- Research
- Open Access
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
- stochastic perturbation
- global existence
- ultimately bounded
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
References
- Skalski, GT, Gilliam, JF: Functional responses with predator interference: viable alternatives to the Holling II model. Ecology 82, 3083-3092 (2001) View ArticleGoogle Scholar
- Hassell, M: Mutual interference between searching insect parasites. J. Anim. Ecol. 40, 473-486 (1971) View ArticleGoogle Scholar
- Beddington, JR: Mutual interference between parasites or predators and its effect on searching efficiency. J. Anim. Ecol. 44, 331-341 (1975) View ArticleGoogle Scholar
- Crowley, P, Martin, P: Functional response and interference within and between year classes of dragonfly. J. North Am. Benthol. Soc. 8, 211-221 (1989) View ArticleGoogle Scholar
- Liu, Z, Yuan, R: Stability and bifurcation in a delayed predator-prey system with Beddington-DeAngelis functional response. J. Math. Anal. Appl. 296, 521-537 (2004) View ArticleMATHMathSciNetGoogle Scholar
- Liu, S, Zhang, J: Coexistence and stability of predator-prey model with Beddington-DeAngelis functional response and stage structure. J. Math. Anal. Appl. 342, 446-460 (2008) View ArticleMATHMathSciNetGoogle Scholar
- Zhao, M, Lv, S: Chaos in a three-species food chain model with a Beddington-DeAngelis functional response. Chaos Solitons Fractals 40, 2305-2316 (2009) View ArticleMATHMathSciNetGoogle Scholar
- Fan, M, Kuang, Y: Dynamics of a non-autonomous predator-prey system with the Beddington-DeAngelis functional response. J. Math. Anal. Appl. 295, 15-39 (2004) View ArticleMATHMathSciNetGoogle Scholar
- Hwang, TW: Global analysis of the predator-prey system with Beddington-DeAngelis functional response. J. Math. Anal. Appl. 281, 395-401 (2003) View ArticleMATHMathSciNetGoogle Scholar
- Hwang, TW: Uniqueness of limit cycles of the predator-prey system with Beddington-DeAngelis functional response. J. Math. Anal. Appl. 290, 113-122 (2004) View ArticleMATHMathSciNetGoogle Scholar
- Guo, G, Wu, J: Multiplicity and uniqueness of positive solutions for a predator-prey model with B-D functional response. Nonlinear Anal. 72, 1632-1646 (2010) View ArticleMATHMathSciNetGoogle Scholar
- Kuang, Y: Delay Differential Equations with Applications in Population Dynamics. Academic Press, New York (1993) MATHGoogle Scholar
- Macdonald, N: Biological Delay Systems: Linear Stability Theory. Cambridge University Press, Cambridge (1989) MATHGoogle Scholar
- Gopalsamy, K: Stability and Oscillations in Delay Differential Equations of Population Dynamics. Kluwer Academic, Boston (1992) View ArticleMATHGoogle Scholar
- Fan, M, Wang, K: Existence and global attractivity of positive periodic solutions of periodic n-species Lotka-Volterra competition systems with several deviating arguments. Math. Biosci. 160, 47-61 (1999) View ArticleMATHMathSciNetGoogle Scholar
- Xu, R, Chaplain, M, Davidson, F: Periodic solutions for a delayed predator-prey model of prey dispersal in two-patch environments. Nonlinear Anal., Real World Appl. 5, 183-206 (2004) View ArticleMATHMathSciNetGoogle Scholar
- Egami, C, Hirano, N: Periodic solutions in a class of periodic delay predator-prey systems. Yokohama Math. J. 51, 45-61 (2004) MATHMathSciNetGoogle Scholar
- Lu, S: On the existence of positive periodic solutions to a Lotka-Volterra cooperative population model with multiple delays. Nonlinear Anal. 68, 1746-1753 (2008) View ArticleMATHMathSciNetGoogle Scholar
