
Citation: Ting Kang, Yanyan Du, Ming Ye, Qimin Zhang. Approximation of invariant measure for a stochastic population model with Markov chain and diffusion in a polluted environment[J]. Mathematical Biosciences and Engineering, 2020, 17(6): 6702-6719. doi: 10.3934/mbe.2020349
[1] | ZongWang, Qimin Zhang, Xining Li . Markovian switching for near-optimal control of a stochastic SIV epidemic model. Mathematical Biosciences and Engineering, 2019, 16(3): 1348-1375. doi: 10.3934/mbe.2019066 |
[2] | Xuehui Ji, Sanling Yuan, Tonghua Zhang, Huaiping Zhu . Stochastic modeling of algal bloom dynamics with delayed nutrient recycling. Mathematical Biosciences and Engineering, 2019, 16(1): 1-24. doi: 10.3934/mbe.2019001 |
[3] | Dongmei Li, Tana Guo, Yajing Xu . The effects of impulsive toxicant input on a single-species population in a small polluted environment. Mathematical Biosciences and Engineering, 2019, 16(6): 8179-8194. doi: 10.3934/mbe.2019413 |
[4] | Linda J. S. Allen, P. van den Driessche . Stochastic epidemic models with a backward bifurcation. Mathematical Biosciences and Engineering, 2006, 3(3): 445-458. doi: 10.3934/mbe.2006.3.445 |
[5] | Zeyan Yue, Sheng Wang . Dynamics of a stochastic hybrid delay food chain model with jumps in an impulsive polluted environment. Mathematical Biosciences and Engineering, 2024, 21(1): 186-213. doi: 10.3934/mbe.2024009 |
[6] | An Ma, Shuting Lyu, Qimin Zhang . Stationary distribution and optimal control of a stochastic population model in a polluted environment. Mathematical Biosciences and Engineering, 2022, 19(11): 11260-11280. doi: 10.3934/mbe.2022525 |
[7] | Maoxiang Wang, Fenglan Hu, Meng Xu, Zhipeng Qiu . Keep, break and breakout in food chains with two and three species. Mathematical Biosciences and Engineering, 2021, 18(1): 817-836. doi: 10.3934/mbe.2021043 |
[8] | Dawid Czapla, Sander C. Hille, Katarzyna Horbacz, Hanna Wojewódka-Ściążko . Continuous dependence of an invariant measure on the jump rate of a piecewise-deterministic Markov process. Mathematical Biosciences and Engineering, 2020, 17(2): 1059-1073. doi: 10.3934/mbe.2020056 |
[9] | H.Thomas Banks, Shuhua Hu . Nonlinear stochastic Markov processes and modeling uncertainty in populations. Mathematical Biosciences and Engineering, 2012, 9(1): 1-25. doi: 10.3934/mbe.2012.9.1 |
[10] | Linard Hoessly, Carsten Wiuf . Fast reactions with non-interacting species in stochastic reaction networks. Mathematical Biosciences and Engineering, 2022, 19(3): 2720-2749. doi: 10.3934/mbe.2022124 |
With the rapid development of industry and agriculture, the environment pollution has caused many serious ecological problems (see [1,2,3]), such as the reduction of species diversity and the extinction of some species. Therefore, it motivates many scholars' interest to study dynamic behavior of population in a polluted environment by establishing mathematical models. The population model in a polluted environment was first proposed by Hallam et al.[4,5]. From then on, more investigations and discussions on the dynamic behavior of the deterministic population model can be found (see [6,7,8,9,10]). But in practical problems, population changes are affected not only by environmental noise but also by sudden changes of temperature and climate. Thus, several scholars have introduced random perturbations into population model to study dynamic behavior. For example, Liu and Wang [11] established the stochastic population model with impulsive toxicant input and obtained sufficient conditions on extinction, persistence, stability in the mean. Subsequently, Yu et al. [12] proved the existence of global positive solution for the stochastic population model with Allee effect under regime switching and established the threshold. In [13], Wei et al. proposed a stochastic population model with partial tolerance, discussed the conditions for population the extinction and proved the stationary distribution with ergodicity by constructing the Lyapunov function. Liu et al. [14] considered the significance of white noise and color noise on population persistence and extinction and studied stochastic population model with Markov switching. More research results on the persistence, extinction, and stability of random population models and others have been presented (see [15,16,17,18,19]). However, the above mentioned references didn't consider the invariant measure of population system with diffusion.
In fact, in the real world, the population and toxins in the ecology spread around the medium such as soil and water. In addition, we also know that the existence and uniqueness of invariant measure is one of the important properties for stochastic population model with Markov switching and diffusion. Nevertheless, if we introduce diffusion into stochastic population model, the corresponding Kolmogorov-Fokker-Planck (KFP) equation will become more complicated. Furthermore, the invariant measure of stochastic population model with Markov switching and diffusion is difficult to obtain. Therefore, it is of great significance to choose an effective numerical approximation method. To the best of our knowledge, the explicit Euler-Maruyama (EM) method has the advantages of easy calculation and small calculation amount. Motivated by [21,20], in the paper, we first develop a new stochastic population model with Markov switching and diffusion. Under suitable regularity assumptions, we discuss the existence and uniqueness of numerical invariant measure generated by the EM method. Subsequently, we prove that numerical invariant measure converges to the invariant measure of exact solution in the Wasserstein distance sense. In particular, the main contributions of the paper are as follows:
● We establish a novel stochastic population model with diffusion and Markov switching in a polluted environment. By using the Chebyshev's inequality, we obtain the existence and uniqueness of invariant measure for the model.
● Under local Lipschitz conditions, we study the approximation of numerical invariant measure generated by the EM method for the newly developed model.
The structure of this article is as follows: In Section2, we introduce some necessary preliminary knowledge results for the following analysis. In Section3, based on the Perron-Frobenius theorem, we study existence and uniqueness of invariant measure for the exact solution. In Section4, we mainly study the existence and uniqueness of numerical invariant measure for the EM scheme. In addition, we also prove that the numerical invariant measure of the EM scheme converges to invariant measure of exact solution. In Section5, the numerical example is given to verify our theoretical results. In Section6, we give the conclusions of this study.
In this paper, we introduce Markov switching and spatial diffusion into the model mentioned by Liu and Wang [15], and obtain the following model
{dX1(t,x)=[k1(t,x)ΔX1(t,x)+β(t,x,X2(t,x),Λt)X1(t,x)]dt−μ(t,x,X2(t,x),Λt)X1(t,x)dt+g(t,X1(t,x),Λt)dWt,in(0,T)×Γ,dX2(t,x)=[k2(t,x)ΔX2(t,x)+K(Λt)X3(t,x)−(l(Λt)+m(Λt))X2(t,x)]dt,in(0,T)×Γ,dX3(t,x)=[k3(t,x)ΔX3(t,x)−M(Λt)X3(t,x)+u(t,x)]dt,in(0,T)×Γ,X1(0,x)=s1(x),X2(0,x)=s2(x),X3(0,x)=s3(x),inx∈Γ,X1(t,x)=0,X2(t,x)=0,X3(t,x)=0,on(0,T]×∂Γ, | (2.1) |
where L:=(0,T)×Γ, Γ is a bounded domain in R3 with smooth boundary ∂Γ, t∈(0,T); X1(t,x) denotes the population density at the location x at time t. X2(t,x) is the concentration of toxicant in the organism at time t and in spatial position x. The concentration of toxicant in the environment at the location x at time t is described by X3(t,x). K(Λt) is the net organismal uptake rate of toxicant from the environment at time t. M(Λt) is the total loss rate of the toxicant from the environment. μ(t,x,X2(t,x),Λt) denotes the decreasing rate function of the population at time t and in spatial position x. ki>0,i=1,2,3 is the diffusion coefficient. β(t,x,X2(t,x),Λt) describes the intrinsic growth rate function of the population at time t and in spatial position x. u(t,x) denotes the exogenous total toxicant input into environment at time t and in spatial position x. l(Λt) is the net organismal excretion rate of toxicant and m(Λt) is depuration rate of toxicant due to metabolic process and other losses.
Throughout the paper, Let (V,‖⋅‖) and (H,|⋅|) be two separable Hilbert spaces, with norm denoted by ‖⋅‖ and |⋅|, respectively. V is viewed as a subspace of H with a continuous dense embedding. V⋐H represents the embedding is compact. V′ and H′ are the dual of V, H. We set H3:=H×H×H. Let (Ω,F,P) be a complete probability space with {Ft}0≤t≤T the natural filtration generated by the Brownian motion Wt, which means Ft=σ{Ws;0≤s≤t} augmented with all P-null sets of F0. To construct such a filtration, we denote by N the collection of P-null sets, i.e. N={B∈F:P(B)=0}. In the paper, C>0 represents different positive constants. Let Λt, t>0, be a right-continuous Markov chain on the probability space taking values in a finite state S={1,2,…,N} for some positive integer N<∞. The generator of {Λt}t>0 is specified by Q=(qij)N×N, such that for a sufficiently small Δ,
P(Λt+Δ=j|Λt=i)={qijΔ+o(Δ),i≠j,1+qiiΔ+o(Δ),i=j, | (2.2) |
where Δ>0, o(Δ) satisfies lim. Here q_{ij} is the transition rate from i to j satisfying q_{ii} = -\sum\limits_{i\neq j}q_{ij} . We assume that the Markov chain \{\Lambda_{t}\} defined on the probability space above is independent of the standard Brownian motion \{W_{t}\}_{t\geq 0} and the Q matrix is irreducible and conservative. Therefore, the Markov chain \{\Lambda_{t}\}_{t\geq 0} has a unique stationary distribution \pi: = (\pi_{1}, \ldots, \pi_{N}) which can be determined by solving the linear equation
\begin{equation*} \pi Q = \mathbf{0} \quad \quad {\rm{subject\; to}}\quad \sum\limits_{i = 1}^{N}\pi_{i} = 1 \; \rm{with}\; \pi_{i} \gt 0. \end{equation*} |
Let \mathcal{P}(H_{3}\times\mathbb{S}) stand for the family of all probability measures on H_{3}\times\mathbb{S} . For \xi = (\xi_{1}, \xi_{2}, \xi_{3})^{\ast} \in H_{3} , \xi\gg \mathbf{0} means each component \xi_{i} > 0 , i = 1, 2, 3 .
Next, let's give some necessary assumptions:
(\mathbb{H}1) Setting X_{k, t}^{s_{k}, i}: = X_{k}^{s_{k}, i}(t, x) , k = 1, 2, 3 , there exists a positive constant \rho_{i} such that for i\in \mathbb S , (t, x)\in \mathcal{L}
\begin{equation} \|g(t, X_{1, t}^{s_{1}, i}, i)-g(t, X_{1, t}^{\bar{s}_{1}, i}, i)\|^{2}\leq \rho_{i} |X_{1, t}^{s_{1}, i}-X_{1, t}^{\bar{s}_{1}, i}|^{2}, \end{equation} | (2.3) |
where s_{1} and \bar{s}_{1} are the different initial values of the first equation for system (2.1).
From (\mathbb{H}1) , for each i\in \mathbb S and X_{1, t}^{s_{1}, i}\in H , we can obtain that for (t, x)\in \mathcal{L}
\begin{equation} \|g(t, X_{1, t}^{s_{1}, i}, i)\|^{2}\leq C+\rho_{i}|X_{1, t}^{s_{1}, i}|^{2}, \end{equation} | (2.4) |
where C depends on the initial value of the function g(t, X_{1, t}^{s_{1}, i}, i) .
