# Dynamics Analysis of a Stochastic Hybrid Logistic Model with Delay and Two-Pulse Perturbations

Authors: Haokun Qi; Hua Guo
Journal: Complexity
 (2020)

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## Abstract

In this paper, we propose and discuss a stochastic logistic model with delay, Markovian switching, Lévy jump, and two-pulse perturbations. First, sufficient criteria for extinction, nonpersistence in the mean, weak persistence, persistence in the mean, and stochastic permanence of the solution are gained. Then, we investigate the lower (upper) growth rate of the solutions. At last, we make use of Matlab to illustrate the main results and give an explanation of biological implications: the large stochastic disturbances are disadvantageous for the persistence of the population; excessive impulsive harvesting or toxin input can lead to extinction of the population.


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## Body

## 1. Introduction

It is universally known that the logistic model is one of the most significant and classical models in mathematical biology. Many scholars have studied it and achieved fruitful results (see [1–8]). The classical logistic equation is expressed by(1)dXt=Xtr−a1Xtdt,where X denotes the population size and r and a1 stand for the intrinsic growth rate and the intraspecific competition rate, respectively. With the improvement of the understanding of biological mathematical models, some factors have been considered, such as random interference, time delay, and so on. Compared with the classical original model, stochastic models (see [9–16]) can better reflect the actual situation. Based on model (1), we obtain the following stochastic model:(2)dXt=Xtr−a1Xt+a2Xt−τ+a3∫−∞0Xt+θdςθdt+σXtdBt+Xt−∫YγvN˜dt,dv,where r and aii=1,2,3 are non-negative constants. τ≥0 is the time delay. ςθ stands for a probability measure on −∞,0. Bt is the independent standard Brownian motion defined on a complete probability space Ω,ℱ,ℱtt≥0,ℙ and σ≥0 is the intensity of the white noise. Xt− is the left limit of Xt. Ndt,dv is a Poisson random measure with characteristic measure λ on a measurable bounded below subset Y of ℝ\0 with λY<∞, γ is a Lévy measure such that N˜dt,dv=Ndt,dv−λdvdt.In addition to the white noise and Lévy noise mentioned above, there are other noises in nature, such as telegraph noise, which can be expressed by continuous-time Markov chain that mainly describes the random switching between two or more environment states [17] and which is different due to rainfall, nutrition, and other factors [18, 19]. Thus, a series of articles about Markovian switching have been investigated (see [20–28]). We focus on the stochastic logistic model with Markovian switching:(3)dXt=Xtrξt−a1ξtXt+a2ξtxt−τ+a3ξt∫−∞0Xt+θdςθdt+σξtXtdBt+Xt−∫Yγξt,vN˜dt,dv,where ξt is a continuous-time Markov chain with values in finite state space M=1,2,…,N. We assume that Bt and ξt are independent.As we all know human activities will have a significant impact on the population system, we must pay attention to the growing influence of human beings on population systems. The main manifestation of human activities is the regular harvesting of species or the regular stocking for the protection of endangered species, which cannot be considered continuously. Therefore, these phenomena can be described more accurately by the stochastic models with impulsive effects (see [29–34]).On the other hand, human activities not only have a direct impact on the population but also have an indirect impact. The toxin produced by environmental pollution has an indirect impact on the species. Environmental pollution caused by human activities has become an important issue that the world has to consider. Environmental pollution not only pollutes the atmosphere but also produces toxins that can enter into animals and plants, causing unimaginable harm to them; the light ones can make some populations die, and the heavy ones may cause species extinction. And these toxins will also accumulate in animals and plants. People transfer toxins in their bodies by eating the animals and plants, which can cause harm to human health. Therefore, it has become an inevitable trend to consider the influence of environmental toxins on the population (see [35–38]).Based on the above discussion, we first consider the following stochastic hybrid logistic model with two-pulse perturbations:(4)dXt=Xtrξt−αξtC0t−a1ξtXt+a2ξtXt−τ+a3ξt∫−∞0Xt+θdςθdt+σξtXtdBt+Xt−∫Yγξt,vN˜dt,dv,dC0t=kCet−g+mC0tdt,dCet=−hCetdt,,t≠tn,t≠nT,n∈ℤ+,ΔXt=δnXt,ΔC0t=0,ΔCet=0,t=tn,n∈ℤ+,ΔXt=0,ΔC0t=0,ΔCet=u,t=nT,n∈ℤ+,where C0t and Cet represent the concentration of toxins in organism and in environment at time t, respectively. α≥0 is the decreasing rate of the growth rate associated with the uptake of the toxins, k>0 stands for the uptake rate of toxicant in the environment, g>0 and m>0 are the excretion rate and depuration rate, respectively, h>0 is the loss rate of toxicant in the environment, and u stands for the toxin input amount at every time. Let 0<t1<t2<⋯,limn⟶+∞tn=+∞,n∈ℤ+, where ℤ+ is the set of positive integers. When δn>0, the impulsive effects imply releasing population, while if δn<0, the impulsive effects indicate harvesting for population. In this paper, we always suppose that 1+δn>0 for all n∈ℤ+.The rest of the paper is organized as follows. In Section2, we give some preliminaries. The existence and uniqueness of the global positive solution of the model are given in Section 3. The sufficient conditions for the stochastic permanence and extinction are studied in Section 4. Some asymptotic properties of the solution are proved in Section 5. Finally, we give some numerical simulations to illustrate our results.


