# Dynamics Analysis of a Stochastic Delay Gilpin-Ayala Model with Markovian Switching

Authors: Qiaoqin Gao; Zhijiang Luo; Guirong Liu
Journal: Discrete Dynamics in Nature and Society
 (2019)

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## Abstract

This paper considers a stochastic delay Gilpin-Ayala model with Markovian switching. Using Lyapunov method, we show existence and uniqueness of global positive solution. Then, by using Chebyshev’s inequality,M-matrix method, and BDG’s inequality, stochastic permanence and asymptotic estimations of solutions are studied. Finally, numerical simulations illustrate the theoretical results. Our results generalize and improve the existing results.


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## Body

## 1. Introduction

Logistic equation is one of the most important and most widely used mathematical models to describe the growth of biological species. In [1], Gilpin and Ayala introduced the following generalized logistic equation (Gilpin-Ayala model):(1)dxtdt=xtr-axλt,where x(t) stands for the population size; r,a, and λ are positive constants. For more details, see [1, 2].Population systems are often subject to environmental noise and there are different types of noise. Therefore, stochastic population systems have been studied extensively. See [3–9] and the references therein. In [8], by using Itô’s formula and the exponential martingale inequality, the authors studied the stationary distribution, ergodicity, and extinction of the following stochastic Gilpin-Ayala model:(2)dxt=xtr-axλtdt+σ1xtdB1t+σ2x1+λtdB2t.Further, [9] obtained the stationary distribution and ergodicity of model (2) and improved the corresponding results in [8]. In addition, there is another type of environment noise, namely, colour noise, which can be demonstrated as a switching between two or more regimes of environment. See [10–18] and the references therein. Reference [11] considered the following stochastic Gilpin-Ayala model under regime switching:(3)dxt=xtrγt-aγtxλtdt+σ1γtxtdB1t+σ2γtx1+θtdB2t,where there exist λ>0,θ>0, and studied persistence, extinction, nonpersistence, and global attractivity of (3). Further, [12] obtained the lower-growth rate and the upper-growth rate of the positive solution of (3). Reference [13] considered the following stochastic Gilpin-Ayala population model with Markovian switching:(4)dxt=xtrγt-aγtxλγttdt+σ1γtxtdBt,where the Gilpin-Ayala parameter λ is also allowed to switch. In [13], the authors established the global stability of the trivial equilibrium state of (4). In addition, they obtained extinction, persistence, and existence of a stationary distribution of (4).On the other hand, many population systems depend on not only present states but also past states. See [19–26]. In [26], the authors considered a delay logistic model under regime switching(5)dxt=xtrγt-aγtxt+bγtxt-τdt+σ1γtxtdBt.They obtained stochastically ultimate boundedness, stochastic permanence, and extinction as well as asymptotic estimations of solutions.Motivated by the works mentioned above, in this paper, we consider a more general delay stochastic Gilpin-Ayala model under Markovian switching(6)dxt=xtrγt-aγtxλt+bγtxλt-τdt+σ1γtxtdB1t+σ2γtx1+θtdB2t,with the initial conditions(7)xs=φs>0,s∈-τ,0;γ0=ι,where τ>0,φ∈C([-τ,0],R+);R+=(0,+∞);λ>0,θ>0;r,a:S→R+;b,σ1,σ2:S→R;S={1,2,…,m}.Throughout this paper, let(Ω,F,{Ft}t≥0,P) be a complete probability space with filtration {Ft}t≥0, where Ft is right continuous and F0 contains all P-null sets. The one-dimensional Brownian motions B1(t) and B2(t) are defined in this space.Letγ(t) be a right-continuous Markov chain on the probability space, taking values in S, and with infinitesimal generator Q=(qij)∈Rm×m given by(8)Pγt+δ=j∣γt=i=qijδ+oδ,ifi≠j,1+qijδ+oδ,ifi=j,where δ>0,qij≥0 is the transitions rate from i to j for i≠j and qii=-∑j≠iqij for each i,j∈S. We assume that the Markov chain γ(t) is irreducible, and B1(t),B2(t), and γ(t) are independent under this condition; the Markov chain has a unique stationary (probability) distribution π=(π1,π2,…,πm)∈R1×m, which can be determined by solving the following equation:(9)πQ=0,subject to ∑i=1mπi=1 and πi>0,i∈S.For convenience, for anyf:S→R, denote(10)f^=maxi∈S⁡fi,fˇ=mini∈S⁡fi,f¯=maxi∈S⁡fi.For (6), we give the following conditions:(A1)
σ2(i)≠0;(A2)
b¯-a(i)<σ2(i)=0;(A3)
b¯-a(i)=σ2(i)=0,λ<1/2,r(i)<1/4σ12(i);(A4)
b¯-a(i)=σ2(i)=0,λ=1/2,b¯+a(i)+r(i)≤1/4σ12(i);(A5)
b¯-a(i)<1/4σ22(i);(A6)
b¯-a(i)=1/4σ22(i),λ<1/2,r(i)<1/4σ12(i);(A7)
b¯-a(i)=1/4σ22(i),λ=1/2,b¯+a(i)+1/2σ22(i)+r(i)≤1/4σ12(i);(A8)
b¯<a(i);(A9)
b¯=a(i),λ<2θ+1/2,σ2(i)≠0;(A10)
b¯=a(i),λ<2θ+1/2,σ2(i)=0,λ<1/2,r(i)<1/4σ12(i);(A11)
b¯=a(i),λ=2θ+1/2,b¯+a(i)<1/4σ22(i);(A12)
b¯=a(i),λ=2θ+1/2,b¯+a(i)=1/4σ22(i),2θ<1/2,r(i)<1/4σ12(i);(A13)
b¯=a(i),λ=2θ+1/2,b¯+a(i)=1/4σ22(i),2θ=1/2,r(i)+1/2σ22(i)<1/4σ12(i).(H1)
λ<2θ; and for every i∈S, one of conditions (A1)-(A4) holds;(H2)
λ=2θ; and for every i∈S, one of conditions (A5)-(A7) holds;(H3)
λ>2θ; and for every i∈S, one of conditions (A8)-(A13) holds;(H4)
λ<2θ; and for every i∈S, (A1) or (A4) holds;(H5)
λ=2θ,b¯<a(i)+1/4σ22(i), i∈S;(H6)
λ>2θ,b¯<aˇ;(H7)
there existsu∈S such that for any i∈S∖{u},qiu>0;(H8)
∑i=1mπiβ(i)>0, where β(i)=r(i)-1/2σ12(i), i∈S;(H9)
b¯<aˇ.


