# Mean-Square Almost Periodic Solutions for Impulsive Stochastic Host-Macroparasite Equation on Time Scales

Authors: Pan Wang; Qingmei Lin; Yongkun Li
Journal: Discrete Dynamics in Nature and Society
 (2015)

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## Abstract

We consider an impulsive stochastic host-macroparasite equation on
time scales. By use of the Banach fixed point theorem and Gronwall-Bellman’s
inequality technique on time scales, we obtain the existence and exponential stability
of mean-square almost periodic solutions for the host-macroparasite equation on time
scales. Finally, we give an example to illustrate the feasibility of our results.


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## Body

## 1. Introduction

Many important human diseases, particularly in tropical and subtropical regions, arise from infection by macroparasites or metazoan organisms. These organisms tend to have much larger generation times and more complex life cycles than microparasites. In life cycles, there are two or more obligatory host species together with the final host (humans). Macroparasitic infections are generally chronic in form and they are more a cause of morbidity than mortality and tend to be persistent in character in areas where they are endemic. The final hosts of parasites are usually humans (the hosts in which the parasite attains reproductive maturity) and they gain entry to the definitive host as a consequence of developmental changes which normally occur before the organism arrives at its preferred site and attains reproductive maturity. Because of this fact, the host-macroparasite system has been attracting the attention of many researchers (see [1–4]). A time delay, therefore, exists between entry to the definitive host and the point when the parasite begins the production of eggs or larvae for transmission to other hosts (see [5–7]). In [7], by means of a continuation theorem in coincidence degree theory, the authors considered the oscillation and global attractivity of the nonlinear delay host-macroparasite model with periodic coefficients.On the one hand, the theory of impulsive differential equations is now being recognized to not only be richer than the corresponding theory of differential equations without impulses, but also represent a more natural framework for mathematical modelling of many real-world phenomena, such as population dynamical models and neural networks. Since many dynamical processes are characterized by the fact that, at certain moments of time, they undergo abrupt changes of state, with the development of the theory of impulsive differential equations (see [8, 9]), various models of impulsive differential equations have been proposed and studied extensively (see [10–14]). For example, authors of [14] considered the nonlinear impulsive delay host-macroparasite model with periodic coefficients: (1)M′t=-βtMt+ptMtqt+Mt-nωn+1,t≠tk,Mtk+=1+bkMtk,t=tk,k∈N,where m is a positive integer, q(t)=1/δ(t), p(t)=α(t)/δn+1(t), and β(t) are ω-periodic functions. By use of the continuation theorem of coincidence degree, some sufficient conditions are obtained for the global attractivity and oscillation of positive periodic solutions.In fact, both continuous and discrete systems are very important in implementation and applications. But it is troublesome to study the dynamical properties for continuous and discrete systems, respectively. Therefore, it is significant to study that on time scales which can unify the continuous and discrete cases. In [15], the author considered the following host-macroparasite equation on time scales: (2)xΔt=-atxt+∑i=1mbitxt-τit1+xt-τitNi+1.By using the contraction principle and Gronwall-Bellman’s inequality on time scale, some sufficient conditions are obtained for the existence and exponential stability of almost periodic solutions.On the other hand, as a matter of fact, population systems are often subject to environmental noise; that is, due to environmental fluctuations, parameters involved in population models are not absolute constants, and they may fluctuate around some average values. Based on these factors, more and more people began to be concerned about stochastic population systems (see [16–20]). Meanwhile, almost periodicity is universal than periodicity, and the mean-square almost periodicity is important in probability for investigating stochastic processes. To the best of our knowledge, there exist few results for mean-square almost periodic solutions for impulsive stochastic process models with delays; one can see some results in [21–23]. However, there exists no result on the existence and uniqueness of mean-square almost periodic solutions for impulsive stochastic host-macroparasite equation on time scales. Motivated by the above, we consider the following impulsive stochastic host-macroparasite equation on time scales: (3)Δxt=-atxt+∑i=1mbitxt-τit1+xt-τitNi+1Δt+∑i=1mδit,xt-ηtΔwt,t≠tk,Δ~xtk=xtk+0-xtk-0=αkxtk+Ikxtk+νk,t=tk,k∈N,where Δx(t) denotes a Δ-stochastic differential of x(t),a(t),bi(t),τi(t),η(t)∈PCrd(T,R+), Ni>0, δi is Borel measurable, i=1,2,…,m, k∈Z, {tk}∈B={{tk}∣tk∈R,tk<tk+1,limt→∞tk=+∞}, αk,νk∈R, and Ik∈C(R,R). Let (Ω,F,P) be a complete probability space furnished with a complete family of right continuous increasing sub-σ-algebras {Ft}t≥0 satisfying Ft⊂F. w(t) is a standard Brownian motion over (Ω,F,P).Throughout this paper, we assume the following.(H1) a, bi, τi, η:T→R+ are all mean-square almost periodic functions, a(t)≥λ>0, t-τi(t)∈T, t-η(t)∈T, for t∈T, δi:T×R→R+ is mean-square almost periodic in t uniformly for x∈R, δi(t,0)=0, and there exists constant li such that (4)Eδit,x-δit,y2≤liEx-y2,t∈T,x,y∈PCrdT,L2R,i=1,2,…,m.(H2) The sequences αk, νk are almost periodic in k and the sequences {Ik(x)} are almost periodic in k uniformly for x∈PCrd(T,L2(R)) and there exists constant L such that (5)EIkx-Iky2≤LEx-y2,t∈T,x,y∈PCrdT,L2R,i=1,2,…,m.(H3) The set of sequences {tkj},  tkj=tk+j-tk, k,j∈N, are equipotentially almost periodic and infktk1=θ>0, where L2(R) will be introduced in Section 2.For convenience, we denote(6)f¯=supt∈R⁡ft,g¯=supt,x∈T×L2R⁡gt,x,where f(t) is a mean-square almost periodic function and g(t,x) is a mean-square uniformly almost periodic function on time scales.Letφ0:(t0-τ,t0)T→L2(R) be a mean-square almost periodic function. Denote by x(t)=x(t;t0,φ0) the solution of (3) satisfying the initial conditions: (7)xt;t0,φ0=φ0t,t∈t0-τ,t0T,xt0+;t0,φ0=φ0t0,where τ=supt∈T{maxi|τi(t)|,|η(t)|}.Our main purpose of this paper is to study the existence and exponential stability of mean-square almost periodic solutions to (3) by means of the Banach fixed point theorem and Gronwall-Bellman’s inequality technique.