## 2. Preliminaries

Denote the generatorΓ=γklN×N of the Markov chain ξt given by(5)ℙξt+Δ=lξt=k=γklΔ+oΔ,ifk≠l,1+γkkΔ+oΔ,ifk=l,where Δ>0,γkl≥0 if k≠l while ∑l=1Nγkl=0. When ξt is irreducible, then ξt has a unique stationary distribution π=π1,π2,…,πN which is the solution of(6)πΓ=0,∑k=1Nπk=1,πk>0,∀k∈M.Consider a stochastic differential delay equation (SDDE) with Markovian switching and Lévy noise (see [12]) as follows:(7)dxt=fxt,xt−τt,t,ξtdt+gxt,xt−τt,t,ξtdBt+∫YJxt−,t,ξt,vN˜dt,dv,x0=x0∈Lℱ0−τ,0;ℝn,ξ0=ξ0∈M,where xt=x1t,x2t,…,xntT∈ℝn is the state vector and xt−τt=x1t−τt,x2t−τt,…,xnt−τtT∈ℝn is the delayed state vector. xt−=lims⟶txs. The time-varying delay τ⋅:ℝ+⟶0,τ is a Borel measurable function. f⋅,⋅,⋅,⋅:ℝn×ℝn×ℝ+×M⟶ℝn is the drift coefficient vector, g⋅,⋅,⋅,⋅:ℝn×ℝn×ℝ+×M⟶ℝn×n is the diffusion coefficient matrix, and J⋅,⋅,⋅,⋅:ℝn×ℝn×M×Y⟶ℝn.For eachk∈M, let Vx,y,t,k∈C1,2ℝn×ℝn×ℝ+×M;ℝ+ be any twice continuously differentiable function; the operator ℒVx,y,t,k:ℝn×ℝn×ℝ+×M⟶ℝ can be defined by(8)ℒVx,y,t,k=Vtx,t,k+Vxx,t,kfx,y,t,k+12tracegTx,y,t,kVxxx,t,kgx,y,t,k+∫YVx+Jx,t,k,v,t,k−Vx,t,k−Jx,t,k,vVxx,t,kλdv+∑l=1NγklVx,t,k,where(9)Vtx,t,k=∂Vx,t,k∂t,Vxx,t,k=∂Vx,t,k∂x1,∂Vx,t,k∂x2,…,∂Vx,t,k∂xn,Vxxx,t,k=∂2Vx,t,k∂xi∂xjn×n.Then, one has the generalized Itô’s formula:(10)dV=ℒVx,y,t,kdt+Vxx,t,kfx,y,t,kdBt+∫Yln1+Jx,t,k,vN˜dt,dv.Hypothesis 1.
(locally Lipschitz condition). For any integerh≥1, there exists a constant Lk>0 such that(11)fx,y,t,k−fx¯,y¯,t¯,k∨gx,y,t,k−gx¯,t¯,k≤Lkx−x¯+y−y¯,for those x,y,x¯,y¯∈ℝn with x∨y∨x¯∨y¯≤h and any t,k∈0,T×M.Hypothesis 2.
(linear growth condition). There is a constantL>0 such that(12)fx,y,t,k∨gx,y,t,k≤L1+x+y,for any x,y,t,k∈ℝn×ℝn×ℝ+×M.Hypothesis 3.
For eachξ>0, there is a constant Lξ that depends on ξ such that(13)∫YJx,t,k,v−Jy,t,k,v2λdv≤Lξx−y2,with x∨y≤ξ.
Here,Hypotheses 1–3 are the conservative conditions to check the existence and uniqueness of the global solution of (7). In this paper, Hypotheses 1–3 are always satisfied.
For simplicity, denote some notations(14)ft=1t∫0tfsds,t>0,g^=mink∈Mgk,gˇ=maxk∈Mgk,f∗=liminft⟶+∞ft,f∗=limsupt⟶+∞ft,bk=rk−σ2k2−∫Yγk,v−ln1+γk,vλdv,b¯k=2rk−σ2k−2∫Yγk,vλdv−2αkC0∗,η∗=limsupt⟶+∞1t∑0<tn<tln1+δn+∑k∈Mπkbk,η∗=liminft⟶+∞1t∑0<tn<tln1+δn+∑k∈Mπkbk.
In order to give the proof in this paper, we provide some assumptions.A.1: there exists a constant K1>0 such that(15)∫Yln1+γξt,v2λdv≤K1.A.2: 1+γξt,v>0, and there exists constant K2>0 such that(16)∫Yγξt,v−ln1+γξt,vλdv<K2.A.3: let the initial value X=β be positive and β∈Cg (see [31, 39]), which is defined by(17)Cg=ψ∈C−∞,0;ℝ+:ψCg=sup−∞<θ≤0erθφθ<+∞.There exists a probability measureρ and a constant r>0 such that(18)ρr=∫−∞0e−2rθdςθ<+∞.A.4: a=a^1−aˇ2−aˇ3>0.A.5: there exist two positive constants m and M such that(19)m≤∏0<tn<t1+δn≤M.A.6: ma^1>maxMaˇ2+aˇ3,Maˇ2eτ+aˇ3ρr.
We give some useful inequality in [40].(1)
(Exponential martingale inequality) LetT, α, β be any positive numbers. Then,(20)ℙsup0≤t≤T∫0tgsdBs−α2∫0tgs2ds>β≤e−αβ.(2)
(Chebyshev’s inequality) Ifc>0, p>0, X∈LpΩ,ℝd which is the family of ℝd-valued random variables X with EXp<∞. Then,(21)ℙω:Xω≥c≤c−pEXp.
Next, we consider the following subsystem of system (4):(22)dC0t=kCet−g+mC0tdt,dCet=−hCetdt,,t≠nT,n∈ℤ+,ΔC0t=0,ΔCet=u,t=nT,n∈ℤ+.
From [38], we can get the following lemma.Lemma 1 (see [38]).
System (22) has a unique globally asymptotical stable positive T-periodic solution C˜0t,C˜etT. If C00>C˜00, Ce0>C˜e0, then C0t>C˜0t, Cet>C˜et for all t≥0, where(23)C˜0t=C˜00e−g+mt−nT+kue−g+mt−nT−e−ht−nTh−g−m1−e−hT,C˜00=kue−g+mT−e−hTh−g−m1−e−g+mT1−e−hT,C˜et=ue−ht−nT1−e−hT,C˜e0=u1−e−hT,for t∈nT,n+1T, n∈ℤ+. In addition,(24)limt⟶+∞C0t=kuhg+mT≜C¯0.From Lemma1, system (4) can be replaced by the dynamical behaviors of the following limiting system:(25)dXt=Xtrξt−αξtC0t−a1ξtXt+a2ξtXt−τ+a3ξt∫−∞0Xt+θdςθdt+σξtXtdBt+xt−∫Yγξt,vN˜dt,dv,,t≠tn,n∈ℤ+,ΔXt=δnXt,t=tn,n∈ℤ+.