(\mathbb{H}2) For each i\in \mathbb S , there exist positive constants \bar{M} , \bar{\beta} and \bar{\mu} such that
\begin{equation} \begin{cases} \begin{split} &\bar{M}: = \max\limits_{i}\{M(\Lambda_{i})\}, \quad 0 \lt \bar{M} \lt \infty; \\ &0\leq\beta(t, x, X_{2, t}^{s_{2}, i}, \Lambda_{i})\leq\bar{\beta} \lt \infty;\\ &0\leq\mu_{0}\leq\mu(t, x, X_{2, t}^{s_{2}, i}, \Lambda_{i})\leq\bar{\mu} \lt \infty. \end{split} \end{cases} \end{equation} | (2.5) |
(\mathbb{H}3) u(t, x) is non-negative measurable in \mathcal{L} , there exists a positive constant \bar{u} such that
\begin{equation} 0\leq u_{0}\leq u(t, x)\leq\bar{u} \lt \infty. \end{equation} | (2.6) |
We replace ((X_{1, t}, X_{2, t}, X_{3, t}), \Lambda_{t}) with ((X_{1, t}^{s_{1}, i}, X_{2, t}^{s_{2}, i}, X_{3, t}^{s_{3}, i}), \Lambda_{t}^{i}) , especially the initial value
((X_{1}^{0}, X_{2}^{0}, X_{3}^{0}), \Lambda_{0}) = ((s_{1}, s_{2}, s_{3}), i). |
For any p\in (0, 1] , we set s: = (s_{1}, s_{2}, s_{3}) and define a metric on H_{3}\times\mathbb{S} as follows
\begin{equation*} \begin{split} d_{p}((s, i), (\bar{s}, i)): = \int_{H_{3}}\sum\limits_{k = 1}^{3}|s_{k}-\bar{s}_{k}|^{p}+I_{\{i\neq j\}}, \quad (s, i), (\bar{s}, i)\in H_{3}\times\mathbb{S}, \end{split} \end{equation*} |
where I_{A} denotes the indicator function of the set A , and \bar{s}: = (\bar{s}_{1}, \bar{s}_{2}, \bar{s}_{3}) is different initial value. For p\in (0, 1] , we define the Wassertein distance between \nu\in\mathcal{P}(H_{3}\times\mathbb{S}) and \nu'\in\mathcal{P}(H_{3}\times\mathbb{S}) by
W_{p}(\nu, \nu') = \inf\mathbb Ed_{p}(X_{k}, X_{k'}), |
where the infimum is taken over all pairs of random variables X_{k} , X_{k'} on H_{3} \times\mathbb{S} with respective laws \nu , \nu' . Let \mathbb{P}_{t}((s_{1}, s_{2}, s_{3}), i;\cdot) be the transition probability kernel of the pair ((X_{1, t}^{s_{1}, i}, X_{2, t}^{s_{2}, i}, X_{3, t}^{s_{3}, i}), \Lambda_{t}^{i}) , a time homogeneous Markov process (see [22]). Recall that \pi\in \mathcal{P}(H_{3}\times\mathbb{S}) is called an invariant measure of ((X_{1, t}^{s_{1}, i}, X_{2, t}^{s_{2}, i}, X_{3, t}^{s_{3}, i}), \Lambda_{t}^{i}) if
\begin{equation} \pi(A\times \{i\}) = \sum\limits_{j = 1}^{N}\int_{H_{3}} \mathbb{P}_{t}((s_{1}, s_{2}, s_{3}), j; A\times \{i\})\pi(d(s_{1}, s_{2}, s_{3}) \times \{j\}), t\geq 0, A \in H_{3}, i\in \mathbb{S} \end{equation} | (2.7) |
holds. For any p > 0 , let
\begin{equation} {\rm{diag}}(\rho)\triangleq {\rm{diag}}(\rho_{1}, \ldots, \rho_{N}), \quad Q_{p}\triangleq Q+\frac{p}{2}{\rm{diag}}(\rho), \quad \eta_{p}\triangleq-\max\limits_{\gamma}Re\gamma. \end{equation} | (2.8) |
where \rho_{i} is introduced in the assumptions and \gamma \in \rm{spec}(Q_{p}) , \rm{spec}(Q_{p}) denotes the spectrum of Q_{p} (i.e., the multi-set of its eigenvalues). Re\gamma is the real part of \gamma and {\rm{diag}}(\rho_{1}, \ldots, \rho_{N}) denotes the diagonal matrix whose diagonal entries are \rho_{1}, \ldots, \rho_{N} , respectively.
In this section, we mainly prove the existence and uniqueness of the invariant measure for the exact solution, under the assumption conditions (\mathbb{H}1) – (\mathbb{H}3) . Firstly, in order to prove the existence and uniqueness of the underlying invariant measure, we prepare the following lemma.
Lemma 3.1. (see [22]) Let N < \infty and assume further that
\begin{equation} \sum\limits_{i = 1}^{N}\mu_{i}\rho_{i} \lt 0, \end{equation} | (3.1) |
where \mu_{i} is the stationary distribution of Markov chain \{\Lambda_{t}\}_{t\geq 0} , and \rho_{i} is introduced in the assumption (\mathbb{H}1) . Then
(1) \eta_{p} > 0 if \max\limits_{i\in \mathbb{S}}\rho_{i}\leq0 ;
(2) \eta_{p} > 0 for p < \max\limits_{i\in \mathbb{S}, \rho_{i} > 0}\{-2q_{ii}/\rho_{i}\} if \max\limits_{i\in \mathbb{S}}\rho_{i} > 0 .
Remark 1: The system (2.1) is said to be attractive " in average " if Eq (3.1) holds. The Lemma 3.1 provides great convenience to study the existence and uniqueness of invariant measure for exact solution, i.e., the proof of Theorem 3.1.
Theorem 3.1. Let N < \infty and assume further that (\mathbb{H}1) – (\mathbb{H}3) hold with \max_{i\in S}\rho_{i} > 0 . Then the exact solution of system (2.1) admits a unique invariant measure \pi\in \mathcal{P}(H_{3}\times\mathbb{S}) .
Proof. The key point of proof is to divide the whole proof into two parts of existence and uniqueness.
(I) Existence of invariant measure. Let ((Y^{s_{1}, i}_{1, t}, Y^{s_{2}, i}_{2, t}, Y^{s_{3}, i}_{3, t}), \Lambda_{t}^{i}) be the exact solution of system (2.1) with ((s_{1}, s_{2}, s_{3}), i) as initial value, where ((s_{1}, s_{2}, s_{3}), i)\in H_{3}\times \mathbb{S} . A simple application of the Feynman-Kac formula show that let Q_{p, t} = e^{tQ_{p}} , where Q_{p} is given in Eq (2.8). Then, the spectral radius Ria (Q_{p, t}) (i.e., Ria (Q_{p, t}) = \sup_{\lambda\in \rm{spec}(Q_{p, t})}|\lambda| ) of Q_{p, t} equals to e^{-\eta_{p}t}. Since all coefficients of Q_{p, t} are positive, by the Perron-Frobenius theorem (see [23]) yields that -\eta_{p} is a simple eigenvalue of Q_{p} , all other eigenvalues have a strictly smaller real part. Note that the eigenvector of Q_{p, t} corresponding to e^{-\eta_{p}t} is also an eigenvector of Q_{p} corresponding to -\eta_{p} . According to Perron-Frobenius theorem, for Q_{p} it can be found that there is a positive eigenvector \xi^{(p)} = (\xi_{1}^{(p)}, \ldots, \xi_{N}^{(p)})\gg {\mathbf{0}} corresponding to the eigenvalue -\eta_{p} , and \xi^{(p)}\gg \mathbf{0} means that each component \xi_{i}^{(p)} > 0 . Let
\begin{equation} p_{0} = 1\wedge \min\limits_{i\in\mathbb{S}, \rho_{i} \gt 0}\{-2q_{ii}/\rho_{i}\}, \end{equation} | (3.2) |
where 1\wedge \min\limits_{i\in\mathbb{S}, \rho_{i} > 0}\{-2q_{ii}/\rho_{i}\}: = \min\{1, \min\limits_{i\in\mathbb{S}, \rho_{i} > 0}\{-2q_{ii}/\rho_{i}\}\} . Combined with Lemma 3.1, we can get
\begin{equation} Q_{p}\xi_{i}^{(p)} = -\eta_{p}\xi_{i}^{(p)}\ll \mathbf{0}. \end{equation} | (3.3) |
In order to investigate the existence and uniqueness of invariant measure for exact solution, we need to prove the boundedness of exact solution for system (2.1). In other words, we need to prove whether the following inequality holds.
