Stability of a stochastic logistic model under regime switching
- Meng Liu^{1, 2}Email author and
- Li Yu^{3}
https://doi.org/10.1186/s13662-015-0666-5
© Liu and Yu 2015
Received: 3 July 2015
Accepted: 8 October 2015
Published: 21 October 2015
Abstract
In this letter, we consider a stochastic generalized logistic equation with Markovian switching. We obtain a critical value which has the property that if the critical value is negative, then the trivial solution of the model is stochastically globally asymptotically stable; if the critical value is positive, then the solution of the model is positive recurrent and has a unique ergodic stationary distribution. We find out that the critical value has a close relationship with the stationary probability distribution of the Markov chain.
Keywords
MSC
1 Introduction
In recent years, population systems under Markovian switching have received much attention. Some nice and interesting properties, for example, stochastic boundedness, extinction, stochastic permanence, positive recurrent, invariant distribution, have been obtained (see, e.g., [6–20]). Especially, Takeuchi et al. [10] considered a two-dimensional autonomous Lotka-Volterra predator-prey model with Markovian switching and revealed a significant effect of Markovian switching on the population dynamics: the subsystems of the system develop periodically, but switching makes the system become neither permanent nor dissipative.
In the study of deterministic population models, people always seek for the equilibria and then investigate their stability. For example, model (1) has two equilibria: the trivial equilibrium state 0 and the positive equilibrium state \(N^{\ast}=(r/a)^{1/\theta}\). If \(r<0\), then 0 is globally asymptotically stable; if \(r>0\), then \(N^{\ast}\) is globally asymptotically stable. Note that system (3) does not have positive equilibrium state, then its solution will not tend to a positive constant. Thus an interesting question arises naturally: whether model (3) still has some structured stability? In this letter, we shall investigate this issue. In Section 2, we show that there is a critical value which has a close relationship with the stationary probability distribution of the Markov chain \(\beta(t)\). If the critical value is negative, then the trivial equilibrium state of system (3) is stochastically globally asymptotically stable; if the critical value is positive, then the solution of system (3) is positive recurrent and has a unique ergodic stationary distribution. We shall give some numerical simulations and conclusions in the last section.
2 Main results
- (A1)
\(\beta(t)\) is independent of the Brownian motion.
- (A2)
\(\beta(t)\) is irreducible, which means that system (3) can switch from any regime to any other regime. Hence \(\beta(t)\) has a unique stationary distribution \(\pi=(\pi_{1},\ldots,\pi_{m})\) which can be obtained by solving the equation \(\pi Q=0\) subject to \(\sum_{i=1}^{m}\pi_{i}=1\) and \(\pi_{i}>0\), \(i=1,\ldots,m\).
Definition 1
([21], p.204)
- (I)The trivial solution of Eq. (5) is said to be stable in probability if for \(\varepsilon\in(0,1)\) and \(\varsigma>0\), there exists \(\delta>0\) such that if \((X_{0},\beta(0))\in S_{\delta}\times S\), where \(S_{\delta}=\{x\in R^{n}: |x|<\delta\}\), then$$P \bigl\{ \bigl\vert X\bigl(t;X_{0},\beta(0)\bigr)\bigr\vert < \varsigma \mbox{ for all } t\geq0 \bigr\} \geq1-\varepsilon. $$
- (II)The trivial solution of Eq. (5) is said to be stochastically asymptotically stable if it is stable in probability and, moreover, for every \(\varepsilon\in(0, 1)\), there exists \(\delta_{0}>0\) such that if \((X_{0},\beta(0))\in S_{\delta_{0}}\times S\), then$$P \Bigl\{ \lim_{t\rightarrow+\infty}X\bigl(t;X_{0},\beta(0)\bigr)=0 \Bigr\} \geq 1-\varepsilon. $$
- (III)The trivial solution of Eq. (5) is said to be stochastically asymptotically stable in the large or stochastically globally asymptotically stable (SGAS) if it is stochastically asymptotically stable and, moreover,$$P \Bigl\{ \lim_{t\rightarrow+\infty}X\bigl(t;X_{0},\beta(0)\bigr)=0 \Bigr\} =1, \quad \forall \bigl(X_{0},\beta(0)\bigr)\in R^{n} \times S. $$
Lemma 1
([21], Theorem 5.37)
Definition 2
([22])
Lemma 2
([22], Theorem 3.13)
Lemma 3
([22], Theorems 4.3 and 4.4)
Lemma 4
([23], Lemma 2.3)
Lemma 5
For any initial value \((N(0),\beta(0))\in (0,+\infty)\times S\), there is a unique global positive solution \(N(t)\) to model (3) almost surely.
