A stochastic predator-prey model with delays
- Bo Du^{1, 2}Email author,
- Yamin Wang^{3} and
- Xiuguo Lian^{1}
https://doi.org/10.1186/s13662-015-0483-x
© Du et al.; licensee Springer. 2015
Received: 27 January 2015
Accepted: 23 April 2015
Published: 7 May 2015
Abstract
A stochastic delay predator-prey system is considered. Sufficient criteria for global existence, stochastically ultimately bounded in mean and almost surely asymptotic properties are obtained.
Keywords
1 Introduction
In addition, throughout the present paper, let \((\Omega,\mathcal{F},\{\mathcal {F}_{t}\}_{t\geq0},P)\) be a complete probability space with a filtration \(\{\mathcal{F}_{t}\}_{t\geq0}\) satisfying the usual conditions (i.e., it is right continuous and \(\mathcal{F}_{0}\) contains all P-null sets). Let \(|\cdot|\) denote the Euclidean norm in \(R^{n}\). For a given constant \(\tau>0\), let \(C([-\tau,0],R_{+}^{n})\) denote the family of all continuous \(R_{+}^{n}\)-valued functions ξ with its norm \(\|\xi\|=\sup\{|\xi(\theta)|:\theta\in[-\tau,0]\}\), where \(R_{+}=[0,+\infty)\). Also, denote by \(C_{\mathcal {F}_{0}}^{b}([-\tau,0];R_{+}^{n})\) the family of bounded, \(\mathcal {F}_{0}\)-measurable, \(C_{\mathcal{F}_{0}}^{b}([-\tau,0];R_{+}^{n})\)-valued random variables.
Remark 1.1
When \(m=n=0\), system (SM) becomes (1.4), so system (SM) is a more general stochastic system. For system (SM), so far as our knowledge is concerned, the work on a predator-prey model with stochastic perturbations seems rare. In this paper, we study system (SM) which is rather general, and some well-known systems may be viewed as its special cases, and obtain some properties of solutions to system (SM).
Remark 1.2
System (SM) is based on assuming that the noise affects parameters \(a_{11}\) and \(a_{22}\). In fact, the noise may affect other parameters in (SM), which results in other types of stochastic models, which are the future research topics; for more details, see [36].
Remark 1.3
For the analysis of population dynamical problems of (SM), two difficult issues arise: (i) how to handle the delays in the given model and (ii) how to handle the nonlinear terms in (SM). To deal with these problems, the construction of a Lyapunov functional V is quite crucial, and it is introduced in Sections 3 and 4.
2 Positive and global solutions
In order for a stochastic differential delay equation to have a unique global (i.e., no explosion in a finite time) solution for any given initial data, the coefficients of the equation are generally required to satisfy the linear growth condition and local Lipschitz condition [36]. However, the coefficients of system (SM) do not satisfy the linear growth condition, though they are locally Lipschitz continuous, so the solution of system (SM) may explode at a finite time. In this section we shall show that under simple hypothesis the solution of system (SM) is not only positive but will also not explode to infinity at any finite time.
Theorem 2.1
For any given initial data \(\{(x_{1}(\theta),x_{2}(\theta))^{\top}:-\tau\leq\theta\leq0\}=\xi\in C_{\mathcal{F}_{0}}^{b}([-\tau,0]; R_{+}^{0}\times R_{+}^{0})\), where \(R_{+}^{0}=(0,+\infty)\). If \(a_{21}\leq4\), there is a unique positive local solution \((x_{1}(t),x_{2}(t))\) to (SM) on \(t\geq-\tau\) with satisfying initial condition ξ and the solution will remain in \(R_{+}^{2}\) with probability 1.
Proof
Remark 2.1
It is well known that systems (1.2) and (1.3) may explode to infinity at a finite time for some system parameters, see [37]. However, the explosion will no longer happen as long as there is noise. In other words, Theorem 2.1 reveals the important property that the environmental noise suppresses the explosion for the delay equation.
3 Stochastically ultimate boundedness
In this section we shall investigate the stochastically ultimate boundedness of system (SM). The following theorem gives a sufficient criterion for the stochastically ultimate boundedness of population.
Lemma 3.1
Proof
Theorem 3.1
Let \(\theta\in(0, 1)\) and \(\theta a_{21}\geq n\). System (SM) is stochastically ultimately bounded.
Proof
4 Almost surely asymptotic properties
In this section, we study the pathwise properties of system (SM).
Theorem 4.1
Proof
Remark 4.1
Theorem 4.2
Proof
5 Numerical simulations
6 Conclusions and future directions
There are still many interesting and challenging questions that need to be studied. In this paper, we only consider the growth rate \(a_{11}\), \(a_{22}\) to be stochastic; other parameters, for example, \(r_{i}\), \(i=1,2\), is stochastic, which is not studied. We wish that such questions will be investigated by some authors.
Declarations
Acknowledgements
This paper is supported by Postdoctoral Foundation of Jiangsu (1402113C) and Postdoctoral Foundation of China (2014M561716).
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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