DDNS Discrete Dynamics in Nature and Society 1607-887X 1026-0226 Hindawi 10.1155/2019/1302648 1302648 Research Article Dynamics Analysis of a Stochastic Delay Gilpin-Ayala Model with Markovian Switching Gao Qiaoqin 1 Luo Zhijiang 1 http://orcid.org/0000-0001-9362-5859 Liu Guirong 2 Consolo Giancarlo 1 Department of Mathematics Luliang University Lishi Shanxi 033001 China llhc.edu.cn 2 School of Mathematical Sciences Shanxi University Taiyuan Shanxi 030006 China sxu.edu.cn 2019 462019 2019 01 02 2019 16 05 2019 462019 2019 Copyright © 2019 Qiaoqin Gao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

National Natural Science Foundation of China 11471197 Natural Science Foundation of Shanxi Province 201801D21009 Natural Science Foundation of Lvliang University ZRXN201610
1. Introduction

Logistic equation is one of the most important and most widely used mathematical models to describe the growth of biological species. In , 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  and the references therein. In , 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,  obtained the stationary distribution and ergodicity of model (2) and improved the corresponding results in . 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  and the references therein. Reference  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,  obtained the lower-growth rate and the upper-growth rate of the positive solution of (3). Reference  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 , 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 . In , 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,φC([-τ,0],R+);R+=(0,+);λ>0,θ>0;r,a:SR+;b,σ1,σ2:SR;S={1,2,,m}.

Throughout this paper, let (Ω,F,{Ft}t0,P) be a complete probability space with filtration {Ft}t0, 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.

Let γ(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=jγt=i=qijδ+oδ,ifij,1+qijδ+oδ,ifi=j,where δ>0,qij0 is the transitions rate from i to j for ij and qii=-jiqij for each i,jS. 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,iS.

For convenience, for any f:SR, denote(10)f^=maxiSfi,fˇ=miniSfi,f¯=maxiSfi.

For (6), we give the following conditions:

σ2(i)0;

b¯-a(i)<σ2(i)=0;

b¯-a(i)=σ2(i)=0,λ<1/2,r(i)<1/4σ12(i);

b¯-a(i)=σ2(i)=0,λ=1/2,b¯+a(i)+r(i)1/4σ12(i);

b¯-a(i)<1/4σ22(i);

b¯-a(i)=1/4σ22(i),λ<1/2,r(i)<1/4σ12(i);

b¯-a(i)=1/4σ22(i),λ=1/2,b¯+a(i)+1/2σ22(i)+r(i)1/4σ12(i);

b¯<a(i);

b¯=a(i),λ<2θ+1/2,σ2(i)0;

b¯=a(i),λ<2θ+1/2,σ2(i)=0,λ<1/2,r(i)<1/4σ12(i);

b¯=a(i),λ=2θ+1/2,b¯+a(i)<1/4σ22(i);

b¯=a(i),λ=2θ+1/2,b¯+a(i)=1/4σ22(i),2θ<1/2,r(i)<1/4σ12(i);

b¯=a(i),λ=2θ+1/2,b¯+a(i)=1/4σ22(i),2θ=1/2,r(i)+1/2σ22(i)<1/4σ12(i).

λ<2θ; and for every iS, one of conditions (A1)-(A4) holds;

λ=2θ; and for every iS, one of conditions (A5)-(A7) holds;

λ>2θ; and for every iS, one of conditions (A8)-(A13) holds;

λ<2θ; and for every iS, (A1) or (A4) holds;

λ=2θ,b¯<a(i)+1/4σ22(i), iS;

λ>2θ,b¯<aˇ;

there exists uS such that for any iS{u},qiu>0;

i=1mπiβ(i)>0, where β(i)=r(i)-1/2σ12(i), iS;

b¯<aˇ.