Proof
The proof is a slight modification of that in [12] by applying Itô’s formula to \(\sqrt{x}-1-0.5\ln x\), \(x>0\) and hence is omitted. □
Now we are in the position to give the main result of this letter.
Theorem 1
- (i)
If \(\bar{b}<0\), where \(\bar{b}=\sum_{i=1}^{m}\pi_{i}b_{i}\), \(b_{i}=r(i)-0.5\sigma_{1}^{2}(i)\), then the trivial solution is SGAS.
- (ii)
If \(\bar{b}>0\), then the solution \(N(t)\) is positive recurrent with respect to the domain \(\mathcal{U}=(0,l)\times S\) and has a unique ergodic asymptotically invariant distribution (UEAID), where l is a positive number to be specified later.
Proof
To finish this section, let us consider the subsystem (4) of system (3).
3 Conclusions and numerical simulations
By Corollary 1, if \(b_{i}<0\) for some \(i\in S\), then the trivial solution of subsystem (4) is SGAS. Hence Theorem 1 means that if the trivial solution of every individual subsystem of system (3) is SGAS, then, as the result of Markovian switching, the trivial solution of system (3) is still SGAS. On the other hand, if \(b_{i}>0\) for some \(i\in S\), then the solution of subsystem (4) is positive recurrent and has a UEAID. Thus Theorem 1 shows that if the solution of every subsystem of system (3) is positive recurrent and has a UEAID, then, as the result of Markovian switching, the solution of system (3) is still positive recurrent and has a UEAID. However, Theorem 1 indicates a much more interesting result: If the solution of some subsystems in system (3) is positive recurrent and has a UEAID while the trivial solution of some subsystems is SGAS, then, as the results of Markovian switching, the solution of system (3) may be positive recurrent and has a UEAID or tends to its trivial solution, depending on the sign of \(\bar{b}=\sum_{i=1}^{m}\pi_{i}b_{i}\). If \(\bar{b}>0\), then the solution of system (3) is positive recurrent and has a UEAID; if \(\bar{b}<0\), then the trivial solution of system (3) is SGAS.
Case 1: To begin with, we let the generator of \(\beta(t)\) be \(Q=\bigl ( {\scriptsize\begin{matrix}{}-0.3 & 0.3 \cr 0.7 & -0.7 \end{matrix}} \bigr )\). It is easy to see that \(\beta(t)\) has a unique stationary distribution \(\pi=(0.7,0.3)\). Compute that \(\pi_{1}b_{1}+\pi _{2}b_{2}<0\). By virtue of Theorem 1, as the result of Markovian switching, the trivial solution of system (13) is SGAS, see Figure 1(c).
Case 2: Next we choose \(Q=\bigl ( {\scriptsize\begin{matrix}{} -0.5 & 0.5 \cr 0.5 & -0.5\end{matrix}} \bigr )\). Hence \(\pi=(\pi_{1},\pi_{2})=(0.5,0.5)\) and \(\pi_{1}b_{1}+\pi_{2}b_{2}>0\). It then follows from Theorem 1 that, as the result of Markovian switching, the solution of system (13) is positive recurrent and has a UEAID, see Figure 1(d) which is the density of solution of (13).
Declarations
Acknowledgements
We are very grateful to two anonymous referees for their careful reading and very valuable comments, which led to an improvement of our paper. Project funded by The National Natural Science Foundation of China (Nos. 11301207, 11171081 and 11571136), China Postdoctoral Science Foundation (2015M571349), Natural Science Foundation of Jiangsu Province (No. BK20130411), Natural Science Research Project of Ordinary Universities in Jiangsu Province (No. 13KJB110002), Qing Lan Project of Jiangsu Province (2014).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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