Abstract
In this paper, we propose and investigate a stochastic two-prey one-predator model. Firstly, under some simple assumptions, we show that for each species x i , i=1,2,3, there is a π i which is represented by the coefficients of the model. If π i <1, then x i goes to extinction (i.e., lim t→+∞ x i (t)=0); if π i >1, then x i is stable in the mean (i.e., \(\lim_{t\rightarrow+\infty}t^{-1} \int_{0}^{t}x_{i}(s)\,\mathrm {d}s=\mbox{a positive constant}\)). Secondly, we prove that there is a stationary distribution to this model and it has the ergodic property. Thirdly, we establish the sufficient conditions for global asymptotic stability of the positive solution. Finally, we introduce some numerical simulations to illustrate the theoretical results.
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Ahmad, S., Lazer, A.C.: Average conditions for global asymptotic stability in a nonautonomous Lotka–Volterra system. Nonlinear Anal. 40, 37–49 (2000)
Ahmad, S., Stamova, I.M.: Almost necessary and sufficient conditions for survival of species. Nonlinear Anal. 5, 219–229 (2004)
Bahar, A., Mao, X.: Stochastic delay population dynamics. Int. J. Pure Appl. Math. 11, 377–400 (2004)
Barbalat, I.: Systems d’equations differentielles d’oscillations nonlineaires. Rev. Roum. Math. Pures Appl. 4, 267–270 (1959)
Beddington, J.R., May, R.M.: Harvesting natural populations in a randomly fluctuating environment. Science 197, 463–465 (1977)
Braumann, C.A.: Itô versus Stratonovich calculus in random population growth. Math. Biosci. 206, 81–107 (2007)
Chen, L., Chen, J.: Nonlinear Biological Dynamical System. Science Press, Beijing (1993)
Cheng, S.: Stochastic population systems. Stoch. Anal. Appl. 27, 854–874 (2009)
Feng, W.: Coexistence, stability, and limiting behavior in a one-predator–two-prey model. J. Math. Anal. Appl. 179, 592–609 (1993)
Freedman, H., Waltman, P.: Mathematical analysis of some three-species food-chain models. Math. Biosci. 33, 257–276 (1977)
Freedman, H., Waltman, P.: Persistence in models of three interacting predator–prey populations. Math. Biosci. 68, 213–231 (1984)
Gard, T.: Introduction to Stochastic Differential Equations. Dekker, New York (1988)
Has’minskii, R.: Stochastic Stability of Differential Equations. Sijthoff & Noordhoff, Alphen aan den Rijn (1980)
Higham, D.: An algorithmic introduction to numerical simulation of stochastic differential equations. SIAM Rev. 43, 525–546 (2001)
Hu, G., Wang, K.: The estimation of probability distribution of SDE by only one sample trajectory. Comput. Math. Appl. 62, 1798–1806 (2011)
Hutson, V., Vickers, G.: A criterion for permanent co-existence of species, with an application to a two-prey one-predator system. Math. Biosci. 63, 253–269 (1983)
Jiang, D., Shi, N., Li, X.: Global stability and stochastic permanence of a non-autonomous logistic equation with random perturbation. J. Math. Anal. Appl. 340, 588–597 (2008)
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)
Karatzas, I., Shreve, S.: Brownian Motion and Stochastic Calculus. Springer, Berlin (1991)
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)
Li, X., Gray, A., Jiang, D., Mao, X.: Sufficient and necessary conditions of stochastic permanence and extinction for stochastic logistic populations under regime switching. J. Math. Anal. Appl. 376, 11–28 (2011)
Liu, M., Wang, K.: Survival analysis of a stochastic cooperation system in a polluted environment. J. Biol. Syst. 19, 183–204 (2011)
Liu, M., Wang, K., Wu, Q.: Survival analysis of stochastic competitive models in a polluted environment and stochastic competitive exclusion principle. Bull. Math. Biol. 73, 1969–2012 (2011)
Luo, Q., Mao, X.: Stochastic population dynamics under regime switching II. J. Math. Anal. Appl. 355, 577–593 (2009)
Øsendal, B.: Stochastic Differential Equations: An Introduction with Applications, 6th edn. Springer, Berlin (2003)
Sridhara, R., Watson, R.: Stochastic three species systems. J. Math. Biol. 28, 595–607 (1990)
Strang, G.: Linear Algebra and Its Applications. Thomson Learning, New York (1988)
Takeuchi, Y., Adachi, N.: Existence of bifurcation of stable equilibrium in two-prey, one-predator communities. Bull. Math. Biol. 45, 877–900 (1983)
Ton, T.V.: Survival of three species in a nonautonomous Lotka–Volterra system. J. Math. Anal. Appl. 362, 427–437 (2010)
Vance, R.: Predation and resource partitioning in one predator–two prey model communities. Am. Nat. 112, 797–813 (1978)
Zhu, C., Yin, G.: Asymptotic properties of hybrid diffusion systems. SIAM J. Control Optim. 46, 1155–1179 (2007)
Zhu, C., Yin, G.: On hybrid competitive Lotka–Volterra ecosystems. Nonlinear Anal. 71, e1370–e1379 (2009)
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The authors thank the editors and reviewers for their important and valuable comments. Authors were supported by the NSFC of P.R. China (Nos. 11171081 and 11171056).
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Communicated by P.K. Maini.
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Liu, M., Wang, K. Dynamics of a Two-Prey One-Predator System in Random Environments. J Nonlinear Sci 23, 751–775 (2013). https://doi.org/10.1007/s00332-013-9167-4
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DOI: https://doi.org/10.1007/s00332-013-9167-4
Keywords
- Two-prey one-predator model
- Random perturbations
- Extinction
- Stationary distribution
- Global asymptotic stability