Asymptotic stability of stochastic age-dependent population equations with Markovian switching

doi:10.1016/j.amc.2013.11.006

Applied Mathematics and Computation

Volume 227, 15 January 2014, Pages 309-319

https://doi.org/10.1016/j.amc.2013.11.006 Get rights and content

Abstract

The main aim of this paper is to discuss the almost surely asymptotic stability of the stochastic age-dependent population equations with Markovian switching. An example will be discussed to illustrate the theory.

Introduction

Stochastic differential equations can be found in many applications in such areas as economics, biology, finance, ecology and other sciences [1], [2], [3]. Recently, the stochastic age-dependent population equations have received a great deal of attention. For example, Zhang [4] showed the exponential stability of numerical solutions to a stochastic age-structured population system with diffusion. Wang and Wang [5] gave the convergence of the semi-implicit Euler method for stochastic age-dependent population equations with Poisson jumps. Li et al. [6] studied the convergence of numerical solutions to stochastic age-dependent population equations with Markovian switching. Ma et al. [7] investigated numerical analysis for stochastic age-dependent population equations with fractional Brownian motion.

However, to the best of our knowledge, there is little work on the asymptotic stability for stochastic age-dependent population equations. The main aim of this paper is to consider asymptotic stability for stochastic age-dependent population equations with Markovian switching of the form $\{\begin{matrix} d_{t} P = - \frac{\partial P}{\partial a} dt - μ (t, a) Pdt + f (P, r (t)) dt + g (P, r (t)) {dW}_{t}, & in Q, \\ P (0, a) = P_{0} (a), r (0) = i_{0} & in [0, A], \\ P (t, 0) = \int_{0}^{A} β (t, a) P (t, a) da, & in [0, T], \end{matrix}$ where $T > 0, A > 0, Q = (0, A) \times (0, T), d_{t} P$ is the differential of P relative to t, i.e., $d_{t} P = \frac{\partial P}{\partial t} dt$ . $P = P (t, a)$ denotes the population density of age a at time $t, β (t, a)$ denotes the fertility rate of females of age a at time $t, μ (t, a)$ denotes the mortality rate of age a at time t and state $r (t)$ . $f (P, r (t)) + g (P, r (t)) \frac{{dW}_{t}}{dt}$ denotes the stochastically perturbation, effecting of external environment for population system, such as epidemics, earthquakes, hurricanes, etc.

In this paper, we shall extend the idea from the papers [8], [9], [11] to the asymptotic stability for stochastic age-dependent population equations with Markovian switching.

The paper is organized as follows. In Section 2, we introduce the stochastic age-dependent population equations with Markovian switching. We investigate the almost surely asymptotic stability for the stochastic age-dependent population equations with Markovian switching under the non-linear growth condition in Section 3. In Section 4 an example is discussed to illustrate the theory.

Section snippets

Preliminaries

Throughout this paper, let $(Ω, F, P)$ be a complete probability space with a filtration ${F_{t}}_{t ⩾ 0}$ satisfying the usual conditions (i.e., it is increasing and right continuous while $F_{0}$ contains all $P$ -null sets).

Let ${r (t), t ⩾ 0}$ be a right-continuous Markov chain on the probability space taking values in a finite state $S = {1, 2, \dots, N}$ with the generator $Γ = {(γ_{ij})}_{N \times N}$ given by $P {r (t + Δ) = j | r (t) = i} = \{\begin{matrix} γ_{ij} Δ + o (Δ) & if i \neq j, \\ 1 + γ_{ij} Δ + o (Δ) & if i = j, \end{matrix}$ where $Δ > 0$ . Here $γ_{ij} ⩾ 0$ is the transition rate from i to j if $i \neq j$ while $γ_{ii} = - \sum_{i \neq j} γ_{ij} .$

Asymptotic stability

With the above notations, we can now state our main result.