## 2. Uniqueness of Global Positive Solution

Asx(t) in (6) denotes population size at time t, it should be nonnegative. Hence, we shall consider the existence and uniqueness of global positive solution of (6).Theorem 1.
If one of conditions (H1)-(H3) holds, then, for any initial valueφ∈C([-τ,0];R+),ι∈S, there exists a unique solution x(t) to (6) on [-τ,+∞) and x(t) will remain in R+ with probability 1.Proof.
Since the coefficients of (6) are locally Lipschitz continuous, then there is a unique maximal local solution x(t) on [-τ,τe), where τe is the explosion time. Let k0>0 satisfying(11)1k0<min-τ≤t≤0φt≤max-τ≤t≤0φt<k0.For each integer k≥k0, define the stopping time(12)τk=inf⁡t∈0,τe:xt∉1k,k.Here, set inf⁡∅=∞. Obviously, τk is increasing as k→∞ and τ∞=limk→+∞τk≤τe a.s. If τ∞=∞ a.s., then x(t)∈R+, for t≥0 and τe=∞ a.s. Thus, to prove this theorem, we need to prove τ∞=∞ a.s. Let(13)Vx=x-1-12ln⁡x,x>0.Clearly, V(x)≥0 for any x>0. Applying Itô’s formula to V(x), we have(14)dVxt=LVxt,xt-τ,γtdt+xt-12σ1γtdB1t+σ2γtxθtdB2t,where LV:R+×R+×S→R is defined by (15)LVx,y,i=-1412x1/2-1σ12i+σ22ix2θ+12x1/2-1ri-aixλ+12bix1/2-1yλ.By Young’s inequality, (16)LVx,y,i≤-1412x1/2-1σ12i+σ22ix2θ+12x1/2-1ri-aixλ+12b¯x1/2yλ+12b¯yλ≤-1412x1/2-1σ12i+σ22ix2θ+12x1/2-1ri-aixλ+12b¯1/21/2+λx1/2+λ+λ1/2+λy1/2+λ+12b¯yλ=-1412x1/2-1σ12i+σ22ix2θ+12x1/2-1ri-aixλ+1/2b¯1+2λx1/2+λ+b¯λ1+2λx1/2+λ+12b¯xλ+b¯λ1+2λy1/2+λ-x1/2+λ+12b¯yλ-xλ≤Fx,i+λb¯1+2λy1/2+λ-x1/2+λ+b¯2yλ-xλ, where (17)Fx,i=-1412x1/2-1σ12i+σ22ix2θ+12rix1/2-1+12b¯+aixλ+12b¯-aix1/2+λ.If one of conditions (H1)-(H3) holds, then F(x,i) is bounded above; i.e., there exists M1>0 such that F(x,i)≤M1,x∈R+,i∈S. Substituting these into (14) yields(18)dVxt≤M1+λb¯1+2λx1/2+λt-τ-x1/2+λt+b¯2xλt-τ-xλtdt+12x1/2t-1σ1γtdB1t+σ2γtxθtdB2t.For any T>0 and t∈[0,T], integrating both sides of (18) from 0 to τk∧t and taking the expectations yields(19)EVxτk∧t≤Vφ0+E∫0τk∧tM1+λb¯1+2λx1/2+λs-τ-x1/2+λs+b¯2xλs-τ-xλsds.In addition,(20)E∫0τk∧tx1/2+λs-τds≤∫-τ0φ1/2+λsds+E∫0τk∧tx1/2+λsds,and (21)E∫0τk∧txλs-τds≤∫-τ0φλsds+E∫0τk∧txλsds.Substituting (20) and (21) into (19) gives(22)EVxτk∧t≤Vφ0+M1T+λb¯1+2λ∫-τ0φ1/2+λsds+b¯2∫-τ0φλsds,t∈0,T.Note that x(τk)=k or x(τk)=1/k. From (22), for any T>0, (23)Vφ0+M1T+λb¯1+2λ∫-τ0φ1/2+λsds+b¯2∫-τ0φλsds≥EVxτk∧T≥EIτk≤TVxτk∧T=EIτk≤TVxτk≥EIτk≤TVk∧V1k=Vk∧V1kEIτk≤T=Vk∧V1kPτk≤T, which implies (24)limsupk→∞⁡Pτk≤T≤limk→∞⁡Vφ0+M1T+λb¯/1+2λ∫-τ0φ1/2+λsds+b¯/2∫-τ0φλsdsVk∧V1/k=0. Then, limk→∞P(τk≤T)=0. Further, P(τ∞≤T)=0. Since T>0 is arbitrary, we must have P(τ∞<∞)=0. That is, τ∞=∞ a.s. Therefore, Theorem 1 holds.Corollary 2.
Assume that (H9) holds. Then, for any initial valueφ∈C([-τ,0];R+),ι∈S, there exists a unique solution x(t) to (6) on [-τ,+∞) and x(t) will remain in R+ with probability 1.Remark 3.
Ifγ(t)≡ς∈S,t∈[0,+∞);r(ς)=r,a(ς)=a,b(ς)=0,σ1(ς)=σ1,σ2(ς)=σ2,θ=λ, then (6) transforms to (2). Clearly, if b(ς)=0,θ=λ, then (H1) holds. Hence, Theorem 1 generalizes Lemma 2 in [9].Remark 4.
Ifb(i)=0,i∈S, then (6) reduces to (3). Clearly, if b(i)=0,i∈S, then one of conditions (H1)-(H3) holds. Hence, Theorem 1 generalizes Theorem 1 in [11].