2. Uniqueness of Global Positive Solution

As x(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 value φC([-τ,0];R+),ιS, 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-τt0φtmax-τt0φt<k0.For each integer kk0, define the stopping time(12)τk=inft0,τe:xt1k,k.Here, set inf=. Obviously, τk is increasing as k and τ=limk+τkτe a.s. If τ= a.s., then x(t)R+, for t0 and τe= a.s. Thus, to prove this theorem, we need to prove τ= a.s. Let(13)Vx=x-1-12lnx,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+×SR 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-xλFx,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,xR+,iS. Substituting these into (14) yields(18)dVxtM1+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 τkt and taking the expectations yields(19)EVxτktVφ0+E0τktM1+λb¯1+2λx1/2+λs-τ-x1/2+λs+b¯2xλs-τ-xλsds.In addition,(20)E0τktx1/2+λs-τds-τ0φ1/2+λsds+E0τktx1/2+λsds,and (21)E0τktxλs-τds-τ0φλsds+E0τktxλsds.Substituting (20) and (21) into (19) gives(22)EVxτktVφ0+M1T+λb¯1+2λ-τ0φ1/2+λsds+b¯2-τ0φλsds,t0,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-τ0φλsdsEVxτkTEIτkTVxτkT=EIτkTVxτkEIτkTVkV1k=VkV1kEIτkT=VkV1kPτkT, which implies (24)limsupkPτkTlimkVφ0+M1T+λb¯/1+2λ-τ0φ1/2+λsds+b¯/2-τ0φλsdsVkV1/k=0. Then, limkP(τkT)=0. Further, P(τT)=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 value φC([-τ,0];R+),ιS, there exists a unique solution x(t) to (6) on [-τ,+) and x(t) will remain in R+ with probability 1.

Remark 3.

If γ(t)ςS,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 .

Remark 4.

If b(i)=0,iS, then (6) reduces to (3). Clearly, if b(i)=0,iS, then one of conditions (H1)-(H3) holds. Hence, Theorem 1 generalizes Theorem 1 in .

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+PxtH1-ε,liminft+Pxtδ1-ε.

Lemma 6 (Lemma 3, [<xref ref-type="bibr" rid="B12">12</xref>]).

If (H7) and (H8) hold, then there exists a constant α>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),iS.

Lemma 7 (Theorem 2.10, [<xref ref-type="bibr" rid="B27">27</xref>]).

If A=(aij)n×nZn×n, then the following results are equivalent.

A is nonsingular M-matrix.

A is semipositive; that is, there is x0 in Rn satisfying Ax0.

Lemma 8.

Assume that one of the following conditions holds:

0<p<1 and (H4) holds

p>0,b¯<a(i)+1/2(1-p)σ22(i),iS and (H5) holds

p>0 and (H6) holds

For any initial value φC([-τ,0];R+),ιS, let x(t) be the solution of (6) with the initial conditions (7). Then there exists K1(p)>0 such that(27)limsupt+ExptK1p.

Proof.

From Theorem 1, 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 iS, let(28)μi=1τlnλ+pai-pb¯+1/2λ+p1-pσ22iλb¯,b¯>01,b¯=0.It follows from (ii) that μ(i)>0,iS. Let β>0 satisfying β<miniS{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,+)×SR 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,ieβtβ+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<miniS{μ(i)}, one can show that(33)12pp-1σ22i+b¯p2λ+p-pai+b¯pλλ+peβτ<0,iS.Hence, there exists M2=M2(p)>0 such that F1(x,i)M2,xR+,iS. Thus,(34)LUx,y,t,iM2eβt+pb¯λλ+peβt-eβτxλ+p+yλ+p.In addition,(35)0teβsxλ+ps-τdseβτ-τ0φλ+psds+eβτ0teβsxλ+psds.From (29), (34), and (35), we have(36)eβtExptφp0+M2βeβt-1+pb¯λλ+peβτ-τ0φλ+psds, which yields(37)limsupt+ExptM2βK1p. That is, (27) holds. If (i) or (iii) holds, the proof is similar and omitted. Hence, Lemma 8 holds.

Remark 9.

If λ=1,σ2(i)=0,iS and (H6) holds, then condition (A1’) in  holds. Hence, Lemma 8 generalizes Theorem 3.3 in .

Remark 10.

If λ>2θ,p>0,b(i)=0,iS, then (H6) holds. Hence, Lemma 8 improves and generalizes Lemma 6 in .

Theorem 11.

Assume that (H7) and (H8) and one of conditions (H4)-(H6) hold. If 0<θ1,0<λ1+θ and b(i)0,iS, then (6) is stochastically permanent.

Proof.