## 3. Positive and Global Solutions

Theorem 1.
For any initial dataX0,ξ0∈ℝ×M, system (25) has a unique positive solution Xt,ξt∈ℝ×M with probability oneProof.
Consider the following SDDEs with Markovian switching and without impulses:(26)dYt=Ytrξt−αξtC0t−a1ξt∏0<tn<t1+δnYt+a2ξt∏0<tn<t−τ1+δnYt−τ+a3ξt∫−∞0∏0<tn<t+θ1+δnYt+θdςθdt+σξtYtdBt+Yt−∫Yγξt,vN˜dt,dv,with initial value Y0,ξ0. By the theory of SDDEs with Markovian switching and Lévy jump, we refer the reader to [12]. System (26) has a unique global positive solution Yt,ξt.
LetXt=∏0<tn<t1+δnYt with initial value X0,ξ0.
SinceXt is continuous on each interval tn,tn+1⊂ℝ+, then(27)dXt=∏0<tn<t1+δndYt=∏0<tn<t1+δnYtrξt−αξtC0t−a1ξt∏0<tn<t1+δnYt+a2ξt∏0<tn<t−τ1+δnYt−τ+a3ξt∫−∞0∏0<tn<t+θ1+δnYt+θdςθdt+σξt∏0<tn<t1+δnYtdBt+∏0<tn<t1+δnYt−∫Yγξt,vN˜dt,dv=Xtrξt−αξtC0t−a1ξtXt+a2ξtXt−τ+a3ξt∫−∞0Xt+θdςθdt+σξtXtdBt+Xt−∫Yγξt,vN˜dt,dv,for t≠tn, n∈ℤ+.
And for∀tn∈ℝ+, we get(28)Xtn+=limt⟶tn+Xt=∏0<tn<t1+δnYtn+=1+δn∏0<tn<t1+δnYtn=1+δnXtn.
Moreover,(29)Xtn−=limt⟶tn−Xt=∏0<tn<t1+δnYtn−=∏0<tn<t1+δnYtn=Xtn.
Thus, system (25) has the unique global positive solution Xt,ξt.


