the rates a and g can been replaced by a⟶a+σ1w˙1t, and g⟶g+σ2w˙2t, respectively, where w1t and w2t are Wiener process (or Brownian motion), and they are independent. σ1 and σ2 indicate the intensities of white noise. The following autonomous stochastic model was built by Lv et al. [13]:(3)N˙t=Nta−bNt−cPtPt+ANt+σ1Ntdw1t,P˙t=Pt−g−hPtfNtPt+ANt−σ2Ptdw2t,and has been investigated, and the system (3) is persistent in mean or die out under some conditions [13].In this paper, we will consider the nonautonomous stochastic modified Bazykin model(4)N˙t=Ntat−btNt−ctPtPt+AtNt+σ1tNtdw1t,P˙t=Pt−gt−htPtftNtPt+AtNt−σ2tPtdw2t,where at,bt,ct,gt,ht,ft,At, and σ1t,σ2t are assumed to be continuous and bounded.It is important for population systems to keep permanence, which means that the species will not extinct in the future. Some definitions and conditions which guarantee that system (4) is stochastic permanent and boundedness are given as follows.Without specification,N0,P0∈R+2 is an initial data by default in the full text, and Yt=Nt,Pt is the solution of system (4) with an initial N0,P0.Definition 1 (see [15]).
Assume any0<ϵ<1, then M>0 and η>0, when following is hold:(5)lim inft⟶+∞ℙYt≤M≥1−ϵ,lim inft⟶+∞ℙYt≥η≥1−ϵ,so solution Yt=Nt,Pt for system (4) is called stochastically permanent.Definition 2 (see [15]).
Assume any0<ϵ<1, then η>0, when following is hold(6)lim supt⟶+∞ℙYt>η<ϵ,so solution Yt=Nt,Pt for system (4) is called stochastically ultimately bounded.
In Section2, main results about the existence and uniqueness of global positive solution and the properties of asymptotic bounded were introduced and then shows us that the solution of system (4) is stochastically ultimate bounded and stochastically permanent. In Section 3, we briefly give conclusions and remark. In Section 4, numerical simulations are carried out to illustrate our results. In the end, some lemmas and proofs of main results are given.


## 2. Main Results

### 2.1. Existence and Globality

Theorem 1.
LetN0,P0∈R+2 be initial value, then the positive solution Yt=Nt,Pt to system (4) exists globally a.s.Corollary 1.
The solutionYt=Nt,Pt of system (4) must stay in R+2 a.s., and that(7)N1t≤Nt≤N2t,P1t≤Pt≤P2t,a.s.,where upper-lower solutions are defined later in (20)–(22) and (51).


### 2.2. Dynamical Behavior of System (4)

In the following, we denote(8)φu=maxt∈0,+∞φt,φl=mint∈0,+∞φt,qt=at−ct−σ12t,λt=gt−ftAt+12σ22t,νt=at−ct−σ12t2,ρt=at−ct−32σ12t,σt=ftAt−gt−32σ22t.The functions above are bounded and continuous on0,+∞.Theorem 2.
Ifbl>0 and gl>0, the solution Nt,Pt to system (4) satisfies(9)limsupt⟶+∞ENt≤aubl,limsupt⟶+∞ENt+clfuPt≤au+gu24blgl.
Furthermore,(10)lim supt⟶+∞EPt≤fuau+gu24blclgl,when cl>0.Theorem 3.
The solutionsNt,Pt of system (4) are stochastically ultimately bounded when bl>0,cl>0, and gl>0.Theorem 4.
The solutionsNt,Pt of system (4) are stochastically permanent, if bl>0,cl>0,gl>0, and ql>0.Theorem 5.
The solutionsNt,Pt of system (4) are stochastically permanent, if bl>0,cl>0,gl>0,ρl>0, and σl>0.


## 2.1. Existence and Globality

Theorem 1.
LetN0,P0∈R+2 be initial value, then the positive solution Yt=Nt,Pt to system (4) exists globally a.s.Corollary 1.
The solutionYt=Nt,Pt of system (4) must stay in R+2 a.s., and that(7)N1t≤Nt≤N2t,P1t≤Pt≤P2t,a.s.,where upper-lower solutions are defined later in (20)–(22) and (51).