## 2. Preliminaries

In this section, we shall recall some basic definitions and lemmas which are used in what follows.A time scaleT is an arbitrary nonempty closed subset of the real numbers; the forward and backward jump operators σ, ρ:T→T and the forward graininess μ:T→R+ are defined, respectively, by(8)σt≔inf⁡s∈T:s>t,ρt≔sup⁡s∈T:s<t,μt=σt-t.A pointt∈T is called left-dense if t>infT and ρ(t)=t, left-scattered if ρ(t)<t, right-dense if t<supT and σ(t)=t, and right-scattered if σ(t)>t. If T has a left-scattered maximum m, then Tk=T∖{m}; otherwise, Tk=T. If T has a right-scattered minimum m, then Tk=T∖{m}; otherwise, Tk=T.A functionf:T→R is right-dense continuous provided it is continuous at right-dense point in T and its left-side limits exist at left-dense points in T. If f is continuous at each right-dense point and each left-dense point, then f is said to be continuous function on T.Fory:T→R and t∈Tk, we define the delta derivative of y(t), yΔ(t), to be the number (if it exists) with the property that, for a given ɛ>0, there exists a neighborhood U of t such that(9)yσt-ys-yΔtσt-s<ɛσt-s,for all s∈U.Ify is continuous, then y is right-dense continuous, and if y is delta differentiable at t, then y is continuous at t.Letf be right-dense continuous; if FΔ(t)=f(t), then we define the delta integral by (10)∫rsftΔt=Fs-Fr,r,s∈T.Lemma 1 (see [24]).
Assumef,g:T→R are delta differentiable at t∈T; then,(i)
(f+g)Δ(t)=fΔ(t)+gΔ(t);(ii)
(fg)Δ(t)=fΔ(t)g(t)+fσ(t)gΔ(t)=f(t)gΔ(t)+fΔ(t)gσ(t);(iii)
ifg(t)gσ(t)≠0, then (f/g)Δ=(fΔtgt-ftgΔt)/g(t)gσ(t);(iv)
iff and fΔ are continuous, then (∫at‍f(t,s)Δs)Δ=f(σ(t),t)+∫at‍fΔ(t,s)Δs.A functionp:T→R is called regressive provided 1+μ(t)p(t)≠0, for all t∈Tk. The set of all regressive and rd-continuous functions p:T→R will be denoted by R=R(T)=R(T,R). We define the set R+=R+(T,R)={p∈R:1+μ(t)p(t)>0,∀t∈T}.Ifr∈R, then the generalized exponential function er is defined by (11)ert,s=exp⁡∫stξμτrτΔτ,for all s,t∈T, with the cylinder transformation (12)ξhz=log⁡1+hzh,h≠0,z,h=0.Letp,q:T→R be two regressive functions; we define(13)p⊕q=p+q+μpq,⊖p=-p1+μp,p⊖q=p⊕⊖q=p-q1+μq.Then, the generalized exponential function has the following properties.Lemma 2 (see [24]).
Assume thatp,q:T→R are two regressive functions; then, (i)
e0(t,s)≡1 and ep(t,t)≡1;(ii)
ep(σ(t),s)=(1+μ(t)p(t))ep(t,s);(iii)
ep(t,s)=1/ep(s,t)=e⊖p(s,t);(iv)
ep(t,s)ep(s,r)=ep(t,r);(v)
ep(t,s)eq(t,s)=ep⊕q(t,s);(vi)
ep(t,s)/eq(t,s)=ep⊖q(t,s);(vii)
(1/ep(t,s))Δ=-p(t)/epσ(t,s).In the following, we will present some basic concepts and results concerning stochastic differential equations on time scales which will be essential to prove our main results. For more details, the reader may want to consult [17].Definition 3 (see [25]).