## 4. Extinction and Persistence

Theorem 2.
WhenA.1–A.4 hold, if η∗<α^C¯0, then(30)limt⟶+∞Xt=0,a.s.Namely, the populationXt of system (25) is extinct.Proof.
Applying Itô’s formula to system (26), we have(31)dlnYt=rξt−σ2ξt2−∫Yγξt,v−ln1+γξt,vλdv−αξtC0t−a1ξt∏0<tn<t1+δnYt+a2ξt∏0<tn<t−τ1+δnYt−τ+a3ξt∫−∞0∏0<tn<t+θ1+δnYt+θdξθdt+σξtdBt+∫Yln1+γξt,vN˜dt,dv.
Integrating both sides of (31) from 0 to t yields(32)lnYt−lnY0=∫0tbξs−αξsC0s−a1ξs∏0<tn<s1+δnYs+a2ξs∏0<tn<s−τ1+δnYs−τ+a3ξs∫−∞0∏0<tn<s+θ1+δnYs+θdξθds+σξsdBs+∫Yln1+γξs,vN˜ds,dv=∫0tbξs−αξsC0s−a1ξsXs+a2ξsXs−τ+a3ξs∫−∞0Xs+θdξθds+∫0tσξsdBs+∫0t∫Yln1+γξs,vN˜ds,dv,owing to(33)∫0ta3ξs∫−∞0Xs+θdξθds=∫0ta3ξs∫−∞−sXs+θdξθ+∫−s0Xs+θdξθds=∫0ta3ξsds∫−∞−sers+θXs+θe−rs+θdξθ+∫−t0dξθ∫−θta3ξsXs+θds≤aˇ3φCg∫0te−rsds∫−∞0e−2rθdξθ+aˇ3∫−∞0dξθ∫0tXsds≤1raˇ3φCgρr1−e−rt+aˇ3∫0tXsds.
Then, we have(34)lnYt−lnY0=∫0tbξs−αξsC0s−a1ξsXs+a2ξsXs−τ+a3ξs∫−∞0Xs+θdξθds+∫0tσξsdBs+∫0t∫Yln1+γξs,vN˜ds,dv≤∫0tbξs−αξsC0s−a^1−aˇ2−aˇ3Xsds+1raˇ3φCgρr1−e−rt+∫−τ0a2ξs+τXsds+∫0tσξsdBs+∫0t∫Yln1+γξs,vN˜ds,dv≤∫0tbξs−αξtC0s−aXsds+1raˇ3φCgρr1−e−rt+∫−τ0a2ξs+τXsds+M1t+M2t,where(35)M1t=∫0tσξsdBs,M2t=∫0t∫Yln1+γξs,vN˜ds,dv.
SinceM1t and M2t are local martingales, the quadratic variations are(36)M1t,M1t=∫0tσ2ξsds≤σˇ12t,M2t,M2t=∫0t∫Yln1+γξs,v2λdvds≤K1t.
Making use of the strong law of large numbers for local martingales (see [41]) yields(37)limt⟶+∞M1tt=0,limt⟶+∞M2tt=0,a.s.
From (34), we can get that(38)∑0<tn<tln1+δn+lnYt≤∑0<tn<tln1+δn+∫0tbξsds−∫0tαξtC0sds−∫0taXsds+1raˇ3φCgρr1−e−rt+∫−τ0a2ξs+τXsds+lnY0+M1t+M2t.
Thus,(39)lnXt≤∑0<tn<tln1+δn+∫0tbξsds−∫0tαξtC0sds−∫0taXsds+1raˇ3φCgρr1−e−rt+∫−τ0a2ξs+τXsds+lnY0+M1t+M2t≤∑0<tn<tln1+δn+∫0tbξsds−α^∫0tC0sds−a∫0tXsds+1raˇ3φCgρr1−e−rt+∫−τ0a2ξs+τXsds+lnY0+M1t+M2t.
Taking superior limit on both sides of (39) and applying the ergodicity of ξ⋅ and (37), we obtain(40)limsupt⟶+∞lnXtt≤limsupt⟶+∞1t∑0<tn<tln1+δn+∑k∈Mπkbk−α^C¯0≔η∗−α^C¯0<0,a.s.Theorem 3.
WhenA.1–A.4 hold, if η∗≥α^C¯0, then(41)limsupt⟶+∞1t∫0tXsds≤η∗−α^C¯0a=X¯∗.
Particularly, ifη∗=α^C¯0, then limt⟶+∞1/t∫0tXsds=0, that is, the population Xt of system (25) is nonpersistent in the mean.Proof.
For∀ε>0, there exists a constant T>0, for all 0<T<n−1≤t≤n, such that(42)1tlnY0+1raˇ3φCgρr1−e−rt+∫−τ0a2ξs+τXsds+M1t+M2t≤ε2,1t∑0<tn<tln1+δn+∫0tbξsds−α^∫0tC0sds≤η∗−α^C¯0+ε2.
Substituting above inequalities into (39), we have(43)lnXt≤∑0<tn<tln1+δn+∫0tbξsds−α^∫0tC0sds−a∫0tXsds+1raˇ3φCgρr1−e−rt+∫−τ0a2ξs+τXsds+lnY0+M1t+M2t≤η∗−α^C¯0+εt−a∫0tXsds,a.s.
Denotegt=∫0tXsds; then, we have dgt/dt=Xt. Taking exponent on both sides of (43) yields(44)eagtdgtdt≤eη∗−α^C¯0+εt.
Integrating (44) from T to t, we can show that(45)eagt≤aη∗−α^C¯0+εeη∗−α^C¯0+εt+eagT0−aη∗−α^C¯0+εeη∗−α^C¯0+εT.
Taking logarithm of (45) yields(46)∫0tXsds≤1alnaη∗−α^C¯0+εeη∗−α^C¯0+εt+eagT−aη∗−α^C¯0+εeη∗−α^C¯0+εT.
Taking superior limit on (46) elicits that(47)limsupt⟶+∞1t∫0tXsds≤limsupt⟶+∞1atlnaη∗−α^C¯0+εeη∗−α^C¯0+εt+eagT−aη∗−α^C¯0+εeη∗−α^C¯0+εT.