## 3. Stochastic Permanence

Definition 5.
Equation (6) is stochastically permanent if for any ε∈(0,1), there exist constants H=H(ε)>0 and δ=δ(ε)>0 such that for any solution x(t) of (6),(25)liminft→+∞⁡Pxt≤H≥1-ε,liminft→+∞⁡Pxt≥δ≥1-ε.Lemma 6 (Lemma 3, [12]).
If (H7) and (H8) hold, then there exists a constantα>0 such that the matrix(26)Aα=diagξ1α,ξ2α,…,ξmα-Qis a nonsingular M-matrix, where ξi(α)=(1+θ)αβ(i)-1/2α2(1+θ)2σ12(i),i∈S.Lemma 7 (Theorem 2.10, [27]).
IfA=(aij)n×n∈Zn×n, then the following results are equivalent.(1)
A is nonsingular M-matrix.(2)
A is semipositive; that is, there is x≫0 in Rn satisfying Ax≫0.Lemma 8.
Assume that one of the following conditions holds:(i)
0<p<1 and (H4) holds(ii)
p>0,b¯<a(i)+1/2(1-p)σ22(i),i∈S and (H5) holds(iii)
p>0 and (H6) holds
For any initial valueφ∈C([-τ,0];R+),ι∈S, let x(t) be the solution of (6) with the initial conditions (7). Then there exists K1(p)>0 such that(27)limsupt→+∞⁡Expt≤K1p.Proof.
From Theorem1, if one of conditions (H4)-(H6) holds, then (6) exists as a unique global positive solution x(t). Now, we prove Lemma 3 if (ii) holds. For i∈S, let(28)μi=1τln⁡λ+pai-pb¯+1/2λ+p1-pσ22iλb¯,b¯>01,b¯=0.It follows from (ii) that μ(i)>0,i∈S. Let β>0 satisfying β<mini∈S{μ(i)}. Applying Itô’s formula to U=eβtxp yields(29)dUxt,t=LUxt,xt-τ,t,γtdt+pσ1γteβtxptdB1t+pσ2γteβtxp+θtdB2t,where LU:R+×R+×[0,+∞)×S→R is defined by(30)LUx,y,t,i=eβtβ+pri+12pp-1σ12ixp-paieβtxλ+p+12pp-1eβtσ22ixp+2θ+pbieβtxpyλ.Applying Young’s inequality and (H5), we have (31)LUx,y,t,i≤eβtβ+pri+12pp-1σ12ixp-paieβtxλ+p+12pp-1eβtσ22ixp+2θ+pb¯eβtpxλ+pλ+p+λyλ+pλ+p=eβtF1x,i+pb¯λλ+peβt-eβτxλ+p+yλ+p, where (32)F1x,i=β+pri+12pp-1σ12ixp+12pp-1σ22ixp+2θ+b¯p2λ+p-pai+b¯pλλ+peβτxλ+p=β+pri+12pp-1σ12ixp+12pp-1σ22i+b¯p2λ+p-pai+b¯pλλ+peβτxλ+p. From condition (ii) and β<mini∈S{μ(i)}, one can show that(33)12pp-1σ22i+b¯p2λ+p-pai+b¯pλλ+peβτ<0,i∈S.Hence, there exists M2=M2(p)>0 such that F1(x,i)≤M2,x∈R+,i∈S. Thus,(34)LUx,y,t,i≤M2eβt+pb¯λλ+peβt-eβτxλ+p+yλ+p.In addition,(35)∫0teβsxλ+ps-τds≤eβτ∫-τ0φλ+psds+eβτ∫0teβsxλ+psds.From (29), (34), and (35), we have(36)eβtExpt≤φp0+M2βeβt-1+pb¯λλ+peβτ∫-τ0φλ+psds, which yields(37)limsupt→+∞⁡Expt≤M2β≕K1p. That is, (27) holds. If (i) or (iii) holds, the proof is similar and omitted. Hence, Lemma 8 holds.Remark 9.
Ifλ=1,σ2(i)=0,i∈S and (H6) holds, then condition (A1’) in [26] holds. Hence, Lemma 8 generalizes Theorem 3.3 in [26].Remark 10.
Ifλ>2θ,p>0,b(i)=0,i∈S, then (H6) holds. Hence, Lemma 8 improves and generalizes Lemma 6 in [11].Theorem 11.
Assume that (H7) and (H8) and one of conditions (H4)-(H6) hold. If0<θ≤1,0<λ≤1+θ and b(i)≥0,i∈S, then (6) is stochastically permanent.Proof.