\mathbb{E}(1+|Y_{1, t}^{s_{1}, i}|^{p}+|Y_{2, t}^{s_{2}, i}|^{p}+|Y_{3, t}^{s_{3}, i}|^{p})\leq C. |
First, using the It \hat{\rm{o}} 's formula (see [24], Theorem 1.45 of p.48), we can have
\begin{equation*} \begin{split} &e^{\eta_{p}t}\mathbb{E}((1+|Y_{1, t}^{s_{1}, i}|^{2}+|Y_{2, t}^{s_{2}, i}|^{2}+|Y_{3, t}^{s_{3}, i}|^{2})^{p/2}\xi_{\Lambda_{t}^{i}}^{(p)})\\ = &(1+|s_{1}|^{2}+|s_{2}|^{2}+|s_{3}|^{2})^{\frac{p}{2}}\xi_{i}^{p}+\mathbb{E}\int_{0}^{t}e^{\eta_{p}\epsilon}(1+|Y_{1, \epsilon}^{s_{1}, i}|^{2}+|Y_{2, \epsilon}^{s_{2}, i}|^{2}+|Y_{3, \epsilon}^{s_{3}, i}|^{2})^{\frac{p}{2}}\bigg\{\eta_{p}\xi_{\Lambda_{\epsilon}^{i}}^{(p)} \\ &+(Q\xi^{(p)})(\Lambda_{\epsilon}^{i})\bigg\}d\epsilon+\frac{p}{2}\mathbb{E}\int_{0}^{t}e^{\eta_{p}\epsilon}(1+|Y_{1, \epsilon}^{s_{1}, i}|^{2}+|Y_{2, \epsilon}^{s_{2}, i}|^{2}+|Y_{3, \epsilon}^{s_{3}, i}|^{2})^{\frac{p}{2}-1}\xi_{\Lambda_{\epsilon}^{i}}^{(p)}\bigg\{2\langle Y_{2, \epsilon}^{s_{2}, i}, K(\Lambda_{\epsilon}^{i})Y_{3, \epsilon}^{s_{3}, i}\\ &-(l(\Lambda_{\epsilon}^{i})+m(\Lambda_{\epsilon}^{i}))Y_{2, \epsilon}^{s_{2}, i}\rangle +2\langle Y_{1, \epsilon}^{s_{1}, i}, k_{1}\Delta Y_{1, \epsilon}^{s_{1}, i}+\beta Y_{1, \epsilon}^{s_{1}, i}-\mu Y_{1, \epsilon}^{s_{1}, i}\rangle+2\langle Y_{3, \epsilon}^{s_{3}, i}, -M(\Lambda_{\epsilon}^{i})Y_{3, \epsilon}^{s_{3}, i}\\ &+u_{\epsilon}+k_{3}\Delta Y_{3, \epsilon}^{s_{3}, i}\rangle+2\langle Y_{2, \epsilon}^{s_{2}, i}, k_{2}\Delta Y_{2, \epsilon}^{s_{2}, i}\rangle\bigg\}d\epsilon+\frac{p}{2}\int_{0}^{t}e^{\eta_{p}\epsilon}\mathbb{E}(1+|Y_{1, \epsilon}^{s_{1}, i}|^{2}+|Y_{2, \epsilon}^{s_{2}, i}|^{2}+|Y_{3, \epsilon}^{s_{3}, i}|^{2})^{\frac{p}{2}-1}\\ &\times\xi_{\Lambda_{\epsilon}^{i}}^{(p)}\bigg\{(p-2)(1+|Y_{1, \epsilon}^{s_{1}, i}|^{2}+|Y_{2, \epsilon}^{s_{2}, i}|^{2}+|Y_{3, \epsilon}^{s_{3}, i}|^{2})^{-1}\|Y_{1, \epsilon}^{s_{1}, i}*g(\epsilon, Y_{1, \epsilon}^{s_{1}, i}, \Lambda_{\epsilon}^{i})\|^{2}+\|g(\epsilon, Y_{1, \epsilon}^{s_{1}, i}, \Lambda_{\epsilon}^{i})\|^{2}\bigg\}d\epsilon. \end{split} \end{equation*} |
Using p(p-2)/2 < 0 , due to p\in (0, p_{0}), and combining with the following inequality,
\begin{equation} \begin{split} &\int_{0}^{t}\int_{\Omega}k(t, x)\Delta Y_{k, \epsilon} Y_{k, \epsilon}dxd\epsilon\\ & = -\int_{0}^{t}\int_{\Omega}k(t, x)\triangledown Y_{k, \epsilon}\triangledown Y_{k, \epsilon}dxd\epsilon\\ &\leq -k_{0}\int_{0}^{t}\|Y_{k, \epsilon}\|^{2}d\epsilon, \quad k = 1, 2, 3 \end{split} \end{equation} | (3.4) |
where 0\leq k_{0}\leq k(t, x) < \infty ( k_{0} is a constant). Further, we have
\begin{equation*} \begin{split} &e^{\eta_{p}t}\mathbb{E}((1+|Y_{1, t}^{s_{1}, i}|^{2}+|Y_{2, t}^{s_{2}, i}|^{2}+|Y_{3, t}^{s_{3}, i}|^{2})^{p/2}\xi_{\Lambda_{t}^{i}}^{(p)})\\ \leq&(1+|s_{1}|^{2}+|s_{2}|^{2}+|s_{3}|^{2})^{\frac{p}{2}}\xi_{i}^{(p)}+\frac{p}{2}\mathbb{E}\int_{0}^{t}e^{\eta_{p}\epsilon}(1+|Y_{1, \epsilon}^{s_{1}, i}|^{2}+|Y_{2, \epsilon}^{s_{2}, i}|^{2}+|Y_{3, \epsilon}^{s_{3}, i}|^{2})^{\frac{p}{2}-1}\bigg\{2\langle Y_{1, \epsilon}^{s_{1}, i}, \beta Y_{1, \epsilon}^{s_{1}, i}\\ &-\mu Y_{1, \epsilon}^{s_{1}, i}\rangle+\varepsilon_{1}(K(\Lambda_{\epsilon}^{i}))^{2}|Y_{2, \epsilon}^{s_{2}, i}|^{2}+\frac{1}{\varepsilon_{1}}|Y_{3, \epsilon}^{s_{3}, i}|^{2}+2(l(\Lambda_{\epsilon}^{i})+m(\Lambda_{\epsilon}^{i}))|Y_{2, \epsilon}^{s_{2}, i}|^{2}+\varepsilon_{2}|Y_{3, \epsilon}^{s_{3}, i}|^{2} +\frac{1}{\varepsilon_{2}}|u_{\epsilon}|^{2}\\ &+2\bar{M}|Y_{3, \epsilon}^{s_{3}, i}|^{2}\bigg\}\xi_{\Lambda_{\epsilon}^{i}}^{(p)}d\epsilon+\frac{p}{2}\int_{0}^{t}e^{\eta_{p}\epsilon}\mathbb{E}(1+|Y_{1, \epsilon}^{s_{1}, i}|^{2}+|Y_{2, \epsilon}^{s_{2}, i}|^{2}+|Y_{3, \epsilon}^{s_{3}, i}|^{2})^{\frac{p}{2}-1}\|g(s, Y_{1, \epsilon}^{s_{1}, i}, \Lambda_{\epsilon}^{i})\|^{2}\xi_{\Lambda_{\epsilon}^{i}}^{(p)}d\epsilon\\ &+\mathbb{E}\int_{0}^{t}e^{\eta_{p}\epsilon}(1+|Y_{1, \epsilon}^{s_{1}, i}|^{2}+|Y_{2, \epsilon}^{s_{2}, i}|^{2}+|Y_{3, \epsilon}^{s_{3}, i}|^{2})^{\frac{p}{2}}\bigg\{\eta_{p}\xi_{\Lambda_{\epsilon}^{i}}^{(p)}+(Q\xi^{(P)})(\Lambda_{\epsilon}^{i})\bigg\}d\epsilon. \end{split} \end{equation*} |
Therefore, based on assumption conditions (\mathbb{H}1) – (\mathbb{H}3) and the inequality 2ab\leq \varepsilon a^{2}+\frac{1}{\varepsilon}b^{2} , \varepsilon > 0 we can obtain
\begin{equation*} \begin{split} &e^{\eta_{p}t}\mathbb{E}((1+|Y_{1, t}^{s_{1}, i}|^{2}+|Y_{2, t}^{s_{2}, i}|^{2}+|Y_{3, t}^{s_{3}, i}|^{2})^{p/2}\xi_{\Lambda_{t}^{i}}^{(p)})\\ \leq& (1+|s_{1}|^{2}+|s_{2}|^{2}+|s_{3}|^{2})^{\frac{p}{2}}\xi_{i}^{(p)}+\frac{p}{2}\mathbb{E}\int_{0}^{t}e^{\eta_{p}\epsilon}(1+|Y_{1, \epsilon}^{s_{1}, i}|^{2}+|Y_{2, \epsilon}^{s_{2}, i}|^{2}+|Y_{3, \epsilon}^{s_{3}, i}|^{2})^{\frac{p}{2}-1}\bigg\{c+[2(\bar{\beta}\\ &-\mu_{0})+\rho_{\Lambda_{\epsilon}^{i}}]|Y_{1, \epsilon}^{s_{1}, i}|^{2}+[2(l(\Lambda_{\epsilon}^{i})+m(\Lambda_{\epsilon}^{i}))+\varepsilon_{1}\bar{K}^{2}]|Y_{2, \epsilon}^{s_{2}, i}|^{2}+[2\bar{M}+\frac{1}{\varepsilon_{2}}+\frac{1}{\varepsilon_{1}}]|Y_{3, \epsilon}^{s_{3}, i}|^{2}+\varepsilon_{2}|u_{\epsilon}|^{2}\bigg\}d\epsilon\\ &+\mathbb{E}\int_{0}^{t}e^{\eta_{p}\epsilon}(1+|Y_{1, \epsilon}^{s_{1}, i}|^{2}+|Y_{2, \epsilon}^{s_{2}, i}|^{2}+|Y_{3, \epsilon}^{s_{3}, i}|^{2})^{\frac{p}{2}}\bigg\{\eta_{p}\xi_{\Lambda_{\epsilon}^{i}}^{(p)}+(Q\xi^{(P)})(\Lambda_{\epsilon}^{i})\bigg\}d\epsilon, \end{split} \end{equation*} |
where \bar{K}: = \max\limits_{i}\{K(\Lambda_{i})\} , for all i\in\mathbb{S} , 0 < \bar{K} < \infty . Then, setting C_{1}: = 2(\bar{\beta}-\mu_{0})+\rho_{0} , \rho_{0}: = \max\limits_{i\in \mathbb{S}}|\rho_{\Lambda_{\epsilon}^{i}}| C_{2}: = \max\limits_{i\in \mathbb{S}}2(l(\Lambda_{\epsilon}^{i})+m(\Lambda_{\epsilon}^{i}))+\varepsilon_{1}\bar{K}^{2} and C_{3}: = 2\bar{M}+\frac{1}{\varepsilon_{2}}+\frac{1}{\varepsilon_{1}} , C_{4}: = \varepsilon_{2}+c are different constants and using the inequality
\begin{equation} \begin{split} (|a|+|b|)^{r}\leq 2^{r-1}(|a|^{r}+|b|^{r}), \quad r\geq1, \quad \forall a, b\in R, \end{split} \end{equation} | (3.5) |
we can further estimate
\begin{equation*} \begin{split} &e^{\eta_{p}t}\mathbb{E}((1+|Y_{1, t}^{s_{1}, i}|^{2}+|Y_{2, t}^{s_{2}, i}|^{2}+|Y_{3, t}^{s_{3}, i}|^{2})^{p/2}\xi_{\Lambda_{t}^{i}}^{(p)})\\ \leq&c(1+|s_{1}|^{p}+|s_{2}|^{p}+|s_{3}|^{p})+\mathbb{E}\int_{0}^{t}e^{\eta_{p}\epsilon}(1+|Y_{1, \epsilon}^{s_{1}, i}|^{2}+|Y_{2, \epsilon}^{s_{2}, i}|^{2}+|Y_{3, \epsilon}^{s_{3}, i}|^{2})^{\frac{p}{2}}\bigg\{\eta_{p}\xi_{\Lambda_{s}^{i}}^{(p)}+(Q\xi^{(P)})(\Lambda_{\epsilon}^{i})\bigg\}d\epsilon\\ &+\frac{p}{2}\mathbb{E}\int_{0}^{t}e^{\eta_{p}\epsilon}(1+|Y_{1, \epsilon}^{s_{1}, i}|^{2}+|Y_{2, \epsilon}^{s_{2}, i}|^{2}+|Y_{3, \epsilon}^{s_{3}, i}|^{2})^{\frac{p}{2}}\bigg\{\frac{C_{1}|Y_{1, \epsilon}^{s_{1}, i}|^{2}+C_{2}|Y_{2, \epsilon}^{s_{2}, i}|^{2}}{(1+|Y_{1, \epsilon}^{s_{1}, i}|^{2}+|Y_{2, \epsilon}^{s_{2}, i}|^{2}+|Y_{3, \epsilon}^{s_{3}, i}|^{2})}\bigg\}d\epsilon\\ &+\frac{p}{2}\mathbb{E}\int_{0}^{t}e^{\eta_{p}\epsilon}(1+|Y_{1, \epsilon}^{s_{1}, i}|^{2}+|Y_{2, \epsilon}^{s_{2}, i}|^{2}+|Y_{3, \epsilon}^{s_{3}, i}|^{2})^{\frac{p}{2}}\bigg\{\frac{C_{3}|Y_{3, \epsilon}^{s_{3}, i}|^{2}+C_{4}|u_{\epsilon}|^{2}}{(1+|Y_{1, \epsilon}^{s_{1}, i}|^{2}+|Y_{2, \epsilon}^{s_{2}, i}|^{2}+|Y_{3, \epsilon}^{s_{3}, i}|^{2})}\bigg\}d\epsilon. \end{split} \end{equation*} |
Finally, by the Gronwall's lemma, we can get the result
\begin{equation} \begin{split} e^{\eta_{p}t}\mathbb{E}((1+|Y_{1, t}^{s_{1}, i}|^{2}+|Y_{2, t}^{s_{2}, i}|^{2}+|Y_{3, t}^{s_{3}, i}|^{2})^{p/2}\xi_{\Lambda_{t}^{i}}^{(p)}) \leq Ce^{CT}, \end{split} \end{equation} | (3.6) |
and further estimates can be obtained as follows
\begin{equation} \sup\limits_{t\geq 0}\mathbb{E}((|Y_{1, t}^{s_{1}, i}|^{p}+|Y_{2, t}^{s_{2}, i}|^{p}+|Y_{3, t}^{s_{3}, i}|^{p})\leq C. \end{equation} | (3.7) |
For \forall t > 0 , we can define a probability measure
\chi_{t}(A) = \frac{1}{t}\int_{0}^{t}\mathbb{P}_{\epsilon}(s, i;A)d\epsilon, \quad A\in (H_{3}\times\mathbb{S}). |
Then, let Y_{t}^{s, i}: = (Y_{1, t}^{s_{1}, i}, Y_{2, t}^{s_{2}, i}, Y_{3, t}^{s_{3}, i}) , for any \varepsilon > 0 , by Eq (3.7) and Chebyshev's inequality, there exists an r > 0 sufficiently large such that
\begin{equation} \chi_{t}(K_{r}\times\mathbb{S} ) = \frac{1}{t}\int_{0}^{t}\mathbb{P}_{\epsilon}(s, i;K_{r}\times\mathbb{S})d\epsilon\geq 1-\frac{\sup\limits_{t\geq0}(E|Y_{t}^{s, i}|^{p})}{r^{p}}\geq1-\varepsilon. \end{equation} | (3.8) |
Hence, \chi_{t} is tight since the compact embedding V\Subset H , then K_{r} = \{s\in H_{3}; |s|\leq r\} is a compact subset of H_{3} (see [25], Definition 2, p.27) for each i\in \mathbb{S} . Combined with the Fellerian property of transition seimgroup for \mathbb{P}_{t}(s, i;\cdot) and according to Krylov-Bogoliubov theorem (see [26]), ((Y_{1, t}^{s_{1}, i}, Y_{2, t}^{s_{2}, i}, Y_{3, t}^{s_{3}, i}), \Lambda_{t}^{i}) has an invariant measure (see [27]). Next, we prove the uniqueness of the invariant measure for ((Y_{1, t}^{s_{1}, i}, Y_{2, t}^{s_{2}, i}, Y_{3, t}^{s_{3}, i}), \Lambda_{t}^{i}) .