For a functionf:[0,1]T→R, one can define the extension f~:[0,1]→R by f~(t)=f(sup[0,t]T), for all t∈[0,1].Let(Ω,F,P) be a probability space, and L2(R) stands for a space that consists of all R-valued random variables x with the norm(14)Ex2=∫wx2dP.Let w be a standard Wiener process and suppose {w(t+h)-w(t):h≥0} is independent of Ft≔σ{w(s):0≤s≤t}, where FR={Ft:t∈R} is a filtration on R, and with σ{·}, we mean the σ-algebra generated by ·. We denote Δ-stochastic integral on [0,1]T, by ∫01‍f(t)Δw(t).Definition 4 (see [25]).
One can say that the random processf:T×Ω→R belongs to the class L2([0,1]T) if the following conditions hold: (i)
f is adapted to FT; that is, f(t,·) is Ft-measurable, for all t∈[0,1]T.(ii)
P(∫01‍|f(t,w)|2Δt<∞)=1.Definition 5 (see [25]).
One can say that the random processf∈L2([0,1]T) has a Δ-stochastic integral on [0,1]T, provided the corresponding process f~ has a stochastic integral in the common sense on [0,1] and then one can have ∫01‍f(t)Δω(t)=∫01‍f~(t)dω(t).Lemma 6 (see [25]).
TheΔ-stochastic integral has the following properties: (i)
Iff1,f2∈L2([0,1]T) and c1,c2∈R, then(15)∫01c1f1t+c2f2tΔwt=c1∫01f1tΔwt+c2∫01f2tΔwt.(ii)
IfE(∫01‍|f(t)|2Δt)<+∞, then E(∫01‍f(t)Δw(t))=0 and the Itô-isometry holds; that is, (16)E∫01ftΔwt2=E∫01f2tΔt.Definition 7 (see [25]).
If, for allt1,t2∈[0,1]T and t1<t2, such that (17)Xt2-Xt1=∫t1t2bt,XΔt+∫t1t2Bt,XΔwt,where b is Lebesgue integrable on [0,1]T and B∈L2([0,1]T), then we say that the process X has a Δ-stochastic differential indicated by the notation (18)ΔXt=bt,XΔt+Bt,XΔwt.We consider the impulsive stochastic dynamic system on[t0,a]T⊂T:(19)ΔXt=btXtΔt+bX,tΔt+BX,tΔwt,t≠tk,Δ~xtk=αkXtk+IkXtk,t=tk,with initial value X(t0)=X0, where X0 is a random variable, independent of w(t).Definition 8 (see [23]).
One can say that a random processX(t) with X(t0)=X0 is a solution of the impulsive stochastic dynamic system (19) on [t0,a]T if the following conditions hold:(i)
X is adapted to the filtration F.(ii)
For allt∈[t0,a]T, we have almost surely(20)Xt=Wt,t0X0+∫t0tWt,σsbXs,sΔs+∫t0tWt,σsBXs,sΔws+∑t0≤tk<tWt,tkIkXtk,whereW(t,s) is the Cauchy matrix of the following system: (21)ΔXt=btXtΔt,t≠tk,Δ~xtk=αkXtk,t=tk.Definition 9 (see [23]).
One can sayφ:T→L2(R) is rd-piecewise continuous with respect to a sequence {tk}⊂T which satisfies tk<tk+1, k∈Z, if φ(t) is continuous on [tk,tk+1)T and rd-continuous on T∖{tk}. Furthermore, [tk,tk+1)T, k∈Z, are called intervals of continuity of the function φ(t).For convenience,PCrd(T,L2(R)) denotes the set of all piecewise continuous functions with respect to a sequence {tk}, k∈Z. For any integers k and j, denote tkj=tk+j-tk.Definition 10 (see [26]).
A time scaleT is called an almost periodic time scale if (22)Π=τ∈R:t±τ∈T,∀t∈T≠0.Definition 11 (see [10]).
For anyɛ>0, let Γɛ⊂Π be a set of real numbers and {tk}⊂T. One can say {tkj}, k,j∈Z, is equipotentially almost periodic on an almost periodic time scale T if, for r∈Γɛ⊂Π, there exists at least one integer q such that (23)tkq-r<ɛ,∀k∈Z.Definition 12 (see [23]).
