Sufficient and necessary conditions on the existence of stationary distribution and extinction for stochastic generalized logistic system
 Lei Liu^{1}Email author and
 Yi Shen^{2}
https://doi.org/10.1186/s136620140345y
© Liu and Shen; licensee Springer 2015
Received: 27 September 2014
Accepted: 25 December 2014
Published: 16 January 2015
Abstract
In this paper, we consider the existence of stationary distribution and extinction for a stochastic generalized logistic system. Sufficient and necessary conditions for the existence of a stationary distribution and extinction are obtained. (a) The system has a unique stationary distribution if and only if the noise intensity is less than twice the intrinsic growth rate. The probability density function has been solved by the stationary FokkerPlanck equation. (b) The system will become extinct when and only when the noise intensity is no less than twice the intrinsic growth rate, and the exponential extinction rate is estimated precisely by two parameters of the systems. A new perspective is provided to explain the recurrence phenomenon in practice. Nontrivial examples are provided to illustrate our results.
Keywords
1 Introduction
In the past few decades, population systems have received a great deal of research attention since they have been successfully used in a variety of application fields, including biology, epidemiology, economics, and neural networks (see [1–7]). Population systems are always subject to environmental noise. It is therefore necessary to reveal how the noise affects the population systems. Recently, the population dynamics under environmental noise has been extensively considered by many authors (see [8–11]). It is well known that when the noise intensity is sufficiently large, the population will become extinct, while it will remain stochastic permanent when the noise intensity is small.
In fact, if we make a great number of records to investigate the dynamic behavior of a permanent population system, we may find that a single record may fluctuate around a fixed point even if the number of records is large. In order to illustrate such biological phenomena clearly, more and more attention has been paid to the existence of stationary distribution and positive recurrence of population systems in recent years (see [12–15]). In this paper, we will concentrate on the stationary distribution and extinction of a stochastic generalized logistic system. The obtained results provide a new perspective to explain such biological phenomena (see Remark 3).
The logistic system is one of the famous population systems due to its universal existence and importance. More recently, the asymptotic behavior of a stochastic logistic system has received a lot of attention (see [16–20]). Jiang et al. [16] showed the stability in time average and stochastic permanence of a nonautonomous logistic equation with random perturbation. Li et al. [18] discussed the stochastic logistic population under regime switching, and sufficient and necessary conditions for stochastic permanence and extinction under some assumptions are obtained. Liu and Wang [20] and Mao [15] studied the stationary distribution of more general stochastic population systems than system (1); the result in [20] and [15] showed that when \(0<\alpha\leq1\), the system (1) has a stationary distribution. Then some questions arise naturally: Is there a stationary distribution to system (1) in the case of \(\alpha>1\)? If yes, can we compute the probability density function of the stationary distribution? And can we compute the mean or variance?
In addition, the existing literature (see [14, 15, 18]) shows clearly that if the noise intensity is more than twice the intrinsic growth rate, the population will become extinct exponentially, whereas it will remain stochastic permanent or has a stationary distribution when the noise intensity is less than twice the intrinsic growth rate. Then one interesting question is: What will happen if the noise intensity equals twice the intrinsic growth rate?

The probability density function of the stationary distribution was obtained by solving the stationary FokkerPlanck equation.

By using some novel techniques, we point out that system (1) will also be extinct when the noise intensity equals twice the intrinsic growth rate.

Sufficient and necessary conditions for the existence of stationary distribution and extinction are established.
The organization of the paper is as follows. Section 2 describes some preliminaries. The main results are stated in Sections 3 and 4. In Sections 3 and 4, we show that system (1) either has a stationary distribution or becomes extinct. The probability density function, mean, and variance of the stationary distribution are obtained in Section 3. The exponential extinction rate is given precisely in Section 4. In Section 5, the sufficient and necessary conditions and some important remarks are stated and three numerical examples are given to illustrate the effectiveness of our results.
