Stochastic Nicholson s blowflies delay differential equation with regime switching

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1 arxiv: v1 [math.pr 12 Jan 219 Stochastic Nicholson s blowflies delay differential equation with regime switching Yanling Zhu, Yong Ren, Kai Wang,, Yingdong Zhuang School of Statistics and Applied Mathematics, Anhui University of Finance and Economics, Bengbu 2333, Anhui, China Department of Mathematics, Anhui Normal University Wuhu 241, Anhui, China Abstract In this paper, we investigate the global existence of almost surely positive solution to a stochastic Nicholson s blowflies delay differential equation with regime switching, and give the estimation of the path. The results presented in this paper extend some corresponding results in Wang et al. stochastic Nicholson s blowflies delayed differential equations, Appl. Math. Lett. 87 (219) Keywords Nicholson s blowflies model, Regime switching, Global solution AMS(2) 34K5; 6H1; 65C3 1 Introduction Gurney et al. (199) [1 proposed the following delay differential equation (known as the famous Nicholson s blowflies model), X (t) = δx(t)+px(t τ)e ax(t τ), (1.1) to model the population of the Australian sheep-blowfly(lucilia cuprina), where p is the maximum per capita daily egg production rate, 1/a is the size at which the blowfly population reproduces at its maximum rate, δ is the per capita daily adult death rate, and τ is the generation time. After that, the dynamics of this equation was studied by Supported by the NSF of China( ,71831),the NSF of Anhui Province(17885MA17),the NSF of Education Bureau of Anhui Province(KJ218A437). Corresponding author. zhyanling99@126.com(y. Zhu), brightry@hotmail.com(y. Ren), wangkai5318@ 163.com(K. Wang) 1

2 many researchers, see Berezansky et al. (21) [2 for an overview of the results on the classical Nicholson s model. Considering that the parameter δ is affected by environmental noises, with δ δ σdb(t), where B(t) is a one-dimensional Brownian motion with B() = defined on a complete probability space (Ω,{F t } t,p), σ denotes the intensity of the noise. Wang et al. (219)[3 presented the following stochastic Nicholson s blowflies delay differential equation dx(t) = [ δx(t)+px(t τ)e ax(t τ) dt+σx(t)db(t), (1.2) with initial conditions X(s) = φ(s), for s [ τ,,φ C([ τ,,[,+ )),φ() >. Under the condition δ > σ 2 /2, they prove that Eq.(1.2) has a unique global solution on [ τ, ), which is positive a.s. on [, ), and give the estimation of limsup t E[X(t) 1 and limsup E[ t X(s)ds, respectively. t But we find from numerical simulation that the condition δ > σ 2 /2 is too strict to the nonexplosion of the solution to equation (1.2). For example, let δ = 1, p = 5, τ = 1, a = 1, σ = 2, and the numerical simulation is showing in figure 1. One can see from figure 1 that the solution X(t) is globally existent on [,+ ), and oscillatory about N = logp/δ, which is the positive equilibrium of the corresponding deterministic equation of equation (1.2), but δ < σ 2 / X()=1 N*=log p/δ X(t) Time t Figure 1. The numerical solution X(t) of equation (1.2) with initial value φ(t) = 1 for t [ 1, and the step = 1e 4. On the other hand, as we know that besides the white noise there is another type of environmental noise in the real ecosystem, that is the telegraph noise, which can be demonstrated as a switching between two or more regimes of environment. The regime switching can be modeled by a continuous time Markov chain (r t ) t taking values in a 2