LetT be an almost periodic time scale and assume that {tk}⊂T satisfying the derived sequence {tkj}, k,j∈Z, is equipotentially almost periodic. One can call a stochastic process φ∈PCrd(T,L2(R)) mean-square almost periodic if (i)
for anyɛ>0, there is a positive number δ=δ(ɛ) such that if the points t′ and t′′ belong to the same interval of continuity and |t′-t′′|<δ, then Eφ(t′)-φ(t′′)2<ɛ;(ii)
for anyɛ>0, there is relative dense set Γ such that if τ∈Γ, then Eφ(t+τ)-φ(t)2<ɛ, for all t∈T, which satisfy the condition |t-τi|>ɛ, i∈Z.Definition 13 (see [23]).
LetT be an almost periodic time scale and assume that {tk}⊂T satisfying the derived sequence {tkj}, k,j∈Z, is equipotentially almost periodic. One can call a stochastic process φ∈PCrd(T×Ω,L2(R)) mean-square almost periodic in t uniformly for x∈Ω if for any ɛ>0 and for each compact subset S of Ω there existed the following:(i)
for anyɛ>0, there is a positive number δ=δ(ɛ,S) such that if the points t′ and t′′ belong to the same interval of continuity and |t′-t′′|<δ, then Eφ(t′,x)-φ(t′′,x)2<ɛ;(ii)
for anyɛ>0, there is relative dense set Γ such that if τ∈Γ, then Eφ(t+τ,x)-φ(t,x)2<ɛ, for all (t,x)∈T×Ω, which satisfy the condition |t-tk|>ɛ, k∈Z,where Ω⊂L2(R) or Ω=L2(R).Definition 14.
The equation (3) is said to be exponentially stable if, for all c>0, there exist α=α(c)>0 and K>0 such that if ut0-vt0≤c, then, for all t≥t0, (24)Eut;t0,ut0-vt;t0,vt02<KEut0-vt02e-αt-t0.In order to study (3), we consider the linear system: (25)Δxt=-atxtΔt,t≠tk,Δ~xtk=αkxtk,t=tk,k∈N.Now let us consider the equations (26)Δxt=-atxtΔt,tk-1<t≤tkand their solutions(27)xt=xse-at,s,tk-1<s≤t≤tk.Then, by [10], the Cauchy matrix of the linear equation (25) is(28)Wt,s=e-at,s,tk-1<s≤t≤tk,∏i=rk+11+αie-at,s,tr-1<s≤tr<tk<t≤tk+1and the solutions of (25) are in the following form: (29)xt;t0,φ0=Wt,t0φ0,φ0∈R.Similar to the proofs of Lemma 3.1 and Lemma 3.2 in [10], one can easily show the following two lemmas, respectively.Lemma 15.
For system (3), let (H1)–(H3) hold. Then, for each ɛ>0, there exist ε1>0, 0<ε1<ɛ, a relative dense set T¯⊂Π of real numbers and Q, such that the following relations are fulfilled:(a)
Eat+τ-at2<ɛ, Ebit+τ-bit2<ɛ, Eτit+τ-τit2<ɛ,   Eξt+τ-ξt2<ɛ, Eδit+τ,x-δit,x2<ɛ, t∈T, τ∈T¯, x∈PCrd(T,L2(R)), |t-tk|>ɛ, k∈Z, i=1,2,…,m;(b)
|αk+q-αk|<ɛ, |Ik+q(x)-Ik(x)|<ɛ, |vk+q-vk|<ɛ, ∀x∈PCrd(T,L2(R)), q∈Q, k∈Z, i=1,2,…,m;(c)
|tkq-τ|<ε1, q∈Q, τ∈T¯, k∈Z.Lemma 16.
For (3), let (H1)–(H3) hold. Then, the Cauchy matrix W(t,s) of (3) satisfies the inequality (30)Wt,s≤Ke-λt-s,t≥sand for any ɛ>0, t,s∈T, |t-tk|>ɛ, |s-tk|>ɛ, k∈Z, there exists a relatively dense set T¯⊂Π of ɛ-almost periods of the function a(t) and a positive constant Γ such that, for r∈T¯, (31)Wt+r,s+r-Wt,s≤ɛΓe-λ/2t-s,t≥s.Definition 17.
The equation (3) is said to be exponentially stable if, for all c>0, there exist α=α(c)>0 and M>0 such that if ut0-vt0≤c, then, for all t≥t0, (32)Eut;t0,ut0-vt;t0,vt02<MEut0-vt02e⊖αt,t0.