- Lu, S: On the existence of positive periodic solutions for neutral functional differential equation with multiple deviating arguments. J. Math. Anal. Appl. 280, 321-333 (2003) View ArticleMATHMathSciNetGoogle Scholar
- Lu, S, Chen, L: The problem of existence of periodic solutions for neutral functional differential system with nonlinear difference operator. J. Math. Anal. Appl. 387, 1127-1136 (2012) View ArticleMATHMathSciNetGoogle Scholar
- Lu, S, Xu, Y, Xia, D: New properties of the D-operator and its applications on the problem of periodic solutions to neutral functional differential system. Nonlinear Anal. 74, 3011-3021 (2011) View ArticleMATHMathSciNetGoogle Scholar
- Ikeda, N, Watanabe, S: Stochastic Differential Equations and Diffusion Processes. Kluwer Academic, Dordrecht (1981) MATHGoogle Scholar
- Ding, X, Jiang, J: Positive periodic solutions in delayed Gause-type predator-prey systems. J. Math. Anal. Appl. 339, 1220-1230 (2008) View ArticleMATHMathSciNetGoogle Scholar
- Kan-on, Y, Mimura, M: Singular perturbation approach to a 3-component reaction-diffusion system arising in population dynamics. SIAM J. Math. Anal. 29, 1519-1536 (1998) View ArticleMATHMathSciNetGoogle Scholar
- Pang, P, Wang, M: Qualitative analysis of a ratio-dependent predator-prey system with diffusion. Proc. R. Soc. Edinb. A 133, 919-942 (2003) View ArticleMATHMathSciNetGoogle Scholar
- May, R: Stability and Complexity in Model Ecosystems. Princeton University Press, New York (1992) Google Scholar
- Liu, M, Wang, K: Extinction and permanence in a stochastic nonautonomous population system. Appl. Math. Lett. 23, 1464-1467 (2010) View ArticleMATHMathSciNetGoogle Scholar
- Liu, M, Wang, K: Stationary distribution, ergodicity and extinction of a stochastic generalized logistic system. Appl. Math. Lett. 25, 1980-1985 (2012) View ArticleMATHMathSciNetGoogle Scholar
- Liu, M, Wang, K: Survival analysis of a stochastic cooperation system in a polluted environment. J. Biol. Syst. 19, 183-204 (2013) View ArticleMathSciNetMATHGoogle Scholar
- Jiang, D, Ji, C, Li, X, O’Regan, D: Analysis of autonomous Lotka-Volterra competition systems with random perturbation. J. Math. Anal. Appl. 390, 582-595 (2012) View ArticleMATHMathSciNetGoogle Scholar
- Ikeda, N, Watanabe, S: A comparison theorem for solutions of stochastic differential equations and its applications. Osaka J. Math. 14, 619-633 (1977) MATHMathSciNetGoogle Scholar
- Huang, LC: Stochastic delay population systems. Appl. Anal. 88, 1303-1320 (2009) View ArticleMathSciNetMATHGoogle Scholar
- Li, X, Mao, X: Population dynamical behavior of non-autonomous Lotka-Volterra competitive system with random perturbation. Discrete Contin. Dyn. Syst. 24, 523-545 (2009) View ArticleMATHMathSciNetGoogle Scholar
- Mao, X, Marion, G, Renshaw, E: Environmental noise suppresses explosion in population dynamics. Stoch. Process. Appl. 97, 95-110 (2002) View ArticleMATHMathSciNetGoogle Scholar
- Liu, M, Qin, H, Wang, K: A remark on a stochastic predator-prey system with time delays. Appl. Math. Lett. 26, 318-323 (2011) View ArticleMathSciNetGoogle Scholar
- Mao, X: Stochastic Differential Equations and Applications. Ellis Horwood, Chichester (1997) MATHGoogle Scholar
- Bahar, A, Mao, X: Stochastic delay Lotka-Volterra model. J. Math. Anal. Appl. 292, 364-380 (2004) View ArticleMATHMathSciNetGoogle Scholar
- Kloeden, PE, Shardlow, T: The Milstein scheme for stochastic delay differential equations without using anticipative calculus. Stoch. Anal. Appl. 30, 181-202 (2012) View ArticleMATHMathSciNetGoogle Scholar