2 Notation
Throughout this paper, unless otherwise specified, let $(\mathrm{\Omega},\mathcal{F},{\{{\mathcal{F}}_{t}\}}_{t\ge 0},\mathbb{P})$ be a complete probability space with a filtration ${\{{\mathcal{F}}_{t}\}}_{t\ge 0}$ satisfying the usual conditions (i.e. it is increasing and right continuous, while ${\mathcal{F}}_{0}$ contains all ℙnull sets). The gamma function \(\Gamma(s)\) is defined for positive real number \(s>0\), which is defined via a convergent improper integral, \(\Gamma(s)=\int^{\infty }_{0}t^{s1}\exp(t)\,dt\).
In the same way as Mao et al. [8] did, we can also show the following result on the existence of global positive solution.
Lemma 2.1
Lemma 2.2
Let condition (2) hold. Then for any \(p>0\), there exists a constant \(K_{p}\) such that \(\sup_{0\leq t\leq\infty} E x(t)^{p}< K_{p}\).
The proof is similar to Liu et al. [19]; it is omitted here.
3 Stationary distribution and its probability density function
Lemma 3.1
[21]
 (i)
in the domain G and some neighborhood therefore, the smallest eigenvalue of the diffusion matrix \(A(x)\) is bounded away from zero;
 (ii)
if \(x\in E^{n}\setminus G\), the mean time τ at which a path issuing from x reaches the set G is finite, and \(\sup_{x\in K} E_{x}\tau<+\infty\) for every compact subset \(K \in E^{n}\) and throughout this paper we set \(\inf\emptyset=\infty\).
 (1)The Markov process \(X(t)\) has a stationary distribution \(\mu(\cdot)\) with density in \(E^{n}\). Let \(f(x)\) be a function integrable with respect to the measure \(\mu(\cdot)\). Then$$\begin{aligned} \mathbb{P}\biggl\{ \lim _{t\rightarrow\infty}\frac{1}{t}\int ^{t}_{0}f\bigl(x(s)\bigr)\,ds=\int _{E^{n}} f(y) \mu(dy) \biggr\} =1. \end{aligned}$$(4)
 (2)The probability density function \(\varphi(y)\) of \(\mu(\cdot)\) is the unique bounded solution to the stationary FokkerPlanck equationsatisfying the additional condition \(\int _{E^{n}}\varphi(y)\,dx=1\).$$\begin{aligned} \frac{1}{2}\sum ^{d}_{i,j=1} \frac{\partial^{2}}{\partial y_{i} \partial y_{j}}\bigl(a_{ij}(y)\varphi\bigr)\sum ^{d}_{i=1} \frac{\partial }{\partial y_{i}}\bigl(b_{i}(y)\varphi\bigr)=0, \end{aligned}$$(5)
Theorem 3.2
 (1)
System (1) has a unique stationary distribution denoted by \(\mu(\cdot)\).
 (2)The probability density function of \(\mu(\cdot)\) denoted by \(\varphi(y)\) has the following form:where \(I_{(0, \infty)}(y)\) is the indicator function for the set \((0, \infty)\). Its mean and variance are \((\frac{\alpha\sigma^{2}}{2a})^{\frac{1}{\alpha}}\frac{\Gamma(\frac {2r}{\alpha\sigma^{2}})}{\Gamma(\frac{2r\sigma^{2}}{\alpha\sigma ^{2}})}\) and \((\frac{\alpha\sigma^{2}}{2a})^{\frac{2}{\alpha}} (\frac {\Gamma(\frac{2r+\sigma^{2}}{\alpha\sigma^{2}})}{\Gamma(\frac{2r\sigma ^{2}}{\alpha\sigma^{2}})}(\frac{\Gamma(\frac{2r}{\alpha\sigma ^{2}})}{\Gamma(\frac{2r\sigma^{2}}{\alpha\sigma^{2}})})^{2})\), respectively.$$\begin{aligned} \varphi(y)=\frac{\alpha}{\Gamma(\frac{2r\sigma^{2}}{\alpha\sigma ^{2}})}\biggl(\frac{2a}{\alpha\sigma^{2}} \biggr)^{\frac{2r\sigma^{2}}{\alpha\sigma ^{2}}}y^{\frac{2r}{\sigma^{2}}2}\exp\biggl(\frac{2a}{\alpha\sigma ^{2}}y^{\alpha} \biggr)I_{(0, \infty)}(y), \end{aligned}$$(6)
Proof
The proof is composed of two parts. The first part is to prove the existence of stationary distribution. The second part is to obtain the probability density function by solving the stationary FokkerPlanck equation. Let \(x(t) = x(t; x_{0})\) for simplicity.
