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Conditions for permanence and ergodicity of certain stochastic predator–prey models

Published online by Cambridge University Press:  24 March 2016

Nguyen Huu Du ,
Dang Hai Nguyen  and
G. George Yin
Nguyen Huu Du
Affiliation:
Department of Mathematics, Mechanics and Informatics, Hanoi National University, 334 Nguyen Trai, Thanh Xuan, Hanoi, Vietnam. Email address: dunh@vnu.edu.vn
Dang Hai Nguyen
Affiliation:
Department of Mathematics, Wayne State University, Detroit, MI 48202, USA. Email address: dangnh.maths@gmail.com
G. George Yin*
Affiliation:
Department of Mathematics, Wayne State University, Detroit, MI 48202, USA.
*
**** Email address: gyin@math.wayne.edu
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Abstract

In this paper we derive sufficient conditions for the permanence and ergodicity of a stochastic predator–prey model with a Beddington–DeAngelis functional response. The conditions obtained are in fact very close to the necessary conditions. Both nondegenerate and degenerate diffusions are considered. One of the distinctive features of our results is that they enable the characterization of the support of a unique invariant probability measure. It proves the convergence in total variation norm of the transition probability to the invariant measure. Comparisons to the existing literature and matters related to other stochastic predator–prey models are also given.

Keywords

Ergodicityextinctionpermanencepredator–preyBeddington–DeAngelis functional responsestationary distribution
Type
Research Papers
Information
Journal of Applied Probability , Volume 53 , Issue 1 , March 2016 , pp. 187 - 202
Copyright
Copyright © Applied Probability Trust 2016 

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References

[1]Beddington, J. R. (1975). Mutual interference between parasites or predators and its effect on searching efficiency. J. Animal Ecol. 44, 331–340. CrossRefGoogle Scholar
[2]Bellet, L. R. (2006). Ergodic properties of Markov processes. In Open Quantum Systems II, Springer, Berlin, pp. 1–39. Google Scholar
[3]DeAngelis, D. L., Goldstein, R. A. and O'Neill, R. V. (1975). A model for tropic interaction. Ecology 56, 881–892. CrossRefGoogle Scholar
[4]Du, N. H. and Sam, V. H. (2006). Dynamics of a stochastic Lotka–Volterra model perturbed by white noise. J. Math. Anal. Appl. 324, 82–97. Google Scholar
[5]Hofbauer, J. and Sigmund, K. (1998). Evolutionary Games and Population Dynamics. Cambridge University Press. CrossRefGoogle Scholar
[6]Holling, C. S. (1959). The components of predation as revealed by a study of small-mammal predation of the European pine sawfly. Canadian Entomologist 91, 293–320. Google Scholar
[7]Ikeda, N. and Watanabe, S. (1989). Stochastic Differential Equations and Diffusion Processes, 2nd edn. North-Holland, Amsterdam. Google Scholar
[8]Ji, C. and Jiang, D. (2011). Dynamics of a stochastic density dependent predator–prey system with Beddington–DeAngelis functional response. J. Math. Anal. Appl. 381, 441–453. Google Scholar
[9]Ji, C., Jiang, D. and Li, X. (2011). Qualitative analysis of a stochastic ratio-dependent predator–prey system. J. Comput. Appl. Math. 235, 1326–1341. Google Scholar
[10]Ji, C., Jiang, D. and Shi, N. (2009). Analysis of a predator–prey model with modified Leslie–Gower and Holling-type II schemes with stochastic perturbation. J. Math. Anal. Appl. 359, 482–498. Google Scholar
[11]Jurdjevic, V. (1997). Geometric Control Theory (Camb. Stud. Adv. Math. 52). Cambridge University Press. Google Scholar
[12]Ichihara, K. and Kunita, H. (1974). A classification of the second order degenerate elliptic operators and its probabilistic characterization. Z. Wahrscheinlichkeitsth. 30, 235–254. Google Scholar
[13]Ichihara, K. and Kunita, H. (1977). Supplements and corrections to the paper: A classification of the second order degenerate elliptic operators and its probabilistic characterization. Z. Wahrscheinlichkeitsth