Dynamics of stochastic non-autonomous plankton-allelopathy system
- Xiaoqing Wen^{1},
- Hongwei Yin^{1} and
- Ying Wei^{2}Email author
https://doi.org/10.1186/s13662-015-0570-z
© Wen et al. 2015
Received: 17 November 2014
Accepted: 10 July 2015
Published: 21 October 2015
Abstract
In this paper, a stochastic non-autonomous plankton-allelopathy system is investigated. The existence and uniqueness of globally positive solution to this system are proved. A sufficient criterion for extinction is established. Globally asymptotical stability of this system is obtained. Some numerical simulations illustrate our main results. Finally, unavoidable conclusions are drawn.
Keywords
1 Introduction
To our knowledge, the stochastic plankton-allelopathy system (1.2) has not yet been investigated. In this paper we make the first attempt to fill this gap and study some mathematical properties of (1.2), such as the existence of a global unique positive solution to (1.2), extinction, and globally asymptotical stability. The rest of the paper is organized as follows. In Section 2, the existence and uniqueness of the global positive solution to (1.2) are proved. In Section 3, sufficient conditions for the extinction of the system (1.2) are established. In Section 4, we show the globally asymptotical stability of the system (1.2). Finally, we present some numerical simulations to confirm our results.
2 Global unique positive solution
Lemma 2.1
There exists a unique positive local solution \((x(t),y(t))\) almost surely (a.s.) \(t\in[0, \tau_{e})\) to the system (1.2) with the initial value \(x_{0},y_{0}>0\), where \(\tau_{e}\) is the explosion time.
Proof
Lemma 2.1 only tells us that there is a unique positive local solution model (2.1). In order to show that this solution is global, we need to show that \(\tau_{e}=\infty \) a.s.
Lemma 2.2
For any given initial value \((x_{0},y_{0})\in\mathbb{R}_{+}^{2}\), there is a unique solution \((x(t),y(t))\) on \(t\geq0\) and the solution will remain in \(\mathbb{R}_{+}^{2}\) a.s., where \(\mathbb{R}_{+}^{2}=\{(x,y)\in\mathbb {R}_{+}^{2}|x,y>0\}\).
Proof
3 Extinction
In this section, we shall address the extinction of the system (1.2). The following theorem shows that the noise may induce extinction of (1.2).
Theorem 3.1
Let the conditions of \(r_{1}(t)-0.5\sigma_{1}^{2}(t)<0\) and \(r_{2}(t)-0.5\sigma _{2}^{2}(t)<0\) hold, then the population x and y will become extinct exponentially a.s.
Proof
4 Globally asymptotical stability
In this section, we will establish a sufficient criterion for the globally asymptotical stability of the system (1.2).
Definition 4.1
Now, we present some important lemmas.
Lemma 4.2
Proof
On the other hand, by the same method we can see that there exists \(G(p)\) such that \(E(y^{p}(t))\leq G(p)\). □
Lemma 4.3
Lemma 4.4
Let \((x(t),y(t))\) be a solution of (1.2) on \(t\geq0\), then almost every sample path of \(x(t),y(t)\) is uniformly continuous.
Proof
Lemma 4.5
[13]
Let f be a non-negative function on \(\mathbb{R}_{+}\) such that f is integrable and is uniformly continuous. Then \(\lim_{t\rightarrow\infty } f(t)=0\).
Theorem 4.6
The system (1.2) is globally asymptotically stable.
Proof
5 Numerical simulations
6 Conclusion
This paper has studied the existence and uniqueness of the positive solution for a stochastic non-autonomous plankton-allelopathy system. Moreover, we have obtained the sufficient conditions of extinction for this system. Furthermore, the globally asymptotical stability of this system has been proved in this paper. Finally, we illustrate some results by numerical simulations for some periodic parameters. These results are useful because the extinction thresholds are very important for the system of species. In further investigation, we may consider some more realistic and complex models such as incorporating spatial diffusion in this stochastic system. The spatial factors can affect the dynamical properties of the stochastic plankton-allelopathy system.
Declarations
Acknowledgements
The authors would like to thank the anonymous referees for their carefulness and helpful comments, which greatly improved this paper. This work was supported by the NSF of Jiangxi (No. 20151BAB212011).
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
- Chattopadhyay, J: Effect of toxic substances on a two-species competitive system. Ecol. Model. 84, 287-289 (1996) View ArticleGoogle Scholar
- Mukhopadhyay, A, Chattopadhyay, J, Tapaswi, PK: A delay differential equations model of plankton allelopathy. Math. Biosci. 149, 167-189 (1998) MATHMathSciNetView ArticleGoogle Scholar
- Zhen, J, Ma, Z: Periodic solutions for delay differential equations model of plankton allelopathy. Comput. Math. Appl. 44, 491-500 (2002) MATHMathSciNetView ArticleGoogle Scholar
- He, M, Chen, F, Li, Z: Almost periodic solution of an impulsive differential equation model of plankton allelopathy. Nonlinear Anal., Real World Appl. 11, 2296-2301 (2010) MATHMathSciNetView ArticleGoogle Scholar
- Jia, Y, Wu, J, Xu, HK: Spatial pattern in a ecosystem of phytoplankton-nutrient from remote sensing. J. Math. Anal. Appl. 402, 23-34 (2013) MATHMathSciNetView ArticleGoogle Scholar
- Li, Z, Chen, FD, He, MX: Asymptotic behavior of the reaction-diffusion model of plankton allelopathy with nonlocal delays. Nonlinear Anal., Real World Appl. 12, 1748-1758 (2011) MATHMathSciNetView ArticleGoogle Scholar
- Tian, C, Zhang, L: Delay-driven irregular spatiotemporal patterns in a plankton system. Phys. Rev. E 88, 012713 (2013) View ArticleGoogle Scholar
- Du, YH, Hsu, SB: On a nonlocal reaction-diffusion problem arising from the modeling of phytoplankton growth. SIAM J. Math. Anal. 42, 1305-1333 (2010) MATHMathSciNetView ArticleGoogle Scholar
- Du, B: Existence, extinction and global asymptotical stability of a stochastic predator-prey model with mutual interference. J. Appl. Math. Comput. 46, 79-91 (2014) MATHMathSciNetView ArticleGoogle Scholar
- Liu, M, Wang, K: Global stability of a nonlinear stochastic predator-prey system with Beddington-Deangelis functional response. Commun. Nonlinear Sci. Numer. Simul. 16, 1114-1121 (2011) MATHMathSciNetView ArticleGoogle Scholar
- Mao, X, Li, X: Population dynamical behavior of non-autonomous Lotka-Volterra competitive system with random perturbation. Discrete Contin. Dyn. Syst. 24, 523-545 (2009) MATHMathSciNetView ArticleGoogle Scholar
- Watanabe, S, Ikeda, N: Stochastic Differential Equations and Diffusion Processes. Elsevier, Amsterdam (1981) MATHGoogle Scholar
- Barbalat, I: Systems d’équations différential d’oscillations nonlinéaires. Rev. Roum. Math. Pures Appl. 4, 267-270 (1959) MATHMathSciNetGoogle Scholar
- Higham, DJ: An algorithmic introduction to numerical simulation of stochastic differential equations. SIAM Rev. 43, 525-546 (2001) MATHMathSciNetView ArticleGoogle Scholar