3 finite state spaces S = {1,2,...,m} and with infinitesimal generator Q = (q ij ) R m m. That is r t satisfies { q ij +o( ), for i j, P(r t+ = j r t = i) = as +, 1+q ij +o( ), for i = j, where q ij is the transition rate from i to j for i j, and q ii = i j q ij for each i S, see Mao et al. (26) [4, Khasminskii et al. (27) [5 and Yin et al. (29) [6 for more details. We assume that the Markov chain r t is independent of the Brownian motion B(t). Considering both the effects of white noise and telegraph noise of environment, we present the following stochastic Nicholson s blowflies delay differential equation with regime switching dx(t) = [ δ rt X(t)+p rt X(t τ rt )e ar t X(t τr t ) dt+σ rt X(t)dB(t) (1.3) with initial conditions X(s) = φ(s), for s [ τ,,φ C([ τ,,r + ), (1.4) where τ = max rt S{τ rt }, R + = [,+ ). The contribution of this paper is as follows, The equation (1.3) considered in this paper includes regime switching, which is more general than equation (1.2); We show the global existence of almost surely positive solution to equation (1.3) without condition δ i > σi 2 /2, i S, which extends the one in [3; We give the estimation of limsup t E[X θ 1 (t) and limsup E[ t t Xθ (s)ds for θ [1,1+2min i S {δ i /σ i }, which is more general than the ones in literature. 2 Main Results In this section, we will show the global existence of almost surely solution of equation (1.3) with initial value (1.4), and give the estimation of the solution. For simplicity, in the following we use the notations: β i = θ[δ i (1 θ)σ2 i, γ i = θp i /(a i e), ρ = min i S {δ i/σ 2 i},m 1 = max i S,x R +{γ ix θ 1 β i x θ }, M 2 = max i S,x R +{(α β i)x θ +γ i x θ 1 }, and < α < min i S β i. Theorem 2.1. For any given initial value (1.4) with φ() >, equation (1.3) has a unique solution X(t) on [ τ, ), which is positive on [, ) a.s.; furthermore, for θ [1,1+2ρ), the solution X(t) of equation (1.3) has the properties: [ lim supe[x θ 1 t (t) M 1, and limsup t t t E X θ (s)ds M 2 /α. 3

4 Proof. Since all the coefficients of equation (1.3) are locally Lipschitz continuous on R +, then for any given initial value (1.4) with φ() >, there is a unique maximal local solution X(t) on t [ τ,λ e ), where Λ e is the explosion time. Step 1. We first show that X(t) is positive almost surely on [,Λ e ). For r t = i S, and t [,τ i, the initial problem (1.3)-(1.4) with φ() > reduces to the following linear stochastic differential equation, dx(t) =[ δ i X(t)+f i1 (t)dt+σ i X(t)dB(t), (2.5) where f i1 (t) = p i X(t τ i )e a ix(t τ i ) on t [,τ i. The solution of equation (2.5) is [ X(t) = e (δ i+σi 2/2)t+σ ib(t) φ()+ e (δ i+σi 2/2)s σ ib(s) f i1 (s)ds > a.s.,t [,τ i, and for t [τ i,2τ i, equation (1.3) reduces to the following initial problem, dx(t) =[ δ i X(t)+f i2 (t)dt+σ i X(t)dB(t), with X(τ i ) > a.s. (2.6) where f i2 (t) = p i X(t τ i )e a ix(t τ i ) > a.s. on t [τ i,2τ i. The solution of equation (2.6) is [ X(t) = e (δ i+σi 2/2)t+σ ib(t) (δi+σ2 X(τ i )+ e i /2)s σib(s) f i2 (s)ds > a.s.,t [τ i,2τ i, τ i Repeating this procedure, we can obtain that X(t) > a.s. on [,Λ e ). Step 2. Now, we prove that this solution X(t) to equation (1.3) is global existence, that is, the solution will not explode in finite time. We only need to show Λ e = a.s. Let k > be sufficiently large that max t [ τ, φ(t) < k, and for each integer k k define the stopping time Λ k = inf{t [,Λ e ) X(t) k)}. It is clear that Λ k is increasing as k. Let Λ = lim k Λ k, whence Λ Λ e a.s. In order to show Λ e =, we only need to prove that Λ = a.s. If it is false, then there is a pair of constants T > and ǫ (,1) such that P{Λ T} > ǫ, which yields that there exists an integer k 1 k such that P{Λ k T} ǫ for all k k 1. Let V(x,i) = x θ with θ [1,1+2ρ), then V x = θxθ 1, 2 V x 2 = θ(θ 1)x θ 2 and by Itô s formula we have dv(x(t)) =[ β i X θ (t)+θp i X θ 1 (t)x(t τ i )e a ix(t τ i ) dt+θσ i X θ (t)db(t) LV(X(t),i)dt+θσ i X θ db(t) (2.7) It follows from X(t) > a.s. on [,Λ e ) and β i > for all i S that there exists a positive constant M 1 such that LV(X(t),i) β i X θ (t)+γ i X θ 1 (t) M 1. 4