Clearly, if one of conditions (H4)-(H6) holds, then one of conditions (H1)-(H3) holds. From Theorem 1, for any given φC([-τ,0],R+),ιS, let x be the solution of the initial value problem (6) and (7). Define(38)V1x=1x1+θ,xR+. 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)T0 such that A(α)w0. Further, for any iS,(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+E0tLV2xs,xs-τ,γsds,where LV2:R+×R+×SR 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+V1xαj=1mqijpj=1+V1xα-2-pi1+θαβi-12pi1+θ2α2σ12i-j=1mqijpjV12x+-pi1+θαri+12piα1+θ2+θσ12i+2j=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 iS,(43)pi1+θαβi-121+θ2α2σ12i-j=1mqijpj-ηpi>0.From the generalized Itô’s formula, we have (44)EeηtV2xt,γt=V2φ0,ι+E0tηeηsV2xs,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+2j=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+V1xeηtF2x,i, where (46)F2x,i=1+V1xα-2-pi1+θαβi-12pi1+θ2α2σ12i-j=1mqijpj-ηpiV12x+-pi1+θαrˇ+12piα1+θ2+θσ12^+2j=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 supxR+,iSF(x,i)M3. Thus, for any iS, (48)piEeηt1+V1xtαpι1+V1φ0α+M3eηt-1η.Further,(49)limsupt+EV1αxtlimsupt+E1+V1xtαM3ηminp1,p2,,pmM4.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-αθEx-α-αθtδ-α-αθ=δα+αθEx-α-αθt.Further, limsupt+P{x(t)<δ}δα+αθM4=ε. This yields(51)liminft+Pxtδ1-ε.

Next we claim that for any ε>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 any ε>0, let H=ε/K1(p)-1/p. Using Chebyshev’s inequality yields(52)Pxt>H=Pxpt>HpH-pExpt,which means that(53)limsupt+Pxt>Hlimsupt+H-pExpt=ε.Hence,(54)liminft+PxtH1-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, [<xref ref-type="bibr" rid="B27">27</xref>]) (BDG’s inequality)).

Let gL2R+;Rm×n. Define, for t0, 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/2Esup0stxspCpEAtp/2for all t0. 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 any φC([-τ,0];R+),ιS, let x(t) be the solution of the initial value problem (6) and (7).

(i) If (H4) holds, then(57)limsupt+lnxtlnt2a.s.

(ii) If (H5) holds and b¯<a(i)+1/2[1-max{2,2λ,1+λ}]σ22(i),iS or (H6) holds, then(58)limsupt+lnxtlnt1a.s.

Proof.

(i) Assume that (H4) holds. Applying the generalized Itô’s formula to etlnx(t) yields(59)etlnxt-lnφ0=0teslnxs+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 nZ+, it follows from exponential martingale inequality that(60)Psup0tvnNit-12e-vnNit,Nit>ρevnlnn1nρ,i=1,2.From the Borel-Cantelli Lemma , for almost all ωΩ, there is a random integer n0=n0(ω) sufficiently large such that for n0>n0(ω),i=1,2,(61)Nit12e-vnNit,Nit+ρevnlnn,0tvn. This, together with (59), implies that for any tvn-3/2,vn-1/2 and for almost all ωΩ, (62)etlnxtlnφ0+0teslnxs+βγs-aγsxλs+bγsxλs-τ+12es-vnσ12γs-121-es-vnσ22γsx2θsds+2ρevnlnnlnφ0+0teslnxs+β^-aˇ-b¯eτxλs+12es-vnσ12γs-121-es-vnσ22γsx2θsds+b¯eτ-τ0φλsds+2ρevnlnnlnφ0+0tesF3xsds+b¯eτ-τ0φλsds+2ρevnlnn,where F3(x)=lnx+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 xR+. Therefore, for almost all ωΩ, if nn0(ω),vn-3/2tvn-1/2, (63)lnxtlntlnφ0etlnt+M51-e-tlnt+b¯eτ-τ0φλsdsetlnt+2ρe-vn-3/2evnlnnlnvn-3/2.This implies (64)limsupt+lnxtlnt2ρe3/2v,which is the required assertion (57) by letting ρ1 and v0.

(ii) Assume that (H5) and b¯<a(i)+1/2[1-max{2,2λ,1+λ}]σ22(i),iS or (H6) holds. It follows from Theorem 1 that for any tτ,x(t)R+ a.s. This, together with (5), yields (65)dxtr^xt+b¯xtxλt-τdt+σ1γtxtdB1t+σ2γtx1+θtdB2t.