Clearly, if one of conditions (H4)-(H6) holds, then one of conditions (H1)-(H3) holds. From Theorem1, for any given φ∈C([-τ,0],R+),ι∈S, let x be the solution of the initial value problem (6) and (7). Define(38)V1x=1x1+θ,x∈R+. Using the generalized Itô’s formula, we have(39)dV1xt=1+θV1xtaγtxλt-bγtxλt-τ-rγtdt+121+θ2+θV1xtσ12γtdt+121+θ2+θσ22γtxθ-1tdt-1+θV1xtσ1γtdB1t-1+θσ2γtx-1tdB2t.From Lemmas 6 and 7, there is α>0,w=(p1,p2,…,pm)T≫0 such that A(α)w≫0. Further, for any i∈S,(40)pi1+θαβi-121+θ2α2σ12i-∑j=1mqijpj>0.Define V2(x,i)=pi(1+V1(x))α. By the generalized Itô’s formula again,(41)EV2xt,γt=V2φ0,ι+E∫0tLV2xs,xs-τ,γsds,where LV2:R+×R+×S→R is defined by(42)LV2x,y,i=αpi1+θ1+V1xα-21+V1xV1xaixλ-biyλ-ri+122+θσ12iV1x+122+θσ22ixθ-1+12α-11+θσ12iV12x+12α-11+θσ22ix-2+1+V1xα∑j=1mqijpj=1+V1xα-2-pi1+θαβi-12pi1+θ2α2σ12i-∑j=1mqijpjV12x+-pi1+θαri+12piα1+θ2+θσ12i+2∑j=1mqijpjV1x+∑j=1mqijpj+piα1+θaiV1xxλ-1-θ+piα1+θaixλ-1-θ+12piα1+θ2+θσ22ixθ-1+12piαα-11+θ2σ22ix-2+12piα1+θ2+θσ22ix-2-piα1+θbiyλV1x1+V1x. It follows from (40) that there exists η>0 such that for any i∈S,(43)pi1+θαβi-121+θ2α2σ12i-∑j=1mqijpj-ηpi>0.From the generalized Itô’s formula, we have (44)EeηtV2xt,γt=V2φ0,ι+E∫0tηeηsV2xs,γs+eηsLV2xs,xs-τ,γsds. In addition, (45)ηeηtV2x,i+eηtLV2x,y,i=eηt1+V1xα-2-pi1+θαβi-12pi1+θ2α2σ12i-∑j=1mqijpj-ηpiV12x+-pi1+θαri+12piα1+θ2+θσ12i+2∑j=1mqijpj+2ηpiV1x+ηpi+∑j=1mqijpj+piα1+θaiV1xxλ-1-θ+piα1+θaixλ-1-θ+12piα1+θ2+θσ22ixθ-1+12piαα-11+θ2σ22ix-2+12piα1+θ2+θσ22ix-2-piα1+θbiyλV1x1+V1x≤eηtF2x,i, where (46)F2x,i=1+V1xα-2-pi1+θαβi-12pi1+θ2α2σ12i-∑j=1mqijpj-ηpiV12x+-pi1+θαrˇ+12piα1+θ2+θσ12^+2∑j=1mqijpj+2ηpiV1x+ηpi+∑j=1mqijpj+piα1+θa^V1xxλ-1-θ+piα1+θa^xλ-1-θ+12piα2+θ2σ22^xθ-1+12piα21+θ2σ22^x-2+12pi2+θ2σ22^x-2.Further, (47)limx→+∞⁡F2x,i=ηpi+∑j=1mqijpj,θ<1,λ<1+θ,ηpi+∑j=1mqijpj+12piα2+θ2σ22^,θ=1,λ<1+θ,ηpi+∑j=1mqijpj+piα1+θa^,θ<1,λ=1+θ,ηpi+∑j=1mqijpj+12piα2+θ2σ22^+piα1+θa^,θ=1,λ=1+θ.Then there is M3>0 such that supx∈R+,i∈SF(x,i)≤M3. Thus, for any i∈S, (48)piEeηt1+V1xtα≤pι1+V1φ0α+M3eηt-1η.Further,(49)limsupt→+∞⁡EV1αxt≤limsupt→+∞⁡E1+V1xtα≤M3ηmin⁡p1,p2,…,pm≕M4.That is, limsupt→+∞E[x-α-αθ(t)]≤M4. For any ε>0, let δ=(ε/M4)1/α+αθ. From Chebyshev’s inequality,(50)Pxt<δ=Px-α-αθt>δ-α-αθ≤Ex-α-αθtδ-α-αθ=δα+αθEx-α-αθt.Further, limsupt→+∞P{x(t)<δ}≤δα+αθM4=ε. This yields(51)liminft→+∞⁡Pxt≥δ≥1-ε.
Next we claim that for anyε>0, there exists H>0 such that liminft→+∞P{x(t)≤H}≥1-ε. Let p∈(0,1/2). From Lemma 8, there exists K1(p)>0 such that limsupt→+∞E(xp(t))≤K1(p).
For anyε>0, let H=ε/K1(p)-1/p. Using Chebyshev’s inequality yields(52)Pxt>H=Pxpt>Hp≤H-pExpt,which means that(53)limsupt→+∞⁡Pxt>H≤limsupt→+∞⁡H-pExpt=ε.Hence,(54)liminft→+∞⁡Pxt≤H≥1-ε.From (51), (54), and Definition 5, Theorem 11 holds.