(II) Uniqueness of invariant measure. First, let ((Y_{1, t}^{s_{1}, i}, Y_{2, t}^{s_{2}, i}, Y_{3, t}^{s_{3}, i}), \Lambda_{t}^{i}) and ((Y_{1, t}^{\bar{s}_{1}, i}, Y_{2, t}^{\bar{s}_{2}, i}, Y_{3, t}^{\bar{s}_{3}, i}), \Lambda_{t}^{i}) be the solutions of the system (2.1) satisfying the initial values ((s_{1}, s_{2}, s_{3}), i) and ((\bar{s}_{1}, \bar{s}_{2}, \bar{s}_{3}), i) , respectively. Under assumption conditions (\mathbb{H}1) – (\mathbb{H}3) , we take \forall\varepsilon \in (0, 1) and use It \hat{\rm{o}} 's formula, combined with Eq (3.4), we have
\begin{equation*} \begin{split} &e^{\eta_{p}t}\mathbb{E}((\varepsilon+|Y_{1, t}^{s_{1}, i}-Y_{1, t}^{\bar{s}_{1}, i}|^{2}+|Y_{2, t}^{s_{2}, i}-Y_{2, t}^{\bar{s}_{2}, i}|^{2}+|Y_{3, t}^{s_{3}, i}-Y_{3, t}^{\bar{s}_{3}, i}|^{2})^{p/2}\xi^{(p)}_{\Lambda_{t}^{i}})\\ \leq &(\varepsilon+|s_{1}-\bar{s}_{1}|^{2}+|s_{2}-\bar{s}_{2}|^{2}|+|s_{3}-\bar{s}_{3}|^{2}|)^{p/2}\xi_{i}^{(p)}\\ &+\frac{p}{2}\mathbb{E}\int_{0}^{t}e^{\eta_{p}\epsilon}(\varepsilon+|Y_{1, \epsilon}^{s_{1}, i}-Y_{1, \epsilon}^{\bar{s}_{1}, i}|^{2}+|Y_{2, \epsilon}^{s_{2}, i}-Y_{2, \epsilon}^{\bar{s}_{2}, i}|^{2}+|Y_{3, \epsilon}^{s_{3}, i}-Y_{3, \epsilon}^{\bar{s}_{3}, i}|^{2})^{p/2-1}\xi^{(p)}_{\Lambda_{\epsilon}^{i}}\\ &\times\bigg\{[2(\bar{\beta}-\mu_{0})+\rho_{\Lambda_{\epsilon}^{i}}]|Y_{1, \epsilon}^{s_{1}, i}-Y_{1, \epsilon}^{\bar{s}_{1}, i}|^{2}+\varepsilon_{1}\bar{K}^{2}|Y_{2, \epsilon}^{s_{2}, i}-Y_{2, \epsilon}^{\bar{s}_{2}, i}|^{2}+[\frac{1}{\varepsilon_{1}}+\frac{1}{\varepsilon_{2}}+2\bar{M}]|Y_{3, \epsilon}^{s_{3}, i}-Y_{3, \epsilon}^{\bar{s}_{3}, i}|^{2}\\ &+2(l(\Lambda_{\epsilon}^{i})+m(\Lambda_{\epsilon}^{i}))|Y_{2, \epsilon}^{s_{2}, i}-Y_{2, \epsilon}^{\bar{s}_{2}, i}|^{2}+\varepsilon_{2}|u_{\epsilon}^{s_{3}, i}-u_{\epsilon}^{{\bar{s}_{3}, i}}|^{2}\bigg\}d\epsilon\\ \leq &(\varepsilon+|s_{1}-\bar{s}_{1}|^{2}+|s_{2}-\bar{s}_{2}|^{2}|+|s_{3}-\bar{s}_{3}|^{2})^{p/2}\xi_{i}^{(p)}\\ &+\frac{p}{2}\mathbb{E}\int_{0}^{t}e^{\eta_{p}\epsilon}(\varepsilon+|Y_{1, \epsilon}^{s_{1}, i}-Y_{1, \epsilon}^{\bar{s}_{1}, i}|^{2}+|Y_{2, \epsilon}^{s_{2}, i}-Y_{2, \epsilon}^{\bar{s}_{2}, i}|^{2}+|Y_{3, \epsilon}^{s_{3}, i}-Y_{3, \epsilon}^{\bar{s}_{3}, i}|^{2})^{p/2}\xi^{(p)}_{\Lambda_{\epsilon}^{i}}\\ &\times\bigg\{\frac{C_{1}|Y_{1, \epsilon}^{s_{1}, i}-Y_{1, \epsilon}^{\bar{s}_{1}, i}|^{2}+C_{2}|Y_{2, \epsilon}^{s_{2}, i}-Y_{2, \epsilon}^{\bar{s}_{2}, i}|^{2}+C_{3}|Y_{3, \epsilon}^{s_{3}, i}-Y_{3, \epsilon}^{\bar{s}_{3}, i}|^{2}+\varepsilon_{2}|u_{\epsilon}^{s_{3}, i}-u_{\epsilon}^{{\bar{s}_{3}, i}}|^{2}}{\varepsilon+|Y_{1, \epsilon}^{s_{1}, i}-Y_{1, \epsilon}^{\bar{s}_{1}, i}|^{2}+|X_{2, \epsilon}^{s_{2}, i}-X_{2, \epsilon}^{\bar{s}_{2}, i}|^{2}+|X_{3, \epsilon}^{s_{3}, i}-X_{3, \epsilon}^{\bar{s}_{3}, i}|^{2}}\bigg\}d\epsilon, \\ \end{split} \end{equation*} |
where C_{i} , i = 1, 2, 3 have been explained before and \rho_{\Lambda_{\epsilon}^{i}} is introduced in the assumption (\mathbb{H}1) . In addition, using the result of Eqs (3.5) and (3.6), we can get
\begin{equation} \begin{split} &e^{\eta_{p}t}\mathbb{E}((\varepsilon+|Y_{1, t}^{s_{1}, i}-Y_{1, t}^{\bar{s}_{1}, i}|^{2}+|Y_{2, t}^{s_{2}, i}-Y_{2, t}^{\bar{s}_{2}, i}|^{2}+|Y_{3, t}^{s_{3}, i}-X_{3, t}^{\bar{s}_{3}, i}|^{2})^{p/2}\xi^{(p)}_{\Lambda_{t}^{i}})\\ \leq & (\varepsilon+|s_{1}-\bar{s}_{1}|^{2}+|s_{2}-\bar{s}_{2}|^{2}+|s_{3}-\bar{s}_{3}|^{2})^{p/2}\xi_{i}^{(p)}\\ &+\frac{p}{2}C \mathbb{E}\int_{0}^{t}e^{\eta_{p}\epsilon}(\varepsilon+|Y_{1, \epsilon}^{s_{1}, i}-Y_{1, \epsilon}^{\bar{s}_{1}, i}|^{2}+|Y_{2, \epsilon}^{s_{2}, i}-X_{2, \epsilon}^{\bar{s}_{2}, i}|^{2}+|X_{3, \epsilon}^{s_{3}, i}-X_{3, \epsilon}^{\bar{s}_{3}, i}|^{2})^{p/2}\xi^{(p)}_{\Lambda_{\epsilon}^{i}}\\ &\times\bigg\{1-\varepsilon(\varepsilon+|Y_{1, \epsilon}^{s_{1}, i}-Y_{1, \epsilon}^{\bar{s}_{1}, i}|^{2}+|Y_{2, \epsilon}^{s_{2}, i}-Y_{2, \epsilon}^{\bar{s}_{2}, i}|^{2}+|X_{3, \epsilon}^{s_{3}, i}-X_{3, \epsilon}^{\bar{s}_{3}, i}|^{2})^{-1}\bigg\}d\epsilon\\ \leq &(\varepsilon+|s_{1}-\bar{s}_{1}|^{2}+|s_{2}-\bar{s}_{2}|^{2}|+|s_{3}-\bar{s}_{3}|^{2})^{p/2}\xi_{i}^{(p)}+C\varepsilon^{p/2}e^{\eta_{p}t}, \end{split} \end{equation} | (3.9) |
when \varepsilon \rightarrow 0 , we can get the following result
\begin{equation} \begin{split} &\mathbb{E}(|Y_{1, t}^{s_{1}, i}-Y_{1, t}^{\bar{s}_{1}, i}|^{p}+|Y_{2, t}^{s_{2}, i}-Y_{2, t}^{\bar{s}_{2}, i}|^{p}+|Y_{3, t}^{s_{3}, i}-Y_{3, t}^{\bar{s}_{3}, i}|^{p})\\ \leq& C(|s_{1}-\bar{s}_{1}|^{p}+|s_{2}-\bar{s}_{2}|^{p}+|s_{3}-\bar{s}_{3}|^{p})e^{-\eta_{p}t}. \end{split} \end{equation} | (3.10) |
Define the stopping time
\tau = \inf\{t\geq 0:\Lambda_{t}^{i} = \Lambda_{t}^{j}\}. |
According to the definition of \mathbb{S} and irreducibility of Q , there exists \theta > 0 such that
\begin{equation} \begin{split} \mathbb{P}(\tau \gt t)\leq e^{-\theta t}, \quad \quad t \gt 0. \end{split} \end{equation} | (3.11) |
Due to p\in (0, p_{0}) , and choose q > 1 such that 0 < pq < p_{0} , where p_{0} is introduced in Eq (3.2). Using H \ddot{o} lder's inequality, we can have