5 Thus, we have dv(x(t)) M 1 dt+θσ i X θ (t)db(t), For t [,τ i, integrating both sides of above inequality on [,Λ n t and taking expectation give then Λn t dv(x(t)) Λn t M 1 dt+ Λn t θσ i X θ (s)db(s), EV(Λ n t) V(φ())+ME(Λ n t) V(φ())+Mτ i, which yields Λ τ i a.s.. Repeating this procedure, we can obtain that Λ mτ i a.s. for any positive integer m, which implies Λ = a.s.. Step 3. Lastly, we give the estimation of the solution X(t). It follows from Itô formula that E[e t V(X(t)) V(φ())+ which yields limsup t E[V(X(t)) M 1. It follows from equation (2.7) that dv(t)+αx θ (t)dt M 2 dt+θσ i X θ (s)db(s) Integrating above inequality from to t gives dv(t)+α X θ (s)ds M 1 e s ds = V(φ())+M 1 (e t 1), M 2 dt+ Taking expectation on both sides of it gives [ αe X θ (s)ds M 2 t+v(φ()), which gives limsup t 1 t E[ Xθ (s)ds M 2 α. 3 Example θσ i X θ (s)db(s) In this section we will give an example to check the results obtained in section 2 by numerical simulation. Let S = {1,2,3}, Q = [ 1,4,6;2, 3,1;3,5, 8, δ = [2,1,4, p = [4,2,8, τ = [1,1,1, a = [.4,.2,.3, σ = [1.5,2,3, and φ(t) = 1, t [ 1,, then the unique stationary distribution of Markov chain r t is π = (.1845,.619,.2136) and one can see that δ 1 > σ 2 1/2, δ 2 = σ 2 2/2, and δ 3 < σ 2 3/2. It follows from Theorem 2.1 that the solution of equation (1.3) is global existence and positive almost surely on R +, see numerical simulation in figure 2 for details. Meanwhile, we see that ρ =.25, M for θ = 1, and M for θ = 1.4, then limsup t E[X(t).8829 and limsup t E[X 7/5 (t)

6 6 r =3 r 4 t 2 X(t) Time t x φ()=1 N*=log p/δ.6931 limsup t E[X(t) Time t Figure 2. The numerical simulation of the processes (X(t),r t ) with initial value φ(t) = 1 for t [ 1, and the step = 1e 4. 4 Conclusion In this paper, we consider a stochastic Nicholson s blowflies delay differential equation with regime switching. To the best of our knowledge it is the first time that regime switching is considered in Nicholson s blowflies model. By choosing suitable V function we prove that the solution of the stochastic Nicholson s blowflies model is globally existent, which is also positive almost surely on R +. The results presented in this paper extend some corresponding results in literature. There are still some interesting problems of equation (1.3): 1) Find conditions for the extinction of the species, that is lim t X(t) = ; 2) Find conditions for the weak persistent of the species, that is limsup t X(t) >. References [1 W.S.C. Gurney, S.P. Blythe, R.M. Nisbet, Nicholsons blowflies revisited, Nature 287 (198) [2 L. Berezansky, E. Braverman, L. Idels, Nicholsons blowflies differential equations revisited: Main results and open problems, Appl. Math. Model. 34 (21) [3 W.T. Wang, L.Q. Wang, W. Chen, Stochastic Nicholsons blowflies delayed differential equations, Appli. Math. Lett. 87 (219) [4 X. Mao, C. Yuan, Stochastic differential equations with Markovian switching. Imperial College Press, 26. [5 R.Z. Khasminskii, C. Zhu, G. Yin, Stability of regime-switching diffusions. Stoch. Process Appl. 117 (27) [6 G.G. Yin, C. Zhu, Hybrid Switching Diffusions, Properties and Applications, Springer New York, 29. 6