## 4. Asymptotic Estimation

In this section, we study some asymptotic properties of (6).Lemma 12 ((Theorem 2.13, [27]) (BDG’s inequality)).
Letg∈L2R+;Rm×n. Define, for t≥0, x(t)=∫0tg(s)dB(s) and A(t)=∫0tg(s)2ds. Then for every p>0, there exist universal positive constants cp,Cp, which are only dependent on p, such that(55)cpEAtp/2≤Esup0≤s≤txsp≤CpEAtp/2for all t≥0. In particular, one may take(56)cp=p2p,Cp=32pp/2,if0<p<2;cp=1,Cp=4,ifp=2;cp=2p-p/2,Cp=pp+12p-1p-1p/2,ifp>2.Theorem 13.
Assume that one of conditions (H4)-(H6) holds. For anyφ∈C([-τ,0];R+),ι∈S, let x(t) be the solution of the initial value problem (6) and (7).
(i) If (H4) holds, then(57)limsupt→+∞⁡ln⁡xtln⁡t≤2a.s.
(ii) If (H5) holds andb¯<a(i)+1/2[1-max⁡{2,2λ,1+λ}]σ22(i),i∈S or (H6) holds, then(58)limsupt→+∞⁡ln⁡xtln⁡t≤1a.s.Proof.
(i) Assume that (H4) holds. Applying the generalized Itô’s formula toetln⁡x(t) yields(59)etln⁡xt-ln⁡φ0=∫0tesln⁡xs+βγs-aγsxλs+bγsxλs-τ-12σ22γsx2θsds+N1t+N2t,where N1(t)=∫0tesσ1(γ(s))dB1(s),N2(t)=∫0tesσ2(γ(s))xθ(s)dB2(s). For any ρ>1,v>0 and n∈Z+, it follows from exponential martingale inequality that(60)Psup0≤t≤vn⁡Nit-12e-vnNit,Nit>ρevnln⁡n≤1nρ,i=1,2.From the Borel-Cantelli Lemma [27], for almost all ω∈Ω, there is a random integer n0=n0(ω) sufficiently large such that for n0>n0(ω),i=1,2,(61)Nit≤12e-vnNit,Nit+ρevnln⁡n,0≤t≤vn. This, together with (59), implies that for any t∈vn-3/2,vn-1/2 and for almost all ω∈Ω, (62)etln⁡xt≤ln⁡φ0+∫0tesln⁡xs+βγs-aγsxλs+bγsxλs-τ+12es-vnσ12γs-121-es-vnσ22γsx2θsds+2ρevnln⁡n≤ln⁡φ0+∫0tesln⁡xs+β^-aˇ-b¯eτxλs+12es-vnσ12γs-121-es-vnσ22γsx2θsds+b¯eτ∫-τ0φλsds+2ρevnln⁡n≤ln⁡φ0+∫0tesF3xsds+b¯eτ∫-τ0φλsds+2ρevnln⁡n,where F3(x)=ln⁡x+β^-(aˇ-b¯eτ)xλ+1/2e-1/2vσ12^-1/21-e-1/2vσ22^x2θ. Clearly, there exists M5>0 such that F3(x)≤M5 for any x∈R+. Therefore, for almost all ω∈Ω, if n≥n0(ω),vn-3/2≤t≤vn-1/2, (63)ln⁡xtln⁡t≤ln⁡φ0etln⁡t+M51-e-tln⁡t+b¯eτ∫-τ0φλsdsetln⁡t+2ρe-vn-3/2evnln⁡nln⁡vn-3/2.This implies (64)limsupt→+∞⁡ln⁡xtln⁡t≤2ρe3/2v,which is the required assertion (57) by letting ρ→1 and v→0.
(ii) Assume that (H5) andb¯<a(i)+1/2[1-max{2,2λ,1+λ}]σ22(i),i∈S or (H6) holds. It follows from Theorem 1 that for any t≥τ,x(t)∈R+ a.s. This, together with (5), yields (65)dxt≤r^xt+b¯xtxλt-τdt+σ1γtxtdB1t+σ2γtx1+θtdB2t.
Further,(66)Esupt≤u≤t+1⁡xu≤Ext+r^∫tt+1Exsds+b¯∫tt+1Exsxλs-τds+Esupt≤u≤t+1⁡∫tuσ1γsxsdB1s+Esupt≤u≤t+1⁡∫tuσ2γsx1+θsdB2s. Using the BDG’s inequality, there exist C1>0 and C2>0 such that (67)Esupt≤u≤t+1⁡∫tuσ1γsxsdB1s≤C1E∫tt+1σ1γsxs2ds1/2≤E12supt≤u≤t+1⁡xu·2C12σ1¯2∫tt+1xsds1/2≤14Esupt≤u≤t+1⁡xu+C12σ1¯2∫tt+1Exsds,and (68)Esupt≤u≤t+1⁡∫tuσ2γsx1+θsdB2s≤C2E∫tt+1σ2γsx1+θs2ds1/2≤E12supt≤u≤t+1⁡xu·2C22σ2¯2∫tt+1x1+2θsds1/2≤14Esupt≤u≤t+1⁡xu+C22σ2¯2∫tt+1Ex1+2θsds. From