## 2.2. Dynamical Behavior of System (4)

In the following, we denote(8)φu=maxt∈0,+∞φt,φl=mint∈0,+∞φt,qt=at−ct−σ12t,λt=gt−ftAt+12σ22t,νt=at−ct−σ12t2,ρt=at−ct−32σ12t,σt=ftAt−gt−32σ22t.The functions above are bounded and continuous on0,+∞.Theorem 2.
Ifbl>0 and gl>0, the solution Nt,Pt to system (4) satisfies(9)limsupt⟶+∞ENt≤aubl,limsupt⟶+∞ENt+clfuPt≤au+gu24blgl.
Furthermore,(10)lim supt⟶+∞EPt≤fuau+gu24blclgl,when cl>0.Theorem 3.
The solutionsNt,Pt of system (4) are stochastically ultimately bounded when bl>0,cl>0, and gl>0.Theorem 4.
The solutionsNt,Pt of system (4) are stochastically permanent, if bl>0,cl>0,gl>0, and ql>0.Theorem 5.
The solutionsNt,Pt of system (4) are stochastically permanent, if bl>0,cl>0,gl>0,ρl>0, and σl>0.


## 3. Conclusion

Theorem5 shows that predator species is permanent a.s., if bl>0,cl>0,gl>0,ρl>0, and σl>0. It is clear that prey species is also permanent a.s., i.e., predator species and prey species are both stochastically permanent. Furthermore, under assumption of Theorem 5, system (4) is persistent according to Theorem 7 in [13]. By our Theorem 5, we can conclude that prey species and predator species in equation (4) are both stochastically permanent. Our work is based on the result of Lv et al. [13] and Wu et al. [15]. At the same time, we have achieved some results about the stochastic model with Markovian switching and Lévy jumps [17, 18].


## 4. Simulations

In this part, some numerical experiments are given to show that our main results as true. In Figure1, we choose(11)a=0.8+0.1sint,b=0.1+0.05cost,c=0.2+sint,A=0.9+0.1sint,g=0.3+0.01sint,h=0.1,f=0.9+0.1cost,σ1=0.49+0.01sint,σ2=0.49+0.1cost.Figure 1
Solution is stochastically permanent ifbl>0,cl>0,gl>0,rl>0,ρl>0,andσl>0.
(a)
(b)We notice thatbl>0,cl>0,gl>0,rl>0,ρl>0,andσl>0, and then the solution of system (4) will be stochastically permanent by virtue of Theorem 5.In Figure2, we choose(12)a=0.8+0.1sint,b=0.1+0.05cost,c=0.2+sint,A=0.9+0.1sint,g=0.3+0.01sint,h=0.1,f=0.9+0.1cost,σ1=0.59+0.01sint,σ2=0.59+0.1cost,where rl>0,ρl<0,andσl<0, and bl>0,cl>0,gl>0,andrl>0, but ρl<0andσl<0. According to Theorem 4, we know that the solution of system (4) is also stochastically permanent.Figure 2
Solution is stochastically permanent ifbl>0,cl>0,gl>0,andrl>0, but ρl<0andσl<0.
(a)
(b)