\begin{equation} \begin{split} &\mathbb{E}(|Y_{1, t}^{s_{1}, i}-Y_{1, t}^{\bar{s}_{1}, j}|^{p}+|Y_{2, t}^{s_{2}, i}-Y_{2, t}^{\bar{s}_{2}, j}|^{p}+|Y_{3, t}^{s_{3}, i}-Y_{3, t}^{\bar{s}_{3}, j}|^{p})\\ = &\mathbb{E}(|Y_{1, t}^{s_{1}, i}-Y_{1, t}^{\bar{s}_{1}, j}|^{p}{\mathbf 1}_{\{\tau \gt t/2\}})+\mathbb{E}(|Y_{1, t}^{s_{1}, i}-Y_{1, t}^{\bar{s}_{1}, j}|^{p}{\mathbf1}_{\{\tau\leq t/2\}})+\mathbb{E}(|Y_{2, t}^{s_{2}, i}-Y_{2, t}^{\bar{s}_{2}, j}|^{p}{\mathbf 1}_{\{\tau \gt t/2\}})\\ &+\mathbb{E}(|Y_{2, t}^{s_{2}, i}-Y_{2, t}^{\bar{s}_{2}, j}|^{p}{\mathbf 1}_{\{\tau\leq t/2\}})+\mathbb{E}(|Y_{3, t}^{s_{3}, i}-Y_{3, t}^{\bar{s}_{3}, j}|^{p}{\mathbf 1}_{\{\tau \gt t/2\}})+\mathbb{E}(|Y_{3, t}^{s_{3}, i}-Y_{3, t}^{\bar{s}_{3}, j}|^{p}{\mathbf1}_{\{\tau\leq t/2\}})\\ \leq&(\mathbb{E}|Y_{1, t}^{s_{1}, i}-Y_{1, t}^{\bar{s}_{1}, j}|^{pq}{\mathbf 1}_{\{\tau \gt t/2\}})^{1/q}(\mathbb{P}(\tau \gt t/2))^{1/p}+\mathbb{E}({\mathbf1}_{\{\tau\leq t/2\}}\mathbb{E}|Y_{1, t-\tau}^{Y_{1, \tau}^{s_{1}, i}, \Lambda_{\tau}^{i}}-Y_{1, t-\tau}^{Y_{1, \tau}^{\bar{s}_{1}, j}, \Lambda_{\tau}^{j}}|^{p})\\ &+(\mathbb{E}|Y_{2, t}^{s_{2}, i}-Y_{2, t}^{\bar{s}_{2}, j}|^{pq}{\mathbf 1}_{\{\tau \gt t/2\}})^{1/q}(\mathbb{P}(\tau \gt t/2))^{1/p}+\mathbb{E}({\mathbf1}_{\{\tau\leq t/2\}}\mathbb{E}|Y_{2, t-\tau}^{Y_{2, \tau}^{s_{2}, i}, \Lambda_{\tau}^{i}}-Y_{2, t-\tau}^{Y_{2, \tau}^{\bar{s}_{2}, j}, \Lambda_{\tau}^{j}}|^{p})\\ &+(\mathbb{E}|Y_{3, t}^{s_{3}, i}-Y_{3, t}^{\bar{s}_{1}, j}|^{pq}{\mathbf 1}_{\{\tau \gt t/2\}})^{1/q}(\mathbb{P}(\tau \gt t/2))^{1/p}+\mathbb{E}({\mathbf1}_{\{\tau\leq t/2\}}\mathbb{E}|Y_{3, t-\tau}^{Y_{3, \tau}^{s_{3}, i}, \Lambda_{\tau}^{i}}-Y_{3, t-\tau}^{Y_{1, \tau}^{\bar{s}_{3}, j}, \Lambda_{\tau}^{j}}|^{p}). \end{split} \end{equation} | (3.12) |
Applying the result of Eq (3.11), we further obtain
\begin{equation} \begin{split} &\mathbb{E}(|Y_{1, t}^{s_{1}, i}-Y_{1, t}^{\bar{s}_{1}, j}|^{p}+|Y_{2, t}^{s_{2}, i}-Y_{2, t}^{\bar{s}_{2}, j}|^{p}+|Y_{3, t}^{s_{3}, i}-Y_{3, t}^{\bar{s}_{3}, j}|^{p})\\ \leq&e^{-\frac{q-1}{2q}\theta t}(\mathbb{E}|Y_{1, t}^{s_{1}, i}-Y_{1, t}^{\bar{s}_{1}, j}|^{pq})^{\frac{1}{q}}+C\mathbb{E}({\mathbf1}_{\{\tau\leq t/2\}}e^{-\eta_{p}(t-\tau)}\mathbb{E}|Y_{1, \tau}^{s_{1}, i}-Y_{1, \tau}^{\bar{s}_{1}, j}|^{p})\\ &+e^{-\frac{q-1}{2q}\theta t}(\mathbb{E}|Y_{2, t}^{s_{2}, i}-Y_{2, t}^{\bar{s}_{2}, j}|^{pq})^{\frac{1}{q}}+C\mathbb{E}({\mathbf1}_{\{\tau\leq t/2\}}e^{-\eta_{p}(t-\tau)}\mathbb{E}|Y_{2, \tau}^{s_{2}, i}-Y_{2, \tau}^{\bar{s}_{2}, j}|^{p})\\ &+e^{-\frac{q-1}{2q}\theta t}(\mathbb{E}|Y_{3, t}^{s_{3}, i}-Y_{3, t}^{\bar{s}_{3}, j}|^{pq})^{\frac{1}{q}}+C\mathbb{E}({\mathbf1}_{\{\tau\leq t/2\}}e^{-\eta_{p}(t-\tau)}\mathbb{E}|Y_{3, \tau}^{s_{3}, i}-X_{3, \tau}^{\bar{s}_{3}, j}|^{p})\\ \leq& e^{-\frac{q-1}{2q}\theta t}(\mathbb{E}|Y_{1, t}^{s_{1}, i}-Y_{1, t}^{\bar{s}_{1}, j}|^{pq})^{\frac{1}{q}}+Ce^{-\frac{\eta_{p}}{2}t}\mathbb{E}|Y_{1, \tau}^{s_{1}, i}-Y_{1, \tau}^{\bar{s_{1}}, j}|^{p}+e^{-\frac{q-1}{2q}\theta t}(\mathbb{E}|Y_{2, t}^{s_{2}, i}-Y_{2, t}^{\bar{s}_{2}, j}|^{pq})^{\frac{1}{q}}\\ &+Ce^{-\frac{\eta_{p}}{2}t}\mathbb{E}|Y_{2, \tau}^{s_{2}, i}-Y_{2, \tau}^{\bar{s_{2}}, j}|^{p}+e^{-\frac{q-1}{2q}\theta t}(\mathbb{E}|Y_{3, t}^{s_{3}, i}-Y_{3, t}^{\bar{s}_{3}, j}|^{pq})^{1/q}+Ce^{-\frac{\eta_{p}}{2}t}\mathbb{E}|Y_{3, \tau}^{s_{3}, i}-Y_{3, \tau}^{\bar{s_{3}}, j}|^{p}\\ \leq& C(1+|s_{1}|^{p}+|\bar{s}_{1}|^{p}+|s_{2}|^{p}+|\bar{s}_{2}|^{p}+|s_{3}|^{p}+|\bar{s}_{3}|^{p})e^{-\sigma t}, \end{split} \end{equation} | (3.13) |
where \sigma: = \frac{(q-1)\theta }{2q}\wedge \frac{\eta_{p}}{2} , and in the last step, it follows from Eqs (3.7) and (3.10) such that
\sup\limits_{t\geq 0}\mathbb{E}(|Y_{1, t}^{s_{1}, i}|^{pq}+|Y_{2, t}^{s_{2}, i}|^{pq}+|Y_{3, t}^{s_{3}, i}|^{pq})\leq C, |
and
\sup\limits_{t\geq 0}\mathbb{E}(|Y_{1, t}^{\bar{s}_{1}, j}|^{pq}+|Y_{2, t}^{\bar{s}_{2}, j}|^{pq}+|Y_{3, t}^{\bar{s}_{3}, j}|^{pq})\leq C. |
Thus, we also have assertion
\lim\limits_{t\rightarrow \infty}\mathbb{E}(|Y_{1, t}^{s_{1}, i}-Y_{1, t}^{\bar{s}_{1}, j}|^{p}+|Y_{2, t}^{s_{2}, i}-Y_{2, t}^{\bar{s}_{2}, j}|^{p}+|Y_{3, t}^{s_{3}, i}-Y_{3, t}^{\bar{s}_{3}, j}|^{p}) = 0. |
Then, according to Eq (3.11), we can get
\begin{equation} \begin{split} \mathbb{P}(\Lambda_{t}^{i}\neq\Lambda_{t}^{j}) = \mathbb{P}(\tau \gt t)\leq e^{-\theta t} \quad t \gt 0. \end{split} \end{equation} | (3.14) |
Next, according to Eqs (3.14) and (3.13) that
\begin{equation} \begin{split} &W_{p}(\delta_{((s_{1}, s_{2}, s_{2}), i)}\mathbb{P}_{t}, \delta_{((\bar{s}_{1}, \bar{s}_{2}, \bar{s}_{3}), j)}\mathbb{P}_{t})\\ &\leq\mathbb{E}(|Y_{1, t}^{s_{1}, i}-Y_{1, t}^{\bar{s}_{1}, j}|^{p}+|Y_{2, t}^{s_{2}, i}-Y_{2, t}^{\bar{s}_{2}, j}|^{p}+|Y_{3, t}^{s_{3}, i}-Y_{3, t}^{\bar{s}_{3}, j}|^{p})+\mathbb{P}(\Lambda_{t}^{i}\neq\Lambda_{t}^{j})\\ &\leq C(1+|s_{1}|^{p}+|\bar{s}_{1}|^{p}+|s_{2}|^{p}+|\bar{s}_{2}|^{p}+|s_{3}|^{p}+|\bar{s}_{3}|^{p})e^{-\sigma t}+e^{-\theta t}\\ &\leq Ce^{-\sigma^{*} t}, \end{split} \end{equation} | (3.15) |
where \sigma^{*}: = \sigma\wedge\theta . Assume \pi , \nu\in \mathcal{P}(H_{3}\times\mathbb{S}) are invariant measures of ((Y_{1, t}^{s_{1}, i}, Y_{2, t}^{s_{2}, i}, Y_{3, t}^{s_{3}, i}), \Lambda_{t}^{i}) , it follows from Eq (3.15) that
\begin{equation*} \label{kop345} \begin{split} W_{p}&(\pi, \nu) = W_{p}(\pi \mathbb{P}_{t}, \nu \mathbb{P}_{t})\\ &\leq\sum\limits_{i, j = 1}^{N}\int_{H_{3}\times\mathbb{S}}\int_{H_{3}\times\mathbb{S}}\pi(d(s_{1}, s_{2}, s_{3})\times \{i\})\nu(d(\bar{s}_{1}, \bar{s}_{2}, \bar{s}_{3})\times\{j\})W_{p}(\delta_{((s_{1}, s_{2}, s_{3}), i)}P_{t}, \delta_{((\bar{s}_{1}, \bar{s}_{2}, \bar{s}_{3}), j)}P_{t}). \end{split} \end{equation*} |
When t\rightarrow \infty , we find W_{p}(\pi, \nu) \rightarrow 0 . Hence, uniqueness of invariant measure follows immediately. The proof of Theorem 3.1 has been completed.
In the following section, we will investigate existence and uniqueness of numerical invariant measure and prove the convergence of numerical invariant measure.