Hölder’s inequality, (69)∫tt+1Exsxλs-τds≤∫tt+1Ex2s1/2Ex2λs-τ1/2ds≤12∫tt+1Ex2sds+12∫tt+1Ex2λs-τds. Hence,(70)Esupt≤u≤t+1xu≤2Ext+2r^∫tt+1Exsds+b¯∫tt+1Ex2sds+b¯∫tt+1Ex2λs-τds+2C12σ1¯2∫tt+1Exsds+2C22σ2¯2∫tt+1Ex1+2θsds.It follows from (70) and Lemma 8 that(71)limsupt→+∞⁡Esupt≤u≤t+1⁡xu≤2+2r^+2C12σ1¯2K11+b¯K12+K12λ+2C22σ2¯2K11+2θ.From (71), there is M6>0 such that(72)Esupk≤t≤k+1⁡xt≤M6,k=1,2,⋯.For any ε>0, from Chebyshev’s inequality, we have(73)Psupk≤t≤k+1⁡xt>k1+ε≤M6k1+ε,k=1,2,⋯.From the Borel-Cantelli Lemma [27], for almost all ω∈Ω, there exists an integer k0(ω) such that for any k≥k0(ω),(74)supk≤t≤k+1⁡xt,ω≤k1+ε.Consequently, for any ε>0 and almost all ω∈Ω, if k≥k0(ω) and k≤t≤k+1, then(75)ln⁡xt,ωln⁡t≤1+εln⁡kln⁡k=1+ε.Therefore(76)limsupt→+∞⁡ln⁡xtln⁡t≤1+εa.s.Letting ε→0, we obtain the desired assertion (58).From Theorem13, it is easy to see that the following results hold.Corollary 14.
Assume thatb(i)=0,i∈S. For any x0>0,ι∈S, let x(t) be the solution of (3) with the initial condition x(0)=x0 and γ(0)=ι. (i) If λ<2θ, then (57) holds; (ii) if λ≥2θ, then (58) holds.Corollary 15.
Assume thatλ=1,θ=1/4,σ2(i)=0,i∈S. For any φ∈C([-τ,0];R+), let x(t) be the solution of the initial value problem (6) and (7). If b¯<aˇ, then (58) holds.Remark 16.
Theorem 6 in [12] shows that (57) holds for any λ>0,θ>0. Hence, Corollary 14 improves Theorem 6 in [12]. In addition, Corollary 15 is corresponding to Lemma 5.1 in [26]. Hence, Theorem 13 generalizes and improves Theorem 6 in [12] and Lemma 5.1 in [26].Theorem 17.
Assume that (H7), (H8), and one of conditions (H4)-(H6) hold. For anyφ∈C([-τ,0];R+),ι∈S, let x(t) be the solution of the initial value problem (6) and (7). If 0<θ≤1,0<λ≤1+θ and b(i)≥0,i∈S, then(77)liminft→+∞⁡ln⁡xtln⁡t≥-11+θαa.s.Proof.
From (39) and the generalized Itô’s formula, (78)d1+V1xtα≤α1+θ1+V1xtα-2-βˇ+12α1+θσ¯12+1V12xt+F4x-α1+θ1+V1xtα-1σ1γtV1xtdB1t+σ2γtx-1tdB2t, where (79)F4x=-V12x+-rˇ+121+θσ¯12V1x+a^V1xxλ-1-θ+a^xλ-1-θ+122+θσ2¯2xθ-1+12α1+θσ2¯2x-2+121+θσ2¯2x-2. Clearly, there exists M7>0 such that F4(x)≤M7 for any x∈(0,+∞). Further,(80)d1+V1xtα≤α1+θ1+V1xtα-2-βˇ+12α1+θσ1¯2+1V12xt+M7dt-α1+θ1+V1xtα-1σ1γtV1xtdB1t+σ2γtx-1tdB2t≤α1+θM81+V1xtαdt-α1+θ1+V1xtα-1σ1γtV1xtdB1t+σ2γtx-1tdB2t,where M8=max⁡-βˇ+1/2α(1+θ)σ1¯2+1,M7 are positive numbers. By (49), there exists M9>0 such that(81)E1+V1xtα≤M9,t≥0.