## 5. Proofs of Main Results

Lemma 1 (see [16]).
LetN2t be a solution of equation (21). If bl>0, then(13)limsupt⟶+∞EN2t≤aubl.Lemma 2 (see [16]).
(14)
E
exp
∫
t
0
t
σ
1
s
d
w
s
=
exp
1
2
∫
t
0
t
σ
1
2
s
d
s
,
0
≤
t
0
≤
t
.Lemma 3.
Ifql>0, then(15)limsupt⟶+∞E1Nt≤buql.Lemma 4.
Ifbl>0 and ql>0, for every ϵ>0, there exists M=>0,η>0; then,(16)lim inft⟶+∞ℙNt≤M≥1−ϵ,lim inft⟶+∞ℙNt≥η≥1−ϵ.Lemma 5.
Ifρl>0 and σl>0, then(17)limsupt⟶+∞E1Pt≤P0−1+fumu2N0−1+2bu2ρl−2.Proof of Theorem 1 and Corollary 1.
The proof process is a standard operation and is omitted here (see [12, 13, 15]).
The following system is obtained by comparison(18)dN1t=N1tat−ct−btN1tdt+σ1tN1tdw1t,N10=N0,dN2t=N2tat−btN2dt+σ1tN2tdw1t,N20=N0,dP1t=P1tftAt−gt−P1tht+ftA2tP1tdt−σ2tP1tdw2t,P10=P0,dP2t=P2t−htP2t+ftAt−gtdt−σ2tP2tdw2t,P20=P0.
They have solutions as follows:(19)N1t=exp∫0tas−cs−σ12s/2ds+∫0tσ1sdw1sN0−1+∫0tbsexp∫0saτ−cτ−σ12τ/2dτ+∫0sσ1τdw1τds,(20)N2t=exp∫0tas−σ12s/2ds+∫0tσ1sdw1sN0−1+∫0tbsexp∫0saτ−σ12τ/2dτ+∫0sσ1τdw1τds,(21)P1t=exp∫0tfs/As−gs−σ22s/2ds−∫0tσ2sdw2sP0−1+∫0thsexp∫0sfr/Ar−gr−σ22s/2dr−∫0sσ2rdw2rds,(22)P2t=exp∫0tfs/As−gs−σ22s/2ds−∫0tσ2sdw2sP0−1+∫0ths+fs/A2sN1sexp∫0sfr/Ar−gr−σ22s/2dr−∫0sσ2rdw2rds.
Consequently(23)N1t≤Nt≤N2t,a.s.,P1t≤Pt≤P2t,a.s.Proof of Theorem 2.
FromNt≤N2tta.s., and Lemma 1, one obtain that (24)limsupt⟶+∞ENt≤aubl.
Let(25)Ft=Nt+clfuPt.
The derivative ofFt along system (4) can be obtained as follows:(26)dFt=at+gtNt−btN2t−gtFtdt+−ct+clfuftNtPtyt+AtNt−htclfuP2tdt+σ1tNtdw1t−clfuσ2tPtdw2t.
Through calculating the integral, we have(27)Ft=F0+∫0tas+gsNs−bsN2s−gsFs−htclfuP2t+−cs+clfufsNsPsPs+AsNsds+∫0tσ1sNsdw1s−∫0tclfuσ2sdw2s.
Then, we calculate the expectation(28)EFt=F0+E∫0tas+gsxs−bsN2s−gsFs−htclfuP2t+−cs+clfufsNsPsPs+AsNsds.
The following formula is easy to be obtained:(29)dEFtdt=at+gtENt−btEN2t−gtEFt−htclfuEP2t+−ct+clfuftENtPtPt+AtNt≤at+gtENt−btEN2t−gtEFt≤au+guENt−blENt2−glEFt.
Note that inequalityK1x−K2x2≤K12/4K2, and then(30)dEFtdt≤au+gu24bl−glEFt.
Therefore, by basic theories of ordinary differential equation, we have(31)0≤limsupt⟶+∞EFt≤au+gu24blgl.
According to the above equation, the following is obtained:(32)lim supt⟶+∞EPt≤−ENt+fuau+gu24blclgl≤fuau+gu24blclgl.Proof of Theorem 3.
By Theorem2,(33)lim supt⟶+∞ENt≤aubl,lim supt⟶+∞EPt≤flau+gu24blclgl.
Now, for anyϵ>0, let H1>au/blϵ and H2>au+gu2fu/4blclglϵ. Then, the following are obtained through the Chebyshev inequality(34)ℙNt>H1≤ENtH1<ϵ,ℙPt>H2≤EPtH2<ϵ.
TakingH=2maxH1,H2, it yields(35)ℙYt>H≤ℙNt+Pt>H≤ENt+PtH<ϵ.