## 3. Existence of Almost Periodic Solution

In this section, we will study the existence and exponential stability of mean-square almost periodic solutions of (3) by using the Banach fixed point theorem and Gronwall-Bellman’s inequality technique on time scales.Theorem 18.
Assume that(H1)–(H3) hold and there exist two positive constants d, N such that the following conditions hold.
(
H
4
)  γ<1, where (33)γ=3K21+μ¯λ2λ2∑i=1mbi¯Mi2+∑i=1mδi¯2+3KL1-e-λθ2,Mi=1Ni1+1/Ni1+Ni,i=1,2,…,m.
(
H
5
)  (⊖λ)⊕p<0, where(34)p=4K21+λμ¯2λ∑i=1mbi¯2+∑i=1mli2.Then,(1)
There exists a unique square-mean almost periodic solutionx(t) of (3) in the region B∗={φ∣φ∈PCrd(T,L2(R)),Eφ(t)2≤d2,t∈T}.(2)
The solutionx(t) in the region B∗ is exponentially stable.Proof.
We define an operatorT in B∗ as(35)Tφ=∫-∞tWt,σsSφsΔs+∫-∞tWt,σsLφsΔws+∑tk<tWt,tkhφk,where(36)Sφs=∑i=1mbisφs-τis1+φs-τisNi+1,Lφs=∑i=1mδis,xs-ηs,hφk=Ikφtk+vk,k∈Z.
For anyφ∈B∗, by means of the proof of Theorem 1 in [19], we have(37)Sφ=sups∈T⁡∑i=1mbisφs-τis1+φs-τisNi+1≤sups∈T⁡∑i=1mbi¯φs-τis1+φs-τisNi+1≤∑i=1mbi¯Mi.
Firstly, we will show thatT is a self-mapping from B∗ to B∗. In fact, for any φ∈B∗, by use of (37), we have(38)ETφ2≤3E∫-∞tWt,σsSφsΔs2+3E∫-∞tWt,σsLφsΔws2+3E∑tk<tWt,tkhφk2≤3K21+μ¯λ2λ2∑i=1mbi¯Mi2+∑i=1mδi¯2+3K1-e-λθsupk∈Z⁡Ik¯+vk2≔d2.
On the other hand, letτ∈T¯ and q∈Q, where the set T¯ is determined in Lemma 15. Then,(39)Tφt+τ-Tφt≤∫-∞tWt+τ,σs+τ-Wt,σsSφs+τΔs+∫-∞tWt,σsSφs+τ-SφsΔs+∫-∞tWt+τ,σs+τ-Wt,σsLφs+τΔws+∫-∞tWt,σsLφs+τ-LφsΔws+∑tk<t+τWt+τ,tkhφk-∑tk<tWt,tkhφk≤∑i=1mbi¯Mi∫-∞tWt+τ,σs+τ-Wt,σsΔs+∫-∞tWt,σsSφs+τ-SφsΔs+∑i=1mδi¯∫-∞tWt+τ,σs+τ-Wt,σsΔws+∫-∞tWt,σsLφs+τ-LφsΔws+∑tk<tWt+τ,tk+q-Wt,tkhφk+q+∑tk<tWt,tkhφk+q-hφk=J1+J2+J3,where(40)J1=∑i=1mbi¯Mi∫-∞tWt+τ,σs+τ-Wt,σsΔs+∫-∞tWt,σsSφs+τ-SφsΔs,J2=∑i=1mδi¯∫-∞tWt+τ,σs+τ-Wt,σsΔws+∫-∞tWt,σsLφs+τ-LφsΔws,J3=∑tk<tWt+τ,tk+q-Wt,tkhφk+q+∑tk<tWt,tkhφk+q-hφk.Since (a+b+c)2≤3a2+3b2+3c2, it follows that (41)ETφt+τ-Tφt2≤3EJ12+3EJ22+3EJ32.By conditions (H1)–(H3) and Lemmas 15 and 16, we can obtain that(42)ETφt+τ-Tφt2≤3c12+c22+c32ɛ,where(43)c1=Γ2+μ¯λλ∑i=1mbi¯Mi+K1+μ¯λλ,c2=Γ2+μ¯λλ∑i=1mδi¯+K1+μ¯λλ,c3=Γ1-e-λ/2θsupk∈Z⁡Ik¯+vk+K1-e-λθ.Consequently, by (39) and (42), we obtain that Tφ∈B∗.