In this section, we mainly discuss existence and uniqueness of numerical invariant measure for system (2.1) under the assumption conditions (\mathbb{H}1) – (\mathbb{H}3) . In order to facilitate the discussion, we consider the numerical solution in the discrete-time for system (2.1). For a given step size \delta \in (0, 1) , we define the discrete-time Euler-Maruyama (EM) scheme associated with model (2.1) as follows
\begin{equation} \begin{cases} \begin{split} &\bar{X}_{1, (n+1)\delta}^{s_{1}, i} = \bar{X}_{1, n\delta}^{s_{1}, i}+[k_{1}(n\delta, x)\Delta \bar{X}_{1, n\delta}^{s_{1}, i}+\beta(n\delta, x, \bar{X}_{2, n\delta}^{s_{2}, i}, \Lambda_{n\delta}^{i})\bar{X}_{1, n\delta}^{s_{1}, i}]\delta\\ &\; \; \; \; \; \; \; \; \; \; \; \; -\mu(n\delta, x, \bar{X}_{2, n\delta}^{s_{2}, i}, \Lambda_{n\delta}^{i})\bar{X}_{1, n\delta}^{s_{1}, i}\delta+g(n\delta, \bar{X}_{1, n\delta}^{s_{1}, i}, \Lambda_{n\delta}^{i})\Delta W_{n}, \\ &\bar{X}_{2, (n+1)\delta}^{s_{2}, i} = \bar{X}_{2, n\delta}^{s_{2}, i}+[k_{2}(n\delta, x)\Delta \bar{X}_{2, n\delta}^{s_{2}, i}+K(\Lambda_{n\delta}^{i})\bar{X}_{3, n\delta}^{s_{3}, i}-(l(\Lambda_{n\delta}^{i})+m(\Lambda_{n\delta}^{i}))\bar{X}_{2, n\delta}^{s_{2}, i}]\delta, \\ &\bar{X}_{3, (n+1)\delta}^{s_{3}, i} = \bar{X}_{3, n\delta}^{s_{3}, i}+[k_{3}(n\delta, x)\Delta \bar{X}_{3, n\delta}^{s_{3}, i}-M(\Lambda_{n\delta}^{i})\bar{X}_{3, n\delta}^{s_{3}, i}+u(n\delta, x)]\delta, \end{split} \end{cases} \end{equation} | (4.1) |
where n\geq 0 and \Delta W_{n}\triangleq W_{(n+1)\delta}-W_{n\delta} denotes Brownian motion increment, \Delta \bar{X}_{k, n\delta}^{s_{k}, i} is the Laplace of \bar{X}_{k, n\delta}^{s_{k}, i} , with the initial data ((\bar{X}_{1}^{0}, \bar{X}_{2}^{0}, \bar{X}_{3}^{0}), \Lambda_{0}) = ((s_{1}, s_{2}, s_{3}), i)\in H_{3}\times\mathbb{S} which is introduced before. Equations (4.1) and (4.2) are the discrete-time EM scheme and continuous-time EM scheme of the corresponding system (2.1), respectively. For convenience, we define the corresponding approximate solution to the system (2.1) on continuous time.
\begin{equation} \begin{cases} \begin{split} &X_{1, t}^{s_{1}, i} = s_{1}+\int_{0}^{t}[k_{1}(\lfloor \epsilon/\delta\rfloor\delta, x)\Delta \bar{X}_{1, \lfloor \epsilon/\delta\rfloor\delta}^{s_{1}, i}+\beta(\lfloor \epsilon/\delta\rfloor\delta, x, \bar{X}_{2, \lfloor \epsilon/\delta\rfloor\delta}^{s_{2}, i}, \Lambda_{\lfloor \epsilon/\delta\rfloor\delta}^{i})\bar{X}_{1, \lfloor \epsilon/\delta\rfloor\delta}^{s_{1}, i}]d\epsilon\\ &\; \; \; \; \; \; -\int_{0}^{t}\mu(\lfloor \epsilon/\delta\rfloor\delta, x, \bar{X}_{2, \lfloor \epsilon/\delta\rfloor\delta}^{s_{2}, i}, \Lambda_{\lfloor \epsilon/\delta\rfloor\delta}^{i})\bar{X}_{1, \lfloor \epsilon/\delta\rfloor\delta}^{s_{1}, i}d\epsilon+\int_{0}^{t}g(\lfloor \epsilon/\delta\rfloor\delta, \bar{X}_{1, \lfloor \epsilon/\delta\rfloor\delta}^{s_{1}, i}, \Lambda_{\lfloor \epsilon/\delta\rfloor\delta}^{i})dW_{\epsilon}, \\ &X_{2, t}^{s_{2}, i} = s_{2}+\int_{0}^{t}[k_{2}(\lfloor \epsilon/\delta\rfloor\delta, x)\Delta \bar{X}_{2, \lfloor \epsilon/\delta\rfloor\delta}^{s_{2}, i}+K(\Lambda_{\lfloor \epsilon/\delta\rfloor\delta}^{i})\bar{X}_{3, \lfloor \epsilon/\delta\rfloor\delta}^{s_{3}, i}]d\epsilon\\ &\; \; \; \; \; \; \; -\int_{0}^{t}(l(\Lambda_{\lfloor \epsilon/\delta\rfloor\delta}^{i})+m(\Lambda_{\lfloor \epsilon/\delta\rfloor\delta}^{i}))\bar{X}_{2, \lfloor \epsilon/\delta\rfloor\delta}^{s_{2}, i}d\epsilon, \\ &X_{3, t}^{s_{3}, i} = s_{3}+\int_{0}^{t}[k_{3}(\lfloor \epsilon/\delta\rfloor\delta, x)\Delta \bar{X}_{3, \lfloor \epsilon/\delta\rfloor\delta}^{s_{3}, i}-M(\Lambda_{\lfloor \epsilon/\delta\rfloor\delta}^{i})\bar{X}_{3, \lfloor \epsilon/\delta\rfloor\delta}^{s_{3}, i}+u(\lfloor \epsilon/\delta\rfloor\delta, x)]d\epsilon, \end{split} \end{cases} \end{equation} | (4.2) |
where t > 0, \Lambda_{0}^{i} = i\in \mathbb{S} , \forall b\geq 0 , \lfloor b\rfloor is the interger part of b . Obviously, by a straightforward calculation, we can have (X_{1, \lfloor \epsilon/\delta\rfloor\delta}^{s_{1}, i}, X_{2, \lfloor \epsilon/\delta\rfloor\delta}^{s_{2}, i}, X_{3, \lfloor \epsilon/\delta\rfloor\delta}^{s_{3}, i}) = (\bar{X}_{1, \lfloor \epsilon/\delta\rfloor\delta}^{s_{1}, i}, \bar{X}_{2, \lfloor \epsilon/\delta\rfloor\delta}^{s_{2}, i}, \bar{X}_{3, \lfloor \epsilon/\delta\rfloor\delta}^{s_{3}, i}).
Let \mathbb{P}_{n\delta}^{\delta}((s_{1}, s_{2}, s_{3}), j; \cdot) be the transition probability kernel of ((\bar{X}_{1, n\delta}^{s_{1}, i}, \bar{X}_{2, n\delta}^{s_{2}, i}, \bar{X}_{3, n\delta}^{s_{3}, i}), \Lambda_{n\delta}^{i}) . If \pi^{\delta}\in\mathcal{P}(H_{3}\times\mathbb{S}) satisfies the following equation
\begin{equation} \pi^{\delta}(A\times \{i\}) = \sum\limits_{j = 1}^{N}\int_{H_{3}} \mathbb{P}_{n\delta}^{\delta}((s_{1}, s_{2}, s_{3}), j; A\times \{i\})\pi^{\delta}(d(s_{1}, s_{2}, s_{3}) \times \{j\}), t\geq 0, A \in H_{3}, i\in \mathbb{S}, \end{equation} | (4.3) |
then we call \pi^{\delta}\in\mathcal{P}(H_{3}\times\mathbb{S}) an invariant measure of ((\bar{X}_{1, n\delta}^{s_{1}, i}, \bar{X}_{2, n\delta}^{s_{2}, i}, \bar{X}_{3, n\delta}^{s_{3}, i}), \Lambda_{n\delta}^{i}) or a numerical invariant measure of ((X_{1, t}^{s_{1}, i}, X_{2, t}^{s_{2}, i}, X_{3, t}^{s_{3}, i}), \Lambda_{t}^{i}) . Let
q_{0}: = \max\limits_{i\in \mathbb{S}}(-q_{ii}), \; \rho_{0} = \max\limits_{i\in \mathbb{S}}|\rho_{i}|, \; \hat{\xi_{0}}\triangleq \max\limits_{i\in \mathbb{S}}\xi_{i}^{(p)}, \; \breve{\xi_{0}}\triangleq(\max\limits_{i\in \mathbb{S}}\xi_{i}^{(p)})^{-1}. |
Our main result in this section is as follows
Lemma 2. Under the conditions of Lemma 3.1 and combining Eq (3.2) with (3.3), it holds that
\begin{equation} \begin{split} &\mathbb{E}(|\bar{X}_{1, n\delta}^{s_{1}, i}-\bar{X}_{1, n\delta}^{\bar{s}_{1}, j}|^{p}+|\bar{X}_{2, n\delta}^{s_{2}, i}-\bar{X}_{2, n\delta}^{\bar{s}_{2}, j}|^{p}+|\bar{X}_{3, n\delta}^{s_{3}, i}-\bar{X}_{3, n\delta}^{\bar{s}_{3}, j}|^{p})\\ &\leq C(1+|s_{1}|^{p}+|s_{2}|^{p}+|s_{3}|^{p}+|\bar{s}_{1}|^{p}+|\bar{s}_{2}|^{p}+|\bar{s}_{3}|^{p})e^{-\eta_{p}n\delta}, \end{split} \end{equation} | (4.4) |
for any p\in(0, p_{0}) , (s, i) = ((s_{1}, s_{2}, s_{3}), i) , (\bar{s}, j) = ((\bar{s}_{1}, \bar{s}_{2}, \bar{s}_{3}), j) \in H_{3}\times\mathbb{S}. p_{0} is given in Eq (3.2).
Lemma 4.1 shows that numerical solution (\bar{X}_{1, n\delta}^{s_{1}, i}, \bar{X}_{2, n\delta}^{s_{2}, i}, \bar{X}_{3, n\delta}^{s_{3}, i}) tends to (\bar{X}_{1, n\delta}^{\bar{s}_{1}, j}, \bar{X}_{2, n\delta}^{\bar{s}_{2}, j}, \bar{X}_{3, n\delta}^{\bar{s}_{3}, j}) when n\rightarrow \infty and \delta \rightarrow 0 under different initial values and states. This lemma provides a great convenience for the proof of Theorem 4.1. Applying a method similar to Theorem 3.1 can prove the conclusion of Lemma 4.1, so it is omitted.
Theorem 4.1. Under the conditions of Theorem 3.1, there exists a sufficiently small \delta^{*} such that for any \delta\in (0, \delta^{*}) , the solutions of the EM method (4.2) converge to a unique invariant measure \pi^{\delta}\in \mathcal{P}(H_{3}\times \mathbb{S}) with some exponential rate \bar{\gamma} > 0 in the Wassertein distance.