Then,(36)lim supt⟶+∞ℙYt>H<ϵ.Proof of Lemma 3.
Combing (21) with Lemma 2, we have(37)E1N1t=N0−1exp−∫0tas−cs−σ12s2dsEexp−∫0tσ1sdw1s+∫0tbsexp−∫staτ−cτ−σ12τ2dτEexp−∫stσ1τdw1τds=N0−1exp−∫0tqsds+∫0tbsexp−∫stqτdτds≤N0−1e−qlt+bu∫0te−qlt−sds≤N0−1e−qlt+buql.
From Corollary1, it has(38)E1Nt≤E1Nt≤N0−1e−qlt+buql.
Therefore,lim supt⟶+∞E1/Nt≤bu/ql.Proof of Lemma 4.
By Theorem2, there exists K>0 s.t. ENt≤K. For each ϵ>0, denote H=K/ϵ; hence, we have(39)ℙNt>H≤ENtH≤ϵ.
Then,(40)ℙNt≤H≥1−ϵ.
By Lemma3, we have(41)lim supt⟶+∞E1Nt≤buql.
Therefore, for anyϵ>0, let δ=ϵrl/bu, we have(42)ℙNt<δ=ℙ1Nt>1δ≤E1/Nt1/δ≤δE1/Nt,which yields(43)lim supt⟶+∞ℙNt<δ≤δE1Nt=δbuql=ϵqlbu⋅buql=ϵ.
This implies(44)lim inft⟶+∞ℙNt≥δ≥1−ϵ.Proof of Theorem 4.
According to the arguments in Lemma4, we know that for any ϵ, existing δ>0,M>0 and the following is true:(45)lim inft⟶+∞ℙNt≥δ≥1−ϵ.
Then, we get(46)lim inft⟶+∞ℙYt≥δ≥lim inft⟶+∞ℙNt≥δ≥1−ϵ.
Similarly, we have(47)lim inft⟶+∞ℙYt≤M≥1−ϵ.
From Definition1, we know the solution of system (4) is stochastically permanent.Proof of Lemma 5.
By Corollary1, it is easy to have(48)P1t<Pt,P1t=exp∫0tfs/As−gs−σ22s/2ds−∫0tσ2sdw2sP0−1+∫0ths+fs/A2sϕsexp∫0sfr/Ar−gr−σ22s/2dr−∫0sσ2rdw2rds.
So,(49)P1−1t=P0−1exp∫0t−fsAs+gs+σ22s2ds+∫0tσ2sdw2s+∫0ths+fsA2sϕsexp∫stgτ−fτAτ+12σ22τdτ+∫stσ2τdw2τds.
By Lemma2 and Hölder inequality, one has(50)EP1−1t≤P0−1exp∫0tgs−fsAs+σ22sds+∫0ths+fsA2sϕsexp∫stdτ−fτ+32σ22τdτEϕ−2s1/2ds.
Combiing inequalitya+b2≤2a2+b2 with (21), one can get(51)EN1−2t≤2x0−2Eexp−2∫0tνsds−2∫0tσ1sdw1s+2E∫0tbsexp−∫stντdτ−∫stσ1τdw1τds2.
Meanwhile,(52)E∫0tbsexp−∫stντdτ−∫stσ1τdw1τds2=∫0t∫0tbsbuexp−∫stντdτexp−∫stντdτ⋅Eexp−∫stσ1τdw1τexp−∫utσ1τdw1τduds.
Furthermore,(53)Eexp−∫stσ1τdw1τexp−∫utσ1τdw1τ≤exp∫stσ12τdτexp∫utσ12τdτ.
Substituting (53) into (52), we have(54)E∫0tbsexp−∫stντdτ−∫stσ1τdw1τds2=∫0tbsexp−∫staτ−cτmτ−32σ12τdτds2.
Therefore, by (51) and (54), one can get(55)EN1−2t≤2N0exp−2ρlt+2buρl2≤2N0+2buρl2.
Finally, we obtain the required assertion through substituting (55) into (51).
In summary, Theorem5 has been obtained.Remark 1.
It is clear that stochastic differential equation which is stochastically permanent implies that its solutions are stochastically ultimately bounded. Lemma4 shows that the prey species will be permanent if bl>0 and ql>0, or it will be extinct. Theorem 4 shows that if bl>0,cl>0,gl>0 and ql>0, system (4) is permanent in probability 1, i.e., the number (or density) is bounded a.s.



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