Next, we prove that the mappingT is a contraction mapping of B∗. In fact, in view of (H1)–(H3), for any φ,ψ∈B∗, we can easily obtain (44)ESφs-Sψs2≤∑i=1mbi¯2Eφ-ψ2,ELφs-Lψs2≤∑i=1mli2Eφ-ψ2.Notice that(45)ETφ-Tψ2≤3E∫-∞tWt,σsSφs-SψsΔs2+3E∫-∞tWt,σsLφs-LψsΔws2+3E∑tk<tWt,tkhφk-hψk2≤3K21+μ¯λ2λ2∑i=1mbi¯2+∑i=1mli2+3KL1-e-λθ2Eφ-ψ2=γEφ-ψ2.This implies that the mapping T is a contraction mapping. Hence, there exists a unique mean-square almost periodic solution of (3) by the Banach fixed point theorem. Suppose that y(t) is an arbitrary solution of (3) with initial conditions: (46)yt;t0,ϕ0=ϕ0t,t∈t0-τ¯,t0,yt0+;t0,ϕ0=ϕ0t0.Since every solution of system (3) can be represented as (47)xt=Wt,t0φ0+∫t0tWt,σsSxsΔs+∫t0tWt,σsLxsΔws+∑t0≤tk<tWt,tkhxk,we obtain that (48)yt-xt=Wt,t0ϕ0-φ0+∫t0tWt,σsSys-SxsΔs+∫t0tWt,σsLys-LxsΔws+∑t0≤tk<tWt,tkhyk-hxk.
Since(a+b+c+d)2≤4a2+4b2+4c2+4d2, it follows that (49)Eyt-xt2≤4K2e-λt-t0Eϕ0-φ02+4K2∑i=1mbi¯2+∑i=1mli2∫-∞te-λt-σsΔs∫t0te-λt-σsEys-xs2Δs+4KL2∑-∞≤tk<te-λt-tk∑t0≤tk<te-λt-tkEytk-xtk2≤4K2e⊖λt,t0Eϕ0-φ02+4K21+λμ¯λ∑i=1mbi¯2+∑i=1mli2∫t0te⊖λt,σsEys-xs2Δs+4KL21-e-λθ∑t0≤tk<te⊖λt,tkEytk-xtk2≤4K2e⊖λt,t0Eϕ0-φ02+4K21+λμ¯2λ∑i=1mbi¯2+∑i=1mli2∫t0te⊖λt,sEys-xs2Δs+4KL21-e-λθ∑t0≤tk<te⊖λt,tkEytk-xtk2.
LetEu(t)2=Ey(t)-x(t)2eλ(t,t0); then, (50)Eut2≤4Eut02+4K21+λμ¯2λ∑i=1mbi¯2+∑i=1mli2∫t0tEus2Δs+4KL21-e-λθ∑t0≤tk<tEutk2.From Gronwall-Bellman’s inequality on time scales, one has (51)Eut2≤4K2Eut02∏t0<tk<t1+4KL21-e-λθept,t0;that is,(52)Eyt-xt2≤Eϕ0-φ02∏t0<tk<t1+4KL21-e-λθe⊖λ⊕pt,t0,where(53)p=4K21+λμ¯2λ∑i=1mbi¯2+∑i=1mli2.Thus, by condition (H5) of the theorem, the mean-quare almost periodic solution of (3) is exponentially stable. This completes the proof.Remark 19.
According to the conditions of Theorem18, we can find that the uniqueness and exponential stability of the mean-square almost periodic solution for the impulsive stochastic host-macroparasite equation on time scales are independent of the magnitude of delays but are dependent on the magnitude of noise and impulse.Remark 20.
It is the first time that the sufficient conditions for the existence and exponential stability of piecewise mean-square almost periodic solutions for the impulsive stochastic host-macroparasite equation on time scales are investigated. The obtained results are essentially new. Without considering impulsive and stochastic effects on (3), then (3) reduce to (2).