Utilizing L’Hospital’s rule results in(48)limsupt⟶+∞1t∫0tXsds≤limsupt⟶+∞η∗−α^C¯0+εa=η∗−α^C¯0a=X¯∗.Theorem 4.
WhenA.1–A.4 hold, if η∗>α^C¯0, then the population Xt of system (25) is weakly persistent a.s.Proof.
DenoteL=ωlimsupt⟶+∞Xt,ω=0; suppose that ℙL>0. Then, it follows from (39) that(49)lnXt−lnX0t≤1t∑0<tn<tln1+δn+∫0tbξsds−α^∫0tC0sds−at∫0tXsds+1rtaˇ3φCgρr1−e−rt+1t∫−τ0a2ξs+τXsds+M1tt+M2tt.
For∀ω∈L, we have limt⟶+∞Xt,ω=0. As a result,(50)limsupt⟶+∞lnXt,ω−lnX0t≤0,limsupt⟶+∞at∫0tXsds=0.
From (49), one has(51)0≥limsupt⟶+∞lnXt,w=η∗−α^C¯0>0,which is a contradiction.Remark 1.
Through Theorems2–4, we find an interesting biological phenomenon: when η∗>α^C¯0, the population Xt is weakly persistent; when η∗<α^C¯0, the population Xt goes to extinction, which means that the persistence and extinction of Xt depend on η∗ and the absorption intensity of toxins α^C¯0.Theorem 5.
WhenA.1–A.3 hold, if η∗>αˇC¯0, then(52)liminft⟶+∞1t∫0tXsds≥η∗−αˇC¯0aˇ1=X¯∗,a.s.That is, the populationXt of system (25) is persistent in the mean a.s.Proof.
Applying Itô’s formula to (26) yields(53)dlnYt=rξt−σ2ξt2−∫Yγξt,v−ln1+γξt,vλdv−αξtC0t−a1ξt∏0<tn<t1+δnYt+a2ξt∏0<tn<t−τ1+δnYt−τ+a3ξt∫−∞0∏0<tn<t+θ1+δnYt+θdξθdt+σξtdBt+∫Yln1+γξt,vN˜dt,dv≥bξt−αˇC0t−aˇ1Xtdt+σξtdBt+∫Yln1+γξt,vN˜dt,dv.
Calculating inequality (53), one has(54)lnXt≥∑0<tn<tln1+δn+∫0tbξsds−αˇ∫0tC0sds−aˇ1∫0tXsds+lnY0+M1t+M2t.
According to the properties of the limit, there existsT1>0 such that for ∀ε1>0,(55)∑0<tn<tln1+δn+∫0tbξsds−αˇ∫0tC0sds≥η∗−αˇC¯0−ε14t,lny0≥−ε14t,M1t≥−ε14t,M2t≥−ε14t.
Then, inequality (54) becomes(56)lnXt≥η∗−αˇC¯0−ε1t−aˇ1∫0tXsds,for t>T1. By using a method similar to Theorem 3, we can obtain that(57)liminft⟶+∞1t∫0tXsds≥liminft⟶+∞η∗−αˇC¯0−ε1aˇ1=η∗−αˇC¯0aˇ1=X¯∗.Theorem 6.
WhenA.1–A.3, A.5, and A.6 hold, if ∑k=1Nπkb¯k>0, then the population Xt of system (25) is stochastically permanent.Proof.
First, we prove that for∀ε>0, there exists a constant b1>0 such that(58)liminft⟶+∞ℙXt≥b1≥1−ε.
Define(59)V1Y=1Y2,V2Y=ϑk1+V1Yϱ,V3Y=eκtV2Y,where ϱ and κ are sufficiently small positive constants and satisfy(60)ϑkϱb¯k−2ϱ2σ2k−∑l=1Nγklϑl>0,1≤k≤N,ϑkϱb¯k−2ϱ2σ2k−∑l=1Nγklϑl−κϑk>0,1≤k≤N,where ϑ=ϑ1,ϑ2,…,ϑNT≫0 such that Aϱϑ≫0.
From (26), we can calculate that(61)dV1Y=2V1Y−rξt+32σ2ξt+αξtC0t+a1ξt∏0<tn<t1+δnYt−a2ξt∏0<tn<t−τ1+δnYt−τ−a3ξt∫−∞0∏0<tn<t+θ1+δnYt+θdςθ+12∫Y11+γξt,v2−1+2γξt,vλdvdt−2V1YσξtdBt+V1Y∫Y11+γξt,v2−1N˜dt,dv≤2V1Y−rξt+32σ2ξt+αξtC0t+a1ξt∏0<tn<t1+δnYt+12∫Y11+γξt,v2−1+2γξt,vλdvdt−2V1YσξtdBt+V1Y∫Y11+γξt,v2−1N˜dt,dv.
Applying the generalized Itô’s formula yields(62)dV2Y≤1+V1Yϱ−2ϱϑk1+V1Y−2V1Yrk−32σ2k−αkC0t−12∫Y11+γk,v2−1+2γk,vλdv+2a1k∏0<tn<t1+δnV11/2Ydt+2ϱϱ−1ϑkσ2k1+V1Yϱ−2V12Y+ϑk∫Y1+V1Y+V1Y11+γk,v2−1ϱ−1+V1Yϱ−ϱ1+V1Yϱ−1V1Y11+γk,v2−1λdvdt+∑l=1Nγklϑl1+V1Yϱdt−2ϱϑkσk1+V1Yϱ−1V1YdBt+∫Y1+V1Y+V1Y11+γk,v2−1ϱ−1+V1YϱN˜dt,dv≤1+V1Yϱ−2−ϑk2rkϱ−σ2kϱ−2ϱ∫Yγk,vλdv−2αkC0∗−2ϱ2σ2k−∑l=1NγklϑlV12Y+2a1kMV13/2Y−ϑk2rkϱ−3σ2kϱ−2ϱ∫Yγk,vλdv−2αkC0∗−∑l=1NγklϑlV1Y+2a1kMV11/2Y+∑l=1Nγklϑldt−2ϱϑkσk1+V1Yϱ−1V1YdBt+∫Yϑk1+V1Y+V1Y11+γk,v2−1ϱ−1+V1YϱN˜dt,dv=1+V1Yϱ−2−ϑkϱb¯k−2ϱ2σ2k−∑l=1NγklϑlV12Y+2a1kMV13/2Y−ϑkϱb¯k−2ϱσ2k−∑l=1NγklϑlV1Y+2a1kMV11/2Y+∑l=1Nγklϑldt−2ϱϑkσk1+V1Yϱ−1V1YdBt+∫Yϑk1+V1Y+V1Y11+γk,v2−1ϱ−1+V1YϱN˜dt,dv,and we use the following equation:(63)∫Y1+V1Y+V1Y11+γk,v2−1ϱ−1+V1Yϱλdv<0.