The proof is completed. □
Remark 1
4 Extinction
In this section, we will show that if the noise is sufficiently large, the solution to system (1) will become extinct with probability 1.
Theorem 4.1
 (i)If \(\sigma^{2}>2r\), the solution \(x(t,x_{0})\) to system (1) has the property thatThat is, the population will become extinct exponentially with probability one and the exponential extinction rate is \((\frac{\sigma^{2}}{2}r)\).$$\begin{aligned} \lim _{t\rightarrow\infty}\frac{\ln x(t,x_{0})}{t}=\biggl(\frac{\sigma^{2}}{2}r\biggr) \quad\textit{a.s.} \end{aligned}$$(8)
 (ii)If \(\sigma^{2}=2r\), the solution \(x(t,x_{0})\) to system (1) has the property thatThat is, system (1) still becomes extinct with zero exponential extinction rate.$$\begin{aligned} \lim _{t\rightarrow\infty} x(t,x_{0})=0 \quad\textit{a.s.},\qquad \lim _{t\rightarrow\infty} \frac{\ln x(t,x_{0})}{t}=0 \quad\textit{a.s.} \end{aligned}$$(9)
To prove Theorem 4.1, let us present three lemmas which are essential to the proof.
Lemma 4.2
[23]
Lemma 4.3
Let condition (2) hold and \(x(t,x_{0})\) be the global solution to system (1) with any positive initial value \(x_{0}\). For any \(\beta>0\), \(x^{\beta}(t,x_{0})\) is uniformly continuous on \([0, \infty)\) a.s.
The proof of this lemma is rather standard; hence it is omitted. For details the reader is referred to [19].
Proof of Theorem 4.1
As the whole proof is very technical, we will divide it into two steps. The first step is to show the exponential extinction of system (1) when \(\sigma^{2}>2r\). The second step is to show the extinction with zero exponential extinction rate in the case of \(\sigma^{2}=2r\). Let \(x(t) = x(t; x_{0})\) for simplicity.
Step 2: Now, let us finally show assertion (9). The proof of this step is composed of two parts. We first show the almost sure convergence of \(x(t)\) to zero as \(t\rightarrow\infty\). Then we show that the exponential extinction rate is zero.
Now we realize this strategy as follows:
Case 2: Now, we turn to the proof that \(J_{2}\subset E_{3}\) a.s. It is sufficient to show \(\mathbb{P}(J_{2}\cap E_{1})=0\) and \(\mathbb{P}(J_{2}\cap E_{2})=0\). We prove by contradiction.
5 Summary and numerical examples
In this paper, we have discussed the existence of a stationary distribution and extinction of system (1), and sufficient conditions have been established in Theorems 3.2 and 4.1. Note that the two sufficient conditions are complementary and mutually exclusive. Thus, there are also the necessary conditions. In conclusion, we formulate the sufficient and necessary conditions as a theorem.
Theorem 5.1
Letcondition (2) hold. There are two mutually exclusive possibilities for systems (1): either a stationary distribution exists, or it becomes extinct. That is, the system is stationary if and only if \(\sigma^{2}<2r\), while it is extinctive if and only if \(r\leq\frac{\sigma^{2}}{2}\).
Remark 3
Example 5.1
(i) \(\sigma=0.5\):
(ii) \(\sigma=2\):
(iii) \(\sigma=1\):
Declarations
Acknowledgements
The project reported here is supported by the National Science Foundation of China (Grant Nos. 61304070, 11271146), the National Key Basic Research Program of China (973 Program) (2013CB228204).
Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.
Authors’ Affiliations
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