Proof. In fact, for any the initial data (s_{1}, s_{2}, s_{3}) , by Eq (4.2) and the Chebyshev's inequality, we derive that \{\delta_{(s_{1}, s_{2}, s_{3})}\mathbb{P}^{\delta}_{n\delta}\} is tight. Therefore, there exists an exact subsequence which converges weakly to an invariant measure denoted by \pi^{\delta}\in\mathcal{P}(H_{3}\times \mathbb{S}) . According to the Eq (3.14), we have the following result
\begin{equation} \mathbb{P}(\Lambda_{n\delta}^{i}\neq\Lambda_{n\delta}^{j}) = \mathbb{P}(\tau^{\delta} \gt n)\leq e^{-\theta n\delta}. \end{equation} | (4.5) |
For any n > 0, combining with Eq (4.4), it is not difficult to get
\begin{equation} \begin{split} &W_{p}(\delta_{((s_{1}, s_{2}, s_{2}), i)}\mathbb{P}_{n\delta}^{\delta}, \delta_{((\bar{s}_{1}, \bar{s}_{2}, \bar{s}_{3}), j)}\mathbb{P}_{n\delta}^{\delta})\\ &\leq\mathbb{E}(|\bar{X}_{1, n\delta}^{s_{1}, i}-\bar{X}_{1, n\delta}^{\bar{s}_{1}, j}|^{p}+|\bar{X}_{2, n\delta}^{s_{2}, i}-\bar{X}_{2, n\delta}^{\bar{s}_{2}, j}|^{p}+|\bar{X}_{3, n\delta}^{s_{3}, i}-\bar{X}_{3, n\delta}^{\bar{s}_{3}, j}|^{p}) +\mathbb{P}(\Lambda_{n\delta}^{i}\neq\Lambda_{n\delta}^{j})\\ &\leq C(1+|s_{1}|^{p}+|s_{2}|^{p}+|s_{3}|^{p}+|\bar{s}_{1}|^{p}+|\bar{s}_{2}|^{p}+|\bar{s}_{3}|^{p})e^{-\bar{\gamma} n\delta}, \end{split} \end{equation} | (4.6) |
where \bar{\gamma}: = \varrho\wedge \theta , and using the Kolmogorov-Chapman equation and Eq (4.6), for any n, m > 0 , we have
\begin{equation} \begin{split} &W_{p}(\delta_{((s_{1}, s_{2}, s_{2}), i)}\mathbb{P}_{n\delta}^{\delta}, \delta_{((s_{1}, s_{2}, s_{2}), i)}\mathbb{P}_{(n+m)\delta}^{\delta})\\ & = W_{p}(\delta_{((s_{1}, s_{2}, s_{2}), i)}\mathbb{P}_{n\delta}^{\delta}, \delta_{((s_{1}, s_{2}, s_{2}), i)}\mathbb{P}_{n\delta}^{\delta}\mathbb{P}_{m\delta}^{\delta})\\ &\leq\int_{H_{3}\times\mathbb{S}}W_{p}(\delta_{((s_{1}, s_{2}, s_{2}), i)}\mathbb{P}_{n\delta}^{\delta}, \delta_{((\bar{s}_{1}, \bar{s}_{2}, \bar{s}_{3}), j)}\mathbb{P}_{n\delta}^{\delta})\mathbb{P}_{m\delta}^{\delta}((s_{1}, s_{2}, s_{2}), i;d(\bar{s}_{1}, \bar{s}_{2}, \bar{s}_{3}), j)\\ &\leq\sum\limits_{j\in\mathbb{S}}\int_{H_{3}}C(1+|s_{1}|^{p}+|s_{2}|^{p}+|s_{3}|^{p}+|\bar{s}_{1}|^{p}+|\bar{s}_{2}|^{p}+|\bar{s}_{3}|^{p})e^{-\bar{\gamma} n\delta}H_{1}\\ &\leq Ce^{-\bar{\gamma} n\delta}, \end{split} \end{equation} | (4.7) |
where H_{1} = \mathbb{P}_{m\delta}^{\delta}((s_{1}, s_{2}, s_{2}), i;d(\bar{s}_{1}, \bar{s}_{2}, \bar{s}_{3}), j) , then taking m\rightarrow \infty such that
\begin{equation} W_{p}(\delta_{((s_{1}, s_{2}, s_{2}), i)}\mathbb{P}_{n\delta}^{\delta}, \pi^{\delta})\rightarrow 0, \quad n\rightarrow \infty, \end{equation} | (4.8) |
in other words, \pi^{\delta} is the unique invariant measure of \{\delta_{(s_{1}, s_{2}, s_{3})}\mathbb{P}^{\delta}_{n\delta}\} . \forall \pi^{\delta}, \nu^{\delta}\in \mathcal{P}(H_{3}\times\mathbb{S}) are invariant measures of ((\bar{X}_{1, n\delta}^{s_{1}, i}, \bar{X}_{2, n\delta}^{s_{2}, i}, \bar{X}_{3, n\delta}^{s_{3}, i}), \Lambda_{n\delta}^{i}) and ((\bar{X}_{1, n\delta}^{\bar{s}_{1}, j}, \bar{X}_{2, n\delta}^{\bar{s}_{2}, j}, \bar{X}_{3, n\delta}^{\bar{s}_{3}, j}), \Lambda_{n\delta}^{j}) , respectively. Further, we have
\begin{equation} \begin{split} &W_{p}(\pi^{\delta}, \nu^{\delta}) = W_{p}(\pi^{\delta} \mathbb{P}_{n\delta}^{\delta}, \nu^{\delta} \mathbb{P}_{n\delta}^{\delta})\\ &\leq\sum\limits_{i, j = 1}^{N}\int_{H_{3}\times\mathbb{S}}\int_{H_{3}\times\mathbb{S}}\pi^{\delta}(d(s_{1}, s_{2}, s_{3})\times \{i\})\nu^{\delta}(d(\bar{s}_{1}, \bar{s}_{2}, \bar{s}_{3})\times\{j\})W_{p}(\delta_{((s_{1}, s_{2}, s_{3}), i)}\mathbb{P}_{n\delta}^{\delta}, \delta_{((\bar{s}_{1}, \bar{s}_{2}, \bar{s}_{3}), j)}\mathbb{P}_{n\delta}^{\delta}). \end{split} \end{equation} | (4.9) |
The uniqueness for the numerical invariant measure have been completed. Therefore, the proof of Theorem 4.1 is complete. To show that the numerical invariant measure \pi^{\delta} converges to the invariant measure of the corresponding exact solution under the Wasserstein distance, the following theorem is given.
Theorem 4.2. Under the assumptions of Theorem 4.1 and Eq (4.8), for \delta\in(0, 1) there exists C > 0 such that
W_{p}(\pi, \pi^{\delta})\leq C\delta^{\frac{p}{2}}, \; \; p\in(0, p_{0}), |
where p_{0} > 0 is defined in Eq (3.2).
Proof. For p\in (0, p_{0}) , due to
W_{p}(\delta_{((s_{1}, s_{2}, s_{2}), i)}\mathbb{P}_{n\delta}, \pi)\leq\int_{H_{3}\times S}\pi(d(\bar{s}_{1}, \bar{s}_{2}, \bar{s}_{3})\times\{j\})W_{p}(\delta_{((s_{1}, s_{2}, s_{2}), i)}\mathbb{P}_{n\delta}^{\delta}, \delta_{((\bar{s}_{1}, \bar{s}_{2}, \bar{s}_{3}), j)} P_{n\delta}^{\delta}), |
and
W_{p}(\delta_{((s_{1}, s_{2}, s_{2}), i)}\mathbb{P}_{n\delta}^{\delta}, \pi^{\delta})\leq\int_{H_{3}\times S}\pi(d(\bar{s}_{1}, \bar{s}_{2}, \bar{s}_{3})\times\{j\})W_{p}(\delta_{((s_{1}, s_{2}, s_{2}), i)}\mathbb{P}_{n\delta}^{\delta}, \delta_{((\bar{s}_{1}, \bar{s}_{2}, \bar{s}_{3}), j)} P_{n\delta}^{\delta}). |
Then based on the assumption conditions of (\mathbb{H}1) – (\mathbb{H}3) and Eq (4.8), there exists a sufficiently small \delta^{*} such that for any \delta\in (0, \delta^{*}) , there is n > 0 sufficiently large such that
\begin{equation} W_{p}(\delta_{((s_{1}, s_{2}, s_{2}), i)}\mathbb{P}_{n\delta}, \pi)+W_{p}(\delta_{((s_{1}, s_{2}, s_{2}), i)}\mathbb{P}_{n\delta}^{\delta}, \pi^{\delta})\leq C\delta^{\frac{p}{2}}, \end{equation} | (4.10) |
For fixed n > 0 and using the triangle inequality, and by the similar way of [22], we can obtain \lim\limits_{\delta\rightarrow 0}W_{p}(\delta_{((s_{1}, s_{2}, s_{2}), i)}\mathbb{P}_{n\delta}, \delta_{((s_{1}, s_{2}, s_{2}), i)}\mathbb{P}_{n\delta}^{\delta}) = 0 . In other words, there exists a positive constant \bar{\nu} such that W_{p}(\delta_{((s_{1}, s_{2}, s_{2}), i)}\mathbb{P}_{n\delta}, \delta_{((s_{1}, s_{2}, s_{2}), i)}\mathbb{P}_{n\delta}^{\delta})\leq Ce^{\bar{\nu} \delta n}\delta^{\frac{p}{2}} . According to Theorem 3.1 and Eq (4.8), we can get the following result
\begin{equation} W_{p}(\delta_{((s_{1}, s_{2}, s_{2}), i)}\mathbb{P}_{n\delta}, \pi)+W_{p}(\delta_{((s_{1}, s_{2}, s_{2}), i)}\mathbb{P}_{n\delta}^{\delta}, \pi^{\delta})\leq Ce^{-\gamma^{*}n\delta}, \end{equation} | (4.11) |
where \gamma^{*}: = \sigma^{*}\wedge\bar{\gamma}. Let \bar{C} be the integer part of constant -p\ln\delta/[2(\bar{\nu}+\gamma^{*})\delta] , obviously, \bar{C}\rightarrow 0 as \delta \rightarrow 0 . On the other hand, we have e^{\bar{\nu} \bar{C}\delta}\delta^{\frac{p}{2}}\leq \delta^{\frac{p\sigma^{*}}{2(\bar{\nu}+\gamma^{*})}}\leq\delta^{\frac{p}{2}} , e^{-\sigma^{*}\bar{C}\delta}\leq e^{\gamma^{*}\delta^{*}}\delta^{\frac{p}{2}} . Therefore, W_{p}(\pi, \pi^{\delta})\leq C\delta^{\frac{p}{2}} holds.
Further, to illustrate the validity of our theory which are discussed in the previous section, we will give a numerical example.
Let \Lambda_{t} be a Markov chain with the state space \mathbb{S} = \{1, 2\} , and the generator
\Gamma = \left( \begin{array}{rrr} 3 & -3 \\ -4 & 4\\ \end{array} \right). |
It is easy to show that its unique stationary distribution \pi = (\pi_{1}, \pi_{2}) is given by \pi_{1} = 1/2 , \pi_{1} = 1/2 . On the other hand, we give the following setting: V(\Lambda_{t}): = l(\Lambda_{t})+m(\Lambda_{t}) , when \Lambda_{t} = 1 , we choose M(1) = \frac{1}{2}\exp(\frac{2}{1+2t}) , K(1) = 0.01\sin(\frac{1}{(3+0.2t)^2}) and V(1) = l(1)+m(1) = 1.99 ; when \Lambda_{t} = 2 , we choose M(2) = \frac{9}{10}(\frac{1}{1+t}) , K(2) = 0.05\sin(\frac{1}{(3+0.2t)^2}) and V(2) = l(2)+m(2) = 1.6 . In the state 1 and 2 , setting T = 1 , t\in (0, 1) , \beta: = \beta(t, x, X_{2}(t, x), \Lambda_{t}) = \frac{1}{2}(1-\frac{0.5X_{2}(t, x)}{0.5(1+X_{2}(t, x))})(1-\frac{x}{5+x}) , \mu: = \mu(t, x, X_{2}(t, x), \Lambda_{t}) = \frac{3}{10}(0.5-\frac{0.8X_{2}(t, x)}{1+0.5X_{2}(t, x)})(1-\frac{x}{0.5+x}) , g: = g(t, x, X_{1}(t, x), \Lambda_{t}) = 0.05+0.3X_{1}(t, x) , and taking k_{1} = 0.005 , k_{2} = k_{3} = 0.05 , s_{1}(x) = s_{2}(x) = \frac{0.2}{(1+x)^2} , s_{3}(x) = \frac{0.2}{(1+1.5x)^2} , the system (2.1) is described as follows
\begin{equation} \begin{cases} \begin{split} &dX_{1}(t, x) = [0.005\Delta X_{1}(t, x)+\beta X_{1}(t, x)-\mu X_{1}(t, x)]dt+gdW_{t}, && in\quad (0, T)\times \Gamma, \\ &dX_{2}(t, x) = [0.05\Delta X_{2}(t, x)+K( \Lambda_{t})X_{3}(t, x)-V( \Lambda_{t})X_{2}(t, x)]dt, && in\quad (0, T)\times \Gamma, \\ &dX_{3}(t, x) = [0.05\Delta X_{3}(t, x)-M( \Lambda_{t})X_{3}(t, x)+u(t, x)]dt, && in\quad (0, T)\times \Gamma, \\ & X_{1}(0, x) = X_{2}(0, x) = \frac{0.2}{(1+x)^2}, \; X_{3}(0, x) = \frac{0.2}{(1+1.5x)^2}, && in \quad x \in \Gamma, \\ &X_{1}(t, x) = 0, \; X_{2}(t, x) = 0, \; X_{3}(t, x) = 0, && on \quad (0, T]\times \partial \Gamma, \\ \end{split} \end{cases} \end{equation} | (5.1) |
First, for the system (5.1), we use the discrete-time EM method for numerical simulation. Figure 1 is a simulation of Markov chain which describes switching between different states.