Letδ>0 satisfy(82)α1+θM8δ+6δσ¯1+σ¯2<12.From (80), for any k∈{1,2,⋯},(83)Esupk-1δ≤t≤kδ⁡1+V1xtα≤E1+V1xk-1δα+α1+θM8Esupk-1δ≤t≤kδ⁡∫k-1δt1+V1xsαds+α1+θEsupk-1δ≤t≤kδ⁡∫k-1δt1+V1xsα-1σ1γsV1xsdB1s+α1+θEsupk-1δ≤t≤kδ⁡∫k-1δt1+V1xsα-1σ2γsx-1sdB2s.In addition, we have(84)Esupk-1δ≤t≤kδ⁡∫k-1δt1+V1xsαds≤E∫k-1δkδ1+V1xsαds≤δEsupk-1δ≤t≤kδ⁡1+V1xtα.From the BDG’s inequality,(85)Esupk-1δ≤t≤kδ⁡∫k-1δt1+V1xsα-1σ1γsV1xsdB1s≤6E∫k-1δkδ1+V1xs2α-2σ12γsV12xsds1/2≤6σ¯1E∫k-1δkδ1+V1xs2αds1/2≤6δσ¯1Esupk-1δ≤t≤kδ⁡1+V1xtα,and(86)Esupk-1δ≤t≤kδ⁡∫k-1δt1+V1xsα-1σ2γsx-1sdB2s≤6E∫k-1δkδ1+V1xs2α-2σ22γsx-2sds1/2≤6σ¯2E∫k-1δkδ1+V1xs2αds1/2≤6δσ¯2Esupk-1δ≤t≤kδ⁡1+V1xtα.From (83)–(86), (87)Esupk-1δ≤t≤kδ⁡1+V1xtα≤E1+V1xk-1δα+α1+θM8δEsupk-1δ≤t≤kδ⁡1+V1xtα+6α1+θδσ¯1+σ¯2Esupk-1δ≤t≤kδ⁡1+V1xtα=E1+V1xk-1δα+α1+θM8δ+6δσ¯1+σ¯2Esupk-1δ≤t≤kδ⁡1+V1xtα.This, together with (81) and (82), yields(88)Esupk-1δ≤t≤kδ⁡1+V1xtα≤2M9.For any ε>0, using Chebyshev’s inequality,(89)Psupk-1δ≤t≤kδ⁡1+V1xtα>kδ1+ε≤2M9kδ1+ε.From the Borel-Cantelli Lemma, for almost all ω∈Ω, there exists k0(ω)>0 such that for any k>k0(ω),(90)supk-1δ≤t≤kδ⁡1+V1xt,ωα≤kδ1+ε,which yields,(91)ln⁡1+V1xtαln⁡t≤1+εln⁡kδln⁡k-1δ<1+εa.s.Therefore(92)limsupt→+∞⁡ln1+V1xtαln⁡t≤1+εa.s. Letting ε→0, we have limsupt→+∞ln(1+V1(x(t)))α/ln⁡t≤1 a.s. Further, (93)limsupt→+∞⁡ln⁡xt-1+θαln⁡t≤1a.s.Hence,(94)limsupt→+∞⁡ln⁡xt-1+θαln⁡t=-1+θαliminft→+∞⁡ln⁡xtln⁡t≤1a.s.This yields (77). Hence, Theorem 17 holds.Theorem 18.
Assume that the conditions of Theorem13 hold. If (H7)-(H9) hold, then for any φ∈C([-τ,0];R+),ι∈S, the solution x(t) of (6) obeys(95)limsupt→+∞⁡1t∫0txλsds≤1aˇ-b¯∑i=1mπiβia.s.Proof.
From Theorems13 and 17,(96)limt→+∞⁡ln⁡xtt=0a.s.Using the generalized Itô’s formula, we obtain(97)ln⁡xt=ln⁡x0+∫0trγs-12σ12γsds-∫0taγsxλsds+∫0tbγsxλs-τds-12∫0tσ22γsx2θsds+M1t+M2t,where M1(t)=∫0tσ1(γ(s)dB1(s),M2(t)=∫0tσ2(γ(s)xθ(s)dB2(s). The quadratic variation of M2(t) is (98)M2t,M2t=∫0tσ22γsx2θsds. From the exponential martingale inequality, for any integer n≥2, (99)Psup0≤t≤n⁡M2t-12M2t,M2t>2ln⁡n≤e-2ln⁡n=1n2.From Borel-Cantelli Lemma, for almost all ω∈Ω, there exists a random integer n0 such that for n≥n0(ω), (100)sup0≤t≤n⁡M2t-12M2t,M2t≤2ln⁡n. Further, for almost all ω∈Ω,n≥n0(ω), 0≤t≤n, (101)M2t≤2ln⁡n+12M2t,M2t=2ln⁡n+12∫0tσ22γsx2θsds. Note that (102)∫0tbγsxλs-τds≤b¯∫0txλs-τds=b¯∫-τt-τxλsds≤b¯∫-τ0xλsds+b¯∫0txλsds. From (97), for any n≥n0,t∈(n-1,n](103)ln⁡xtt≤ln⁡x0t+1t∫0tβγsds+-aˇ+b¯t∫0txλsds+b¯t∫-τ0xλsds+2ln⁡nn-1+M1tt. Let t→+∞, using the strong law of large numbers for martingales and (96), we obtain (104)limsupt→+∞⁡1t∫0txλsds≤1aˇ-b¯∑i=1mπiβia.s. We then obtain assertions (95). Hence, Theorem 18 holds.Remark 19.
Liu et al. [12] have claimed that, under (H7) and (H8), the solution of (2) obeys(105)limsupt→+∞⁡1t∫0txλsds≤1aˇ∑i=1mπiβia.s.It is easy to see that if b(i)=0,i∈S, then (95) becomes (105). Hence, Theorem 18 generalizes Theorem 8 in [12].Remark 20.
Ifλ=1,σ2(i)=0,i∈S, then conditions (H7)-(H9) are corresponding to the conditions (A1”), (A2), and (A3) in [26]. Hence, Theorem 18 generalizes Theorem 5.3 in [26].