Further, (66)Esuptut+1xuExt+r^tt+1Exsds+b¯tt+1Exsxλs-τds+Esuptut+1tuσ1γsxsdB1s+Esuptut+1tuσ2γsx1+θsdB2s. Using the BDG’s inequality, there exist C1>0 and C2>0 such that (67)Esuptut+1tuσ1γsxsdB1sC1Ett+1σ1γsxs2ds1/2E12suptut+1xu·2C12σ1¯2tt+1xsds1/214Esuptut+1xu+C12σ1¯2tt+1Exsds,and (68)Esuptut+1tuσ2γsx1+θsdB2sC2Ett+1σ2γsx1+θs2ds1/2E12suptut+1xu·2C22σ2¯2tt+1x1+2θsds1/214Esuptut+1xu+C22σ2¯2tt+1Ex1+2θsds. From Hölder’s inequality, (69)tt+1Exsxλs-τdstt+1Ex2s1/2Ex2λs-τ1/2ds12tt+1Ex2sds+12tt+1Ex2λs-τds. Hence,(70)Esuptut+1xu2Ext+2r^tt+1Exsds+b¯tt+1Ex2sds+b¯tt+1Ex2λs-τds+2C12σ1¯2tt+1Exsds+2C22σ2¯2tt+1Ex1+2θsds.It follows from (70) and Lemma 8 that(71)limsupt+Esuptut+1xu2+2r^+2C12σ1¯2K11+b¯K12+K12λ+2C22σ2¯2K11+2θ.From (71), there is M6>0 such that(72)Esupktk+1xtM6,k=1,2,.For any ε>0, from Chebyshev’s inequality, we have(73)Psupktk+1xt>k1+εM6k1+ε,k=1,2,.From the Borel-Cantelli Lemma , for almost all ωΩ, there exists an integer k0(ω) such that for any kk0(ω),(74)supktk+1xt,ωk1+ε.Consequently, for any ε>0 and almost all ωΩ, if kk0(ω) and ktk+1, then(75)lnxt,ωlnt1+εlnklnk=1+ε.Therefore(76)limsupt+lnxtlnt1+εa.s.Letting ε0, we obtain the desired assertion (58).

From Theorem 13, it is easy to see that the following results hold.

Corollary 14.

Assume that b(i)=0,iS. For any x0>0,ιS, 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 λ2θ, then (58) holds.

Corollary 15.

Assume that λ=1,θ=1/4,σ2(i)=0,iS. For any φC([-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  shows that (57) holds for any λ>0,θ>0. Hence, Corollary 14 improves Theorem 6 in . In addition, Corollary 15 is corresponding to Lemma 5.1 in . Hence, Theorem 13 generalizes and improves Theorem 6 in  and Lemma 5.1 in .

Theorem 17.

Assume that (H7), (H8), and one of conditions (H4)-(H6) hold. For any φC([-τ,0];R+),ιS, let x(t) be the solution of the initial value problem (6) and (7). If 0<θ1,0<λ1+θ and b(i)0,iS, then(77)liminft+lnxtlnt-11+θαa.s.

Proof.

From (39) and the generalized Itô’s formula, (78)d1+V1xtαα1+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+V1xtαα1+θ1+V1xtα-2-βˇ+12α1+θσ1¯2+1V12xt+M7dt-α1+θ1+V1xtα-1σ1γtV1xtdB1t+σ2γtx-1tdB2tα1+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+V1xtαM9,t0.