Then, taking T = 1 , N = 100 , |W_{i+1}-W_{i}| = \sqrt{\delta } and t\in(0, 1) , step sizes \delta = 0.005 . Among them, the values of X_{3}(t, x) and X_{2}(t, x) do not exceed 0.4. This satisfies the practical significance, i.e., 0\leq X_{2}(t, x)\leq 1 , 0\leq X_{3}(t, x)\leq1 .
As far as we know, the exact solution for system (5.1) is difficult to find. Inspired by [30] and based on the method of [28], we can take the "explicit solution" Y_{1}(t, x) = \exp(\frac{1}{2}-\frac{1}{1-x}-\frac{t^2}{2})(1+\Delta W) , Y_{2}(t, x) = K\int_{0}^{t}Y_{3}(t, x)\exp\{(l+m)(s-t)\}ds+C_{Y_{2}}\exp\{-(l+m)t\} and Y_{3}(t, x) = \int_{0}^{t}u(t, x)\exp\{h(s-t)\}ds+C_{Y_{3}}\exp\{-ht\} replace exact solution, where C_{Y_{2}}, C_{Y_{3}} are initial values of Y_{2} and Y_{3} , respectively. Setting C_{Y_{3}} = \frac{0.2}{(1+1.5x)^2} , C_{Y_{2}} = \frac{0.2}{(1+x)^2} , K = 0.05 , l+m = 1.9 and h = 0.5 , u(t, x) = \frac{1}{5}(\frac{4}{(1+2x)^2}-\frac{1}{2})(1-t)^3 . Then, The simulation results are presented separately in Figure 2(a), Figure 4(a) and Figure 6(a). In Figure 6(b) and Figure 4(b) reflect the numerical simulation of X_{3}(t, x) and X_{2}(t, x) with Markov switching when the step size is 0.005 under the state "1 " and " 2 " switching.
In addition, Figure 7, Figure 5 and Figure 3 show mean-square error between "explicit solutions Y_{3} , Y_{2} and Y_{1} " and the corresponding numerical solutions X_{3} , X_{2} and X_{1} (Figure 6(b), Figure 4(b) and Figure 2(b)) of stochastic population with diffusion and Markov switching in a polluted environment system (5.1), when we take step sizes \delta = 0.005, 0.0001 . Obviously, when the step size \delta changes from 0.005 to 0.0001, the error values decreases from 0.14, 0.4 and 0.04 to 0.012, 0.025 and 0.02, respectively. Combining Figure 7, Figure 5 and Figure 3, we have the assertion that the smaller the step size, the smaller the error. Hence, it is not difficult to conclude that when \delta \rightarrow 0 , the numerical solution X_{3}(t, x) , X_{2}(t, x) , X_{1}(t, x) under discrete-time EM method converges to the explicit solution Y_{3}(t, x) , Y_{2}(t, x) , Y_{1}(t, x) , respectively.
In this paper, we establish a new stochastic population model with Markov chain and diffusion in a polluted environment. Based on the Perron-Frobenius theorem, when the diffusion coefficient satisfies the local Lipschitz, the criterion on the existence and uniqueness of invariant measure for the exact solution is given. Moreover, we also discuss the existence and uniqueness of numerical invariance measure for model (2.1) under the discrete-time Euler-Maruyama scheme, and prove that numerical invariance measure converges to invariance measure of the corresponding exact solution in the Wasserstein distance sense. At the end of this paper, the accuracy of the theoretical results is verified by numerical simulation.
The authors are very grateful to the anonymous reviewers for their insightful comments and helpful suggestions. The research was supported by the Natural Science Foundation of China (Grant numbers 11661064). This research was funded by the "Major Innovation Projects for Building First-class Universities in China's Western Region" (ZKZD2017009).
The authors declare there is no conflict of interest.
[1] | L. Duan, Q. Lu, Z. Yang, L. Chen, Effects of diffusion on a stage-structured population in a polluted environment, Appl. Math. Comput., 02 (2004), 347-359. |
[2] | A. J. Shaw, Ecological genetics of plant populations in polluted environment. Ecological Genetics and Air Pollution, Springer New York, 1991. |
[3] |
G. P. Samanta, A. Maiti, Dynamical model of a single-species system in a polluted environment, J. Appl. Mathe. Comput., 16 (2004), 231-242. doi: 10.1007/BF02936164
![]() |
[4] |
T. G. Hallam, C. E. Clark, R. R. Lassiter, Effects of toxicants on populations: a qualitative approach I. Equilibrium environmental exposure, Ecol. Model., 18 (1983), 291-304. doi: 10.1016/0304-3800(83)90019-4
![]() |
[5] | T. G. Hallam, C. E. Clar, G. S. Jordan, Effects of toxicant on population: a qualitative approach II. First Order Kinetics, J. Math. Biol., 109 (1983), 411-429. |
[6] | D. Mukherjee, Persistence and global stability of a population in a polluted environment with delay, J. Biol. Syst., 10 (2008), 225-232. |
[7] | Z. Ma, G. Cui, W. Wang, Persistence and extinction of a population in a polluted environment, Math. Biosci., 101 (2004), 75-97. |
[8] |
T. G. Hallam, Z. Ma, Persistence in Population models with demographic fluctuations, J. Math. Biol., 24 (1986), 327-339. doi: 10.1007/BF00275641
![]() |
[9] |
J. Pan, Z. Jin, Z. Ma, Thresholds of survival for an n-dimensional Volterra mutualistic system in a polluted environment, J. Math. Anal. Appl., 252 (2000), 519-531. doi: 10.1006/jmaa.2000.6853
![]() |
[10] |
Z. Ma, B. J. Song, T. G. Hallam, The threshold of survival for systems in a fluctuating environment, Bull. Math. Biol., 51 (1989), 311-323. doi: 10.1016/S0092-8240(89)80078-3
![]() |
[11] | M. Liu, K. Wang, Persistence and extinction of a stochastic single-species population model in a polluted environment with impulsive toxicant input, Electron. J. Differ. Equ., 230 (2013), 823-840. |
[12] |
X. Yu, S. Yuan, T. Zhang, Persistence and ergodicity of a stochastic single species model with Allee effect under regime switching, Commun. Nonlinear Sci. Numer. Simul., 59 (2018), 359-374. doi: 10.1016/j.cnsns.2017.11.028
![]() |
[13] |
F. Wei, S. A. H. Geritz, J. Cai, A stochastic single-species population model with partial pollution tolerance in a polluted environment, Appl. Math. Lett., 63 (2017), 130-136. doi: 10.1016/j.aml.2016.07.026
![]() |
[14] | M. Liu, K. Wang, Persistence and extinction of a stochastic single-specie model under regime switching in a polluted environment. J. Theor. Biol., 267 (2010), 283-291. |
[15] | M. Liu, K. Wang, Survival analysis of a stochastic single-species population model with jumps in a polluted environment, Int. J. Biomath., 09 (2016), 1-15. |
[16] |
Y. Zhao, S. Yuan, Q. Zhang, The effect of Lévy noise on the survival of a stochastic competitive model in an impulsive polluted environment, Appl. Math. Model., 40 (2016), 7583-7600. doi: 10.1016/j.apm.2016.01.056
![]() |
[17] |
T. C. Gard, Stability for multi-species population models in random environments, Nonlinear Anal., 10 (1986), 1411-1419. doi: 10.1016/0362-546X(86)90111-2
![]() |
[18] | J. Tong, Z. Zhang, J. Bao, The stationary distribution of the facultative population model with a degenerate noise, Discrete Continuous Dyn. Syst., 83 (2013), 655-664. |
[19] |
M. Liu, K. Wang, Y. Wang, Long term behaviors of stochastic single-species growth models in a polluted environment, Appl. Math. Model., 35 (2011), 4438-4448. doi: 10.1016/j.apm.2011.03.014
![]() |
[20] |
C. Yuan, X. Mao, Stationary distributions of Euler-Maruyama-type stochastic difference equations with Markovian switching and their convergence, J. Differ. Equ. Appl., 11 (2005), 29-48. doi: 10.1080/10236190412331314150
![]() |
[21] |
G. Yin, X. Mao, K. Yin, Numerical approximation of invariant measures for hybrid diffusion systems, IEEE Trans. Automat. Contr., 50 (2005), 934-946. doi: 10.1109/TAC.2005.851437
![]() |
[22] |
J. Bao, J. Shao, C. Yuan, Approximation of invariant measures for regime-switching diffusions, Potential Anal., 44 (2016), 707-727. doi: 10.1007/s11118-015-9526-x
![]() |
[23] |
H. Yang, X. Li, Explicit approximations for nonlinear switching diffusion systems in finite and infinite horizons, J. Differ. Equ., 265 (2018), 2921-2967. doi: 10.1016/j.jde.2018.04.052
![]() |
[24] | X. Mao, C. Yuan, Stochastic differential equations with Markovian switching, Imperial College Press, 2006. |
[25] |
G. Dhariwal, A. Jungel, N. Zamponi, Global martingale solutions for a stochastic population crossdiffusion system, Stoch. Process. their Appl., 129 (2019), 3792-3820. doi: 10.1016/j.spa.2018.11.001
![]() |
[26] | G. Da Prato, J. Zabczyk, Ergodicity for infinite dimensional systems, Cambridge University Press, Cambridge, 1996. |
[27] | G. Yin, C. Zhu, Hybrid swithching diffusion: Properties and Applications, Springer, 2010. |
[28] | Y. Zhao, S. Yuan, Q. Zhang, Numerical solution of a fuzzy stochastic single-species age-structure model in a polluted environment, Appl. Math. Comput., 260 (2015), 385-396. |
[29] | W J. Anderson, Continuous-time Markov chains, Springer, Berlin, 1991. |
[30] | J. Tan, A. Rathinasamy, Y. Pei, Convergence of the split-step θ-method for stochastic agedependent population equations with Poisson jumps, Elsevier Science Inc, 254 (2015), 305-317. |
1. | An Ma, Qimin Zhang, Global attractor and threshold dynamics of a reaction–diffusion population model in a polluted environment, 2023, 69, 1598-5865, 989, 10.1007/s12190-022-01781-4 | |
2. | An Ma, Shuting Lyu, Qimin Zhang, Stationary distribution and optimal control of a stochastic population model in a polluted environment, 2022, 19, 1551-0018, 11260, 10.3934/mbe.2022525 | |
3. | An Ma, Jing Hu, Qimin Zhang, DYNAMIC ANALYSIS AND OPTIMAL CONTROL OF A TOXICANT-POPULATION MODEL WITH REACTION-DIFFUSION, 2024, 14, 2156-907X, 579, 10.11948/20210438 | |
4. | An Ma, Jing Hu, Ming Ye, Qimin Zhang, Investigation of sliding mode dynamics and near-optimal controls for a reaction–diffusion population model in a polluted environment, 2024, 79, 09473580, 101097, 10.1016/j.ejcon.2024.101097 |