Theorem 3.1

Let (i)–(ii) hold, and $E | \frac{\partial P_{s}}{\partial a} | ⩽ m_{1} E | P_{s} |$ . Assume that there are functions $V \in C^{2, 1} (G \times R_{+} \times S; R_{+}), γ \in L^{1} (R_{+}; R_{+})$ and $w \in C (G; R_{+})$ such that $LV (P, t, i) ⩽ γ (t) - w (P)$ for all $(P, t, i) \in G \times R_{+} \times S$ and $w (0) = 0, w (P) > 0, \forall P \neq 0$ and $\lim_{| P | \to \infty} \inf_{(t, i) \in R_{+} \times S} V (P, t, i) = \infty .$ Then for any initial value $P_{0}$ and $i_{0} \in S$ , $\lim_{t \to \infty} P (P_{0}, t, i_{0}) = 0 a . s .$ That is, the solution of Eq. (1) is almost surely asymptotically stable.

To prove this theorem, we can also give the following lemma.

Lemma 3.2

Under the conditions of

Example

In this section we shall discuss an example to illustrate our theory.

Let $B (t)$ be a scalar Brownian motion. Let $r (t)$ be a right continuous Markov chain taking values in $S = {1, 2}$ with the generator $Γ = {(γ_{ij})}_{2 \times 2} = [\begin{matrix} - 2 & 2 \\ 1 & - 1 \end{matrix}] .$ Of course $B (t)$ and $r (t)$ are assumed to be independent. Let us consider a stochastic age-dependent population equation with Markovian switching of the form $\{\begin{matrix} d_{t} P = [- \frac{\partial P}{\partial a} - \frac{2}{{(1 - a)}^{2}} P + f (P, r (t))] dt + g (P, r (t)) dB (t), & (t, a) \in Q, \\ P (0, a) = 1 / 3 \exp (- \frac{1}{1 - a}), r (0) = i_{0} = 1, & a \in [0, \frac{1}{2}], \\ P (t, 0) = \int_{0}^{1} \frac{1}{{(1 - a)}^{2}} P (t, a) da, & t \in [0, T], \end{matrix}$ where, $Q = ($

Acknowledgements

We would like to thank the editor and referee for their very helpful comments and suggestions which improve this paper significantly. We also thank the National Natural Science Foundation of China (Grant Nos. 11061024, 11261043, and 61261044) and Ningxia Natural Science Foundation (No. NZ13052)(China) for their financial support.

References (11)

Q. Zhang
Exponential stability of numerical solutions to a stochastic age-structured population system with diffusion
J. Comput. Appl. Math.
(2008)
L. Wang et al.
Convergence of the semi-implicit Euler method for stochastic age-dependent population equations with Poisson jumps
Appl. Math. Modell.
(2010)
R. Li et al.
Convergence of numerical solutions to stochastic age-dependent population equations with Markovian switching
J. Comput. Appl. Math.
(2009)
W. Ma et al.
Numerical analysis for stochastic age-dependent population equations with fractional Brownian motion
Commun. Nonlinear Sci. Numer. Simul.
(2012)
Z. Qi-Min et al.
Existence, uniqueness and exponential stability for stochastic age-dependent population
Appl. Math. Comput.
(2004)

There are more references available in the full text version of this article.

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Asymptotic properties of the solutions of stochastic population equations have been widely studied in the past decades, particularly the stability theory has been attracted lots of attention. For example, Ma et al. [7] studied the asymptotic stability of stochastic age-dependent population equations with Markovian switching. Zhang [8] showed the exponential stability of numerical solutions to stochastic age-structured population system with diffusion.
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Asymptotic stability of stochastic age-dependent population equations with Markovian switching

Abstract

Introduction

Section snippets

Preliminaries

Asymptotic stability

Example

Acknowledgements

J. Comput. Appl. Math.

Appl. Math. Modell.

J. Comput. Appl. Math.

Commun. Nonlinear Sci. Numer. Simul.

Appl. Math. Comput.