Let δ>0 satisfy(82)α1+θM8δ+6δσ¯1+σ¯2<12.From (80), for any k{1,2,},(83)Esupk-1δtkδ1+V1xtαE1+V1xk-1δα+α1+θM8Esupk-1δtkδk-1δt1+V1xsαds+α1+θEsupk-1δtkδk-1δt1+V1xsα-1σ1γsV1xsdB1s+α1+θEsupk-1δtkδk-1δt1+V1xsα-1σ2γsx-1sdB2s.In addition, we have(84)Esupk-1δtkδk-1δt1+V1xsαdsEk-1δkδ1+V1xsαdsδEsupk-1δtkδ1+V1xtα.From the BDG’s inequality,(85)Esupk-1δtkδk-1δt1+V1xsα-1σ1γsV1xsdB1s6Ek-1δkδ1+V1xs2α-2σ12γsV12xsds1/26σ¯1Ek-1δkδ1+V1xs2αds1/26δσ¯1Esupk-1δtkδ1+V1xtα,and(86)Esupk-1δtkδk-1δt1+V1xsα-1σ2γsx-1sdB2s6Ek-1δkδ1+V1xs2α-2σ22γsx-2sds1/26σ¯2Ek-1δkδ1+V1xs2αds1/26δσ¯2Esupk-1δtkδ1+V1xtα.From (83)–(86), (87)Esupk-1δtkδ1+V1xtαE1+V1xk-1δα+α1+θM8δEsupk-1δtkδ1+V1xtα+6α1+θδσ¯1+σ¯2Esupk-1δtkδ1+V1xtα=E1+V1xk-1δα+α1+θM8δ+6δσ¯1+σ¯2Esupk-1δtkδ1+V1xtα.This, together with (81) and (82), yields(88)Esupk-1δtkδ1+V1xtα2M9.For any ε>0, using Chebyshev’s inequality,(89)Psupk-1δtkδ1+V1xtα>kδ1+ε2M9kδ1+ε.From the Borel-Cantelli Lemma, for almost all ωΩ, there exists k0(ω)>0 such that for any k>k0(ω),(90)supk-1δtkδ1+V1xt,ωαkδ1+ε,which yields,(91)ln1+V1xtαlnt1+εlnkδlnk-1δ<1+εa.s.Therefore(92)limsupt+ln1+V1xtαlnt1+εa.s. Letting ε0, we have limsupt+ln(1+V1(x(t)))α/lnt1 a.s. Further, (93)limsupt+lnxt-1+θαlnt1a.s.Hence,(94)limsupt+lnxt-1+θαlnt=-1+θαliminft+lnxtlnt1a.s.This yields (77). Hence, Theorem 17 holds.

Theorem 18.

Assume that the conditions of Theorem 13 hold. If (H7)-(H9) hold, then for any φC([-τ,0];R+),ιS, the solution x(t) of (6) obeys(95)limsupt+1t0txλsds1aˇ-b¯i=1mπiβia.s.

Proof.

From Theorems 13 and 17,(96)limt+lnxtt=0a.s.Using the generalized Itô’s formula, we obtain(97)lnxt=lnx0+0trγs-12σ12γsds-0taγsxλsds+0tbγsxλs-τds-120tσ22γsx2θsds+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 n2, (99)Psup0tnM2t-12M2t,M2t>2lnne-2lnn=1n2.From Borel-Cantelli Lemma, for almost all ωΩ, there exists a random integer n0 such that for nn0(ω), (100)sup0tnM2t-12M2t,M2t2lnn. Further, for almost all ωΩ,nn0(ω), 0tn, (101)M2t2lnn+12M2t,M2t=2lnn+120tσ22γsx2θsds. Note that (102)0tbγsxλs-τdsb¯0txλs-τds=b¯-τt-τxλsdsb¯-τ0xλsds+b¯0txλsds. From (97), for any nn0,t(n-1,n](103)lnxttlnx0t+1t0tβγsds+-aˇ+b¯t0txλsds+b¯t-τ0xλsds+2lnnn-1+M1tt. Let t+, using the strong law of large numbers for martingales and (96), we obtain (104)limsupt+1t0txλsds1aˇ-b¯i=1mπiβia.s. We then obtain assertions (95). Hence, Theorem 18 holds.

Remark 19.

Liu et al.  have claimed that, under (H7) and (H8), the solution of (2) obeys(105)limsupt+1t0txλsds1aˇi=1mπiβia.s.It is easy to see that if b(i)=0,iS, then (95) becomes (105). Hence, Theorem 18 generalizes Theorem 8 in .

Remark 20.

If λ=1,σ2(i)=0,iS, then conditions (H7)-(H9) are corresponding to the conditions (A1”), (A2), and (A3) in . Hence, Theorem 18 generalizes Theorem 5.3 in .

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+1t0tx2sds2.316a.s.

In Figure 1, (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.

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+1t0tx1.5sds6.994a.s.

In Figure 2, (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.

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+1t0txsds0.383a.s. Figure 3 clearly supports Theorems 11 and 18.

Stochastic trajectories of model (6) with parameters in Example 3.

Assume that the Markov chain γ(t)=1S. 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)=2S. 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 , subsystem (110) is stochastically extinct. Figure 4 clearly supports these results.

Stochastic trajectories of subsystems (109) and (110).

In Figure 4, (i) left panel shows stochastic trajectory of subsystem (109); (ii) right panel shows stochastic trajectory of subsystem (110).

Example 3 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.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

All the authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (no. 11471197), the Natural Science Foundation of Shanxi Province (no. 201801D21009), and the Natural Science Foundation of Lvliang University (no. ZRXN201610).

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