Therefore, we obtain that(64)dV3Y=κeκtV2Ydt+eκtdV2Y≤eκt1+V1Yϱ−2−ϑkϱb¯k−2ϱ2σ2k−∑l=1Nγklϑl−κϑkV12Y+2a1kMV13/2Y−ϑkϱb¯k−2ϱσ2k−∑l=1Nγklϑl−2κϑkV1Y+2a1kMV11/2Y+∑l=1Nγklϑl+κϑkdt−2ϱϑkeκtσk1+V1Yϱ−1V1YdBt+ϑkeκt∫Y1+V1Y+V1Y11+γk,v2−1ϱ−1+V1YϱN˜dt,dv=eκtHYdt−2ϱϑkeκtσk1+V1Yϱ−1V1YdBt+ϑkeκt∫Y1+V1Y+V1Y11+γk,v2−1ϱ−1+V1YϱN˜dt,dv,where(65)HY=1+V1Yϱ−2−ϑkϱb¯k−2ϱ2σ2k−∑l=1Nγklϑl−κϑkV12Y+2a1kMV13/2Y−ϑkϱb¯k−2ϱσ2k−∑l=1Nγklϑl−2κϑkV1Y+2a1kMV11/2Y+∑l=1Nγklϑl+κϑk.
SinceH=supY∈ℝ+HY<+∞,(66)dV3Y≤eκtHdt−2ϱϑkeκtσk1+V1Yϱ−1V1YdBt+ϑkeκt∫Y1+V1Y+V1Y11+γk,v2−1ϱ−1+V1YϱN˜dt,dv.
Calculating (66), it is not difficult to show that(67)Eϑk1+V1Yϱ≤ϑkV2Y0Ee−κt+HκE1−e−κt.
Thus,(68)limsupt⟶+∞EYt−2ϱ≤limsupt⟶+∞E1+V1Ytϱ≤limsupt⟶+∞V2Y0eκt+H1−e−κtκmink∈Mϑk=H¯.
Consequently,(69)limsupt⟶+∞EXt−2ϱ=limsupt⟶+∞E∏0<tn<t1+δn−2ϱYt−2ϱ≤M−2ϱH¯.
Then, for∀ε>0, let b1=Mε/H¯1/2ϱ; using Chebyshev’s inequality, we have(70)limsupt⟶+∞ℙXt<b1=limsupt⟶+∞ℙXt−2ϱ>b1−2ϱ≤limt⟶+∞EXt−2ϱb1−2ϱ≤ε.
In other words,(71)liminft⟶+∞ℙXt≥b1≥1−ε.
Next, we prove that for∀ε>0, there exists b2>0 such that(72)liminft⟶+∞ℙXt≤b2≥1−ε.
Applying generalized Itô’s formula to (26):(73)dYt=Ytrξt−αξtC0t−a1ξt∏0<tn<t1+δnYt+a2ξt∏0<tn<t−τ1+δnYt−τ+a3ξt∫−∞0∏0<tn<t+θ1+δnYt+θdςθdt+σξtYtdBt+Yt−∫Yγξt,vN˜dt,dv≤rξt−α^C0tYt−ma^1Y2t+Maˇ2YtYt−τ+Maˇ3Yt∫−∞0Yt+θdςθdt+σξtYtdBt+etYt−∫Yγξt,vN˜dt,dv≤rξt−α^C0tYt−ma^1Y2t+Maˇ22Y2t+Maˇ22Y2t−τ+Maˇ32Y2t+Maˇ32∫−∞0Y2t+θdςθdt+σξtYtdBt+Yt−∫Yγξt,vN˜dt,dv.
Thus,(74)detYt≤et1+rξt−α^C0tYt−ma^1Y2t+Maˇ22Y2t+Maˇ22Y2t−τ+Maˇ32Y2t+Maˇ32∫−∞0Y2t+θdςθdt+etσξtYtdBt+etYt−∫Yγξt,vN˜dt,dv=etGYt−ma^12Y2t+Maˇ22Y2t−τ+Maˇ32∫−∞0Y2t+θdςθdt+etσξtYtdBt+etYt−∫Yγξt,vN˜dt,dv,where(75)GYt=1+rξt−α^C0tYt−ma^12Y2t+Maˇ22Y2t+Maˇ32Y2t.
According toma^1>Maˇ2+aˇ3, we get(76)G=supY∈ℝ+GY<+∞.
Then,(77)etYt−Y0≤Get−1−ma^12∫0tesY2sds+Maˇ22∫0tesY2s−τds+Maˇ32∫0tes∫−∞0Y2s+θdςθds+∫0tesσξsYsdBs+∫0tes∫Yγξs,vYt−N˜ds,dv.
On the one hand,(78)Maˇ22∫0tesY2s−τds=Maˇ22∫−τt−τes+τY2sds≤Maˇ22∫−τ0es+τY2sds+Maˇ2eτ2∫0tesY2sds,Maˇ32∫0tes∫−∞0Y2s+θdςθds=Maˇ32∫0tes∫−∞−sY2s+θdςθ+∫−s0Y2s+θdςθds=Maˇ32∫0tesds∫−∞−se2rs+θY2s+θe−2rs+θdςθ+Maˇ32∫−t0dςθ∫−θtesY2s+θds=Maˇ32∫0tesds∫−∞−se2rs+θY2s+θe−2rs+θdςθ+Maˇ32∫−t0dςθ∫0t+θes−θY2sds≤Maˇ32ξCg2∫0te1−2rsds∫−∞0e−2rθdςθ+Maˇ32∫−∞0e−θdςθ∫0tesY2sds≤Maˇ32ξCg2ρrt+Maˇ3ρr2∫0tesY2sds.
Therefore,(79)etYt≤Y0+Get−1−ma^1−Maˇ2eτ+aˇ3ρr2∫0tesY2sds+Maˇ22∫−τ0es+τY2sds+Maˇ32ξCg2hrt+∫0tesσξsYsdBs+∫0tes∫Yγξs,vYt−N˜ds,dv.
Taking expectations on both sides of (79) yields(80)limsupt⟶+∞EYt≤G.
This leads to(81)limsupt⟶+∞EXt≤limsupt⟶+∞E∏0<tn<t1+δnYt≤MG.
Then, for∀ε>0, let b2=MG/ε, and we have(82)limsupt⟶+∞ℙXt>b2≤limt⟶+∞EXtb2=ε.
That is,(83)limsupt⟶+∞ℙXt≤b2≥1−ε.
From (71) and (83), Xt of system (25) is stochastically permanent.