## 4. An Example

Consider the following impulsive stochastic host-macroparasite equation on time scales:(54)Δxt=-atxt+∑i=1mbitxt-τit1+xt-τitNi+1Δt+∑i=1mδit,xt-ηtΔωt,t≠tk,Δ~xtk=xtk+0-xtk-0=αkxtk+Ikxtk+νk,t=tk,k∈N,where i=2, tk=k, k∈N, and(55)at=8+cos⁡2t,b1t=0.031+cos⁡3t,b2t=0.051+sin⁡2t,δ1t,x=0.04sin⁡5tx,δ2t,x=0.02sin⁡2tx,αk=-0.06cos⁡k,Ikx=0.06sin⁡kx,νk=0.07,N1=1,N2=2.By calculating, we have(56)a¯=9,b1¯=0.06,b2¯=0.1,δ1¯=l1=0.04,δ2¯=l2=0.02,L=0.06,M1=0.25,M2=427.It is easy to see thatK=θ=1, λ=5; thus, (57)3K21+μ¯λ2λ2∑i=1mbi¯Mi2+∑i=1mδi¯2+3KL1-e-λθ2≈0.046<1,p-λ=4K21+λμ¯2λ∑i=1mbi¯2+∑i=1mli2-λ≈0.841-5=-4.159<0.Hence, (⊖λ)⊕p<0. According to Theorem 18, the impulsive stochastic equation (54) has a unique piecewise mean-square almost periodic solution which is exponentially stable in the region B∗, where (58)B∗=φ∣φ∈PCrdT,L2R,Eφt2≤1,t∈T.


## 5. Conclusion

In this paper, we have investigated the host-macroparasite equation with impulsive and stochastic effects on time scales. By employing the Banach fixed point theorem, some stochastic analysis techniques, and Gronwall-Bellman’s inequality technique on time scales, we have obtained some sufficient conditions ensuring the existence and exponential stability of mean-square almost periodic solutions for impulsive host-macroparasite equation on time scales. These sufficient conditions can be easily checked by simple algebraic methods. An example has been given to demonstrate the effectiveness of the presented results. The methods used in paper can be applied to study square-mean almost periodic problems of many other types of impulsive stochastic models on time scales.



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*Source: 306349-2015-09-09.xml*