## 5. Numerical Simulations

In this section, we make numerical simulations to illustrate our theoretical results. Consider the following examples.Example 1.
In (6), let λ=2,θ=1,τ=1,S={1,2}, q12=-q11=1,q21=-q22=2, r(1)=3,a(1)=4,b(1)=2,σ1(1)=0.2,σ2(1)=0.2,r(2)=1,a(2)=3,b(2)=1.5,σ1(2)=0.15,σ2(2)=0.2 and initial value x(s)=1.5,s∈[-1,0],γ(0)=1.Clearly,aˇ=3,b¯=2,β(1)=2.98,β(2)≈0.989,π1=2/3,π2=1/3,π1β(1)+π2β(2)≈2.316>0. Hence, (H5) and (H7)-(H9) hold. Further, all conditions of Theorems 11 and 18 are satisfied. According to Theorems 11 and 18, (6) is stochastically permanent and its solution x(t) with any positive initial value satisfies (106)limsupt→+∞⁡1t∫0tx2sds≤2.316a.s.In Figure1, (i) left panel shows stochastic trajectories of model (6) with parameters in Example 1; deterministic trajectory of model (6) with r(i)=3,a(i)=4,b(i)=2,σ1(i)=0,σ2(i)=0,λ=2,τ=1,i=1,2; (ii) right panel shows stochastic trajectories of model (6) with parameters in Example 1; deterministic trajectory of model (6) with r(i)=1,a(i)=3,b(i)=1.5,σ1(i)=0,σ2(i)=0,λ=2,τ=1,i=1,2. Figure 1 clearly supports Theorems 11 and 18.Figure 1
Stochastic and deterministic trajectories of model (6).Example 2.
In (6), let λ=1.5,θ=0.7,τ=1,S={1,2}, q11=q22=-1,q12=q21=1, r(1)=0.4,a(1)=0.15,b(1)=0.1,σ1(1)=0.03,σ2(1)=0.013,r(2)=0.3,a(2)=0.2,b(2)=0.09,σ1(2)=0.02,σ2(2)=0.021 and initial value x(s)=1,s∈[-1,0],γ(0)=2.Clearly,aˇ=0.15,b¯=0.1,β(1)=0.39955,β(2)=0.2998,π1=1/2,π2=1/2,π1β(1)+π2β(2)≈0.3497>0. Hence, (H6)-(H9) hold. Further, all conditions of Theorems 11 and 18 are satisfied. From Theorems 11 and 18, (6) is stochastically permanent and its solution x(t) with any positive initial value satisfies (107)limsupt→+∞⁡1t∫0tx1.5sds≤6.994a.s.In Figure2, (i) left panel shows stochastic trajectories of model (6) with parameters in Example 2; deterministic trajectory of model (6) with r(i)=0.4,a(i)=0.15,b(i)=0.1,σ1(i)=0,σ2(i)=0,λ=1.5,τ=1,i=1,2; (ii) right panel shows stochastic trajectories of model (6) with parameters in Example 2; deterministic trajectory of model (6) with r(i)=0.3,a(i)=0.2,b(i)=0.09,σ1(i)=0,σ2(i)=0,λ=2,τ=1,i=1,2. Figure 2 clearly supports Theorems 11 and 18.Figure 2
Stochastic and deterministic trajectories of model (6).Example 3.
In (6), let λ=1,θ=0.25,τ=1,S={1,2}, q12=-q11=1,q21=-q22=2, r(1)=0.2,a(1)=0.5,b(1)=0.3,σ1(1)=0.4,σ2(1)=0.1,r(2)=0,a(2)=0.4,b(2)=0.2,σ1(2)=0.5,σ2(2)=0 and initial value x(s)=1.5,s∈[-1,0],γ(0)=1.Clearly,aˇ=0.4,b¯=0.3,β(1)=0.12,β(2)=-0.125,π1=2/3,π2=1/3,π1β(1)+π2β(2)≈0.0383>0. Hence, (H6)-(H8) hold. Hence, all conditions of Theorem 11 are satisfied. According to Theorem 11, (6) is stochastically permanent and its solution x(t) with any positive initial value satisfies (108)limsupt→+∞⁡1t∫0txsds≤0.383a.s. Figure 3 clearly supports Theorems 11 and 18.Figure 3
Stochastic trajectories of model (6) with parameters in Example 3.Assume that the Markov chainγ(t)=1∈S. Then (6) obeys the following subsystem:(109)dxt=xt0.2-0.5xt+0.3xt-1dt+0.4xtdB1t+0.1x1.25tdB2t.Similarly, assume that the Markov chain γ(t)=2∈S. Then (6) obeys the following subsystem:(110)dxt=xt-0.4xt+0.2xt-1dt+0.5xtdB1t.For subsystem (109), it is easy to check that all conditions of Theorem 11 are satisfied. Hence, subsystem (109) is stochastically permanent. In addition, for subsystem (110), -a(2)+b2=-0.4+0.2=-0.2<0,r(2)-0.5σ12(2)=-0.5×0.52=-0.125<0. From Corollary 4.6 in [26], subsystem (110) is stochastically extinct. Figure 4 clearly supports these results.Figure 4
Stochastic trajectories of subsystems (109) and (110).In Figure4, (i) left panel shows stochastic trajectory of subsystem (109); (ii) right panel shows stochastic trajectory of subsystem (110).Example3 shows that if the overall system consists of permanent subsystems and nonpermanent subsystems, the overall system can be permanent. Hence, switching component can add some interesting behavior to hybrid systems.


## 6. Conclusions

In this paper, we study the stochastic permanence and asymptotic estimations of solution to a general stochastic delay Gilpin-Ayala system with Markovian switching. Compared with the models in the literatures, (6) provides a more realistic system of the population dynamics. However, the delays and Markovian switching mechanism make the task more complicated to deal with. We overcome difficulties by constructing suitable Lyapunov functionals and using some analysis technique. Under some suitable conditions, (6) still retains some well properties, for example, the existence of global positive solution and persistence and asymptotic estimations. Further, the results show that switching component can add some interesting behavior to hybrid systems driven by Markov chain. In addition, some interesting topics merit further consideration. One may propose stability distribution, ergodicity, and extinction.



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