## 5. Asymptotic Properties

Theorem 7.
WhenA.1–A.3, A.5, and A.6 hold and any solution Xt of system (25) have the property that(84)limsupt⟶+∞lnXtlnt≤1,a.s.,and if ∑k=1Nπkb¯k>0,(85)liminft⟶+∞lnXtlnt≥−12ϱ,a.s.Proof.
From (73), we have(86)dYt=Ytrξt−αξtC0t−a1ξt∏0<tn<t1+δnYt+a2ξt∏0<tn<t−τ1+δnYt−τ+a3ξt∫−∞0∏0<tn<t+θ1+δnYt+θdςθdt+σξtYtdBt+Yt−∫Yγξt,vN˜dt,dv.
Then,(87)Elimsupt≤ν≤t+1Yν≤EYt+Esupt≤ν≤t+1∫tt+1Ysds+Elimsupt≤ν≤t+1∫tνσξsYsdBs+Elimsupt≤ν≤t+1∫tν∫Yγξs,vysN˜ds,dv.
By the Burkholder–Davis–Gundy inequality (see [40]), one has(88)Esupt≤ν≤t+1∫tνσξsYsdBs≤32E∫tt+1σ2ξsY2sds1/2≤42σˇEsupt≤ν≤t+1Ys,(89)Elimsupt≤ν≤t+1∫tν∫Yγξs,vYsN˜ds,dv≤32E∫tν∫Yγ2ξs,vY2sNds,dv1/2≤42∫Yγˇ2λdv1/2Esupt≤ν≤t+1Yν.
Substituting (80), (88), and (89) into (87) yields(90)Elimsupt≤ν≤t+1Yν≤EYt+BEsupt≤ν≤t+1Yν,where B=G+42σˇ+42∫Yγˇλdv1/2.
In view of (90), we have(91)Elimsupt≤θ≤t+1Yν≤B¯.
By Chebyshev’s inequality, for∀ε>0, we can obtain(92)ℙωsupζ≤t≤ζ+1Yt>ζ1+ε≤B¯ζ1+ε,ζ=1,2,….
By using the Borel–Cantelli lemma [27], we get(93)supζ≤t≤ζ+1Yt≤ζ1+ε.
Therefore, forζ≤t≤ζ+1 and ζ≥ζ0ω, if t⟶∞, then ζ⟶∞.(94)limsupt⟶+∞lnYtlnt≤lnsupζ≤t≤ζ+1Ytlnt≤lnζ1+εlnt≤1+ε.
Letε⟶0; then, limsupt⟶+∞lnYt/lnt≤1, that is,(95)limsupt⟶+∞lnXtlnt=limsupt⟶+∞∑0<tn<tln1+δn+lnYtlnt≤1.
Now, we prove the lower growth rate of the solution.
From (68), we obtain(96)limsupt⟶+∞E1+V1Yϱ≤H¯,where H¯ is a positive constant.
Similarly,(97)d1+V1Yϱ≤ϱ1+V1Yϱ−2−2rξt−12σ2ξt−ϱσ2ξt−αξtC0t−∫Yγξt,vλdvV12Y+2a1ξt∏0<tn<t1+δnV13/2Y−2rξt−32σ2ξt−αξtC0t−∫Yγξt,vλdvV1Y+2a1ξt∏0<tn<t1+δnV11/2Ydt−2ϱσξt1+V1Yϱ−1V1YdBt+∫Y1+V1Y+V1Y11+γξt,v2−1ϱ−1+V1YϱN˜dt,dv≤ϱ1+V1Yϱ−2D1V12Y+D2V13/2Y+D3V1Y+D2V11/2Ydt−2ϱσξt1+V1Yϱ−1V1YdBt+∫Y1+V1Y+V1Y11+γξt,v2−1ϱ−1+V1YϱN˜dt,dv,where(98)D1=−2rξt−12σ2ξt−ϱσ2ξt−αξtC0t−∫Yγξt,vλdv,D2=2Ma1ξt,D3=−2rξt−32σ2ξt−αξtC0t−∫Yγξt,vλdv.
LetD>0 be sufficiently large for(99)ϱD1V12Y+D2V13/2Y+D3V1Y+D2V11/2Y≤D1+V1Y2.
Then,(100)d1+V1Yϱ≤D1+V1Yϱ+2ϱσξt1+V1Yϱ−1V1YdBt+∫Y1+V1Y+V1Y11+γξt,v2−1ϱ−1+V1YϱN˜dt,dv.
Letζ=1,2,…, and making use of (100) gives(101)Elimsupk−1ζ≤t≤kζ1+V1Ytϱ≤E1+V1Yk−1ζϱ+DElimsupk−1ζ≤t≤kζ∫k−1ζt1+V1Ytϱds+Elimsupk−1ζ≤t≤kζ∫k−1ζt2ϱσξt1+V1Yϱ−1V1YdBt+Elimsupk−1ζ≤t≤kζ∫k−1ζtϱ1+V1Yϱ−1V1Y∫Y11+γξt,v2−1N˜dt,dv.
We compute that(102)DElimsupk−1ζ≤t≤kζ∫k−1ζt1+V1Ysϱds≤DE∫k−1ζt1+V1Ysϱds≤DζElimsupk−1ζ≤t≤kζ1+V1Ytϱ,(103)Elimsupk−1ζ≤t≤kζ∫k−1ζt2ϱσξt1+V1Yϱ−1V1YdBs≤32E∫k−1ζkζ4ϱ2σ2ξt1+V1Y2ϱ−2V12Yds1/2≤82ϱσˇE∫k−1ζkζ1+V1Y2ϱds1/2≤82ζϱσˇElimsupk−1ζ≤t≤kζ1+V1Ytϱ,(104)Elimsupk−1ζ≤t≤kζ∫k−1ζtϱ1+V1Yϱ−1V1Y∫Y11+γξt,v2−1N˜dt,dv≤32E∫k−1ζt∫Yϱ21+V1Y2ϱ−2V12Y∫Y11+γξt,v2−12Nds,dv1/2≤32ϱ∫Y11+γ^2−12λdv1/2E∫k−1ζt∫Y1+V1Y2ϱNds,dv1/2≤42ζϱ∫Y11+γ^2−12λdv1/2Esupk−1ζ≤t≤kζ1+V1Yϱ.
Substituting (102)–(104) into (101) results in(105)Elimsupk−1ζ≤t≤kζ1+V1Ytϱ≤E1+V1Yk−1ζϱ+CElimsupk−1ζ≤t≤kζ1+V1Ytϱ,where C=Dζ+82ζϱσˇ+42ζϱ∫Y1/1+γ^2−12λdv1/2.
Then, from (96) and the above inequality, we get(106)Elimsupk−1ζ≤t≤kζ1+V1Ytϱ≤C¯.
Then, for∀ε>0, we have(107)ℙωsupk−1ζ≤t≤kζ1+V1Ytϱ>kζ1+ε≤C¯kζ1+ε,k=1,2,….
Thus,(108)ln1+V1Ytϱlnt≤1+εlnkζlnk−1ζ.
That is,(109)limsupt⟶+∞ln1+V1Ytϱlnt≤1+ε.
Lettingε⟶0 gives(110)limsupt⟶+∞lnYt−2ϱlnt≤1.
Consequently,(111)liminft⟶+∞lnYtlnt≥−12ϱ.
That is,(112)liminft⟶+∞lnXtlnt≥liminft⟶+∞lnmYtlnt≥−12ϱ.


## 6. Numerical Simulations

Now, we use Matlab to analyze our results. We choose the same initial valueX0,C00,Ce0=1,0.3,0.6 and the same parameter value as follows:(113)r1=1,α1=0.2,a11,a21,a31=0.25,0.1,0.02,r2=0.8,α2=0.3,a12,a22,a32=0.3,0.03,0.05,k=0.6,g=0.26,m=0.05,h=0.23,T=4,τ1=2,λY=1.In system (4), let ξt∈S=1,2 and the generator(114)Γ=−773−3.By solving (6), we get a stationary distribution π=π1,π2=0.3,0.7.Now, we discuss the effects of different parameters on the population.Example 1.
Letγ1=0.5,γ2=1.1,σ1=0.04,σ2=0.1,u=1,δn=e0.3−1. Then, we obtain η∗=0.8771>0.4208=α^C¯0. According to Theorem 4, we know that Xt is weakly persistent (see Figure 1(a)). It can be seen from Figures 1(b) and 1(c) that the toxins in the organism C0t and the toxins in the environment Cet have a periodic solution.Figure 1
(a) Markov chains, (b) the time sequence diagram, and (c) the phase portrait ofC0t and Cet.
(a)(b)(c)


### 6.1. Effect of the White Noiseσ

Choosing the same parameters as in Example1 but σ: by comparing Figure 1(a) and Figures 2(a) and 2(b), we found that the disturbance of population in the stochastic model is positively correlated with the intensity of white noise. By comparing Figures 2(a)–2(c), the population will go extinct with increasing intensity of white noise. When the intensity of white noise is less than a certain level, the population still be persistent, while the white noise with large intensity may cause population extinction.Figure 2
Effect of the white noise: (a)σ1=0.2,σ2=0.2; (b) σ1=0.5,σ2=0.5; (c) σ1=1,σ2=1.
(a)(b)(c)


### 6.2. Effect of the Lévy Noiseγ

Choosing the same parameters as in Example1 but σ1=0.4,σ2=0.4, and γ: from Figure 3, it is concluded that the Lévy noise has a large impact on the persistence of population. It is shown that the number of the population will decrease when the intensity of the Lévy noise increases. When the intensity of the Lévy noise became larger, the population became extinct. Therefore, Lévy noise will not only reduce the number of species but also lead to population extinction.Figure 3
Effect of the Lévy noise: (a)γ1=3,γ2=3; (b) γ1=10,γ2=10.
(a)(b)


### 6.3. Effect of Telegraph Noise

Choosing the same parameters as in Example1 but δn=0: from Figure 4, we find that all parameter values are the same except the Markov chain. Since telegraph noise is described by a Markov chain, choosing a different Markov chain will produce different results.Figure 4
Effect of telegraph noise.
(a)(b)


### 6.4. Effect of the Exogenous Total Toxicant Inputu

Choosing the same parameters as in Example1 but u: from Figure 5, we find that the population can still be persistent when the total input of exogenous toxicant is small (see Figures 5(a) and 5(b)), while the population will go extinct when the total input of exogenous toxicant is strong (see Figure 5(c)).Figure 5
Effect of the exogenous total toxicant input: (a)u=1; (b) u=2; (c) u=3.
(a)(b)(c)


### 6.5. Effect of Impulseδn

Choosing the same parameters as in Example1 but δn: in Figure 6, we describe the impact of pulse harvest on the population. By comparing Figures 6(a)–6(c), we find that with the increase of the harvest, the population will gradually decrease until it becomes completely extinct.Figure 6
Effect of impulse: (a)δn=e−0.3−1; (b) δn=e−0.7−1; (c) δn=e−2−1.
(a)(b)(c)


## 6.1. Effect of the White Noiseσ

Choosing the same parameters as in Example1 but σ: by comparing Figure 1(a) and Figures 2(a) and 2(b), we found that the disturbance of population in the stochastic model is positively correlated with the intensity of white noise. By comparing Figures 2(a)–2(c), the population will go extinct with increasing intensity of white noise. When the intensity of white noise is less than a certain level, the population still be persistent, while the white noise with large intensity may cause population extinction.Figure 2
Effect of the white noise: (a)σ1=0.2,σ2=0.2; (b) σ1=0.5,σ2=0.5; (c) σ1=1,σ2=1.
(a)(b)(c)


## 6.2. Effect of the Lévy Noiseγ

Choosing the same parameters as in Example1 but σ1=0.4,σ2=0.4, and γ: from Figure 3, it is concluded that the Lévy noise has a large impact on the persistence of population. It is shown that the number of the population will decrease when the intensity of the Lévy noise increases. When the intensity of the Lévy noise became larger, the population became extinct. Therefore, Lévy noise will not only reduce the number of species but also lead to population extinction.Figure 3
Effect of the Lévy noise: (a)γ1=3,γ2=3; (b) γ1=10,γ2=10.
(a)(b)


## 6.3. Effect of Telegraph Noise

Choosing the same parameters as in Example1 but δn=0: from Figure 4, we find that all parameter values are the same except the Markov chain. Since telegraph noise is described by a Markov chain, choosing a different Markov chain will produce different results.Figure 4
Effect of telegraph noise.
(a)(b)


## 6.4. Effect of the Exogenous Total Toxicant Inputu

Choosing the same parameters as in Example1 but u: from Figure 5, we find that the population can still be persistent when the total input of exogenous toxicant is small (see Figures 5(a) and 5(b)), while the population will go extinct when the total input of exogenous toxicant is strong (see Figure 5(c)).Figure 5
Effect of the exogenous total toxicant input: (a)u=1; (b) u=2; (c) u=3.
(a)(b)(c)


## 6.5. Effect of Impulseδn

Choosing the same parameters as in Example1 but δn: in Figure 6, we describe the impact of pulse harvest on the population. By comparing Figures 6(a)–6(c), we find that with the increase of the harvest, the population will gradually decrease until it becomes completely extinct.Figure 6
Effect of impulse: (a)δn=e−0.3−1; (b) δn=e−0.7−1; (c) δn=e−2−1.
(a)(b)(c)


## 7. Conclusion

In this paper, we explore the dynamics of a stochastic delay hybrid logistic model with two-pulse perturbations. First, by using Itô’s formula, exponential martingale inequality, Chebyshev’s inequality, and other mathematical skills, we establish some sufficient conditions for extinction, nonpersistence in the mean, weak persistence, persistence in the mean, and stochastic permanence. Then, the asymptotic properties of the lower growth rate and the upper growth rate of the solution are estimated.Now, we give the key results as follows:(I) (1) Ifη∗<α^C¯0, then the population Xt is extinct.(2) Ifη∗=α^C¯0, then the population Xt is nonpersistent in the mean.(3) Ifη∗>α^C¯0, then the population Xt is weakly persistent.(4) Ifη∗>αˇC¯0, then the population Xt is persistent in the mean.(5) If∑k=1Nπkb¯k>0, then the population Xt is stochastically permanent.(II) The solutionXt obeys(115)X¯∗=η∗−αˇC¯0aˇ1≤liminft⟶+∞1t∫0tXsds≤limsupt⟶+∞1t∫0tXsds≤η∗−α^C¯0a=X¯∗,a.s.(III) The solution satisfies(116)limsupt⟶+∞lnXtlnt≤1,liminft⟶+∞lnXtlnt≥−12ϱ,a.s.From our results and analysis, we can obtain the following conclusions. (1) Both white noise and Lévy noise tend to have negative effects on the persistence of population. However, the Lévy jump may have a greater influence than white noise on the persistence of population. (2) If the choice of telegraph noise is different, the persistence of the population will produce different results. (3) The total input of exogenous toxicant has a great disadvantaged influence on persistence of the population. With the increase of the total input of exogenous toxicant, the number of population will decrease, which enlightens us to reduce pollutant emissions to protect the ecological system. (4) The pulse release of the population is beneficial to the growth of the population; the pulse harvest of the population is beneficial to the growth of the population under reasonable conditions, but once overharvested, it will cause the population to become extinct.These give us significant enlightenment: (1) when fishermen are fishing, they must maintain reasonable fishing to avoid overfishing that can lead to population extinction; (2) the discharge of factory sewage, exhaust gas, domestic sewage, etc., should be strictly controlled, and we should do our best to reduce the discharge of polluted toxins; and (3) humans should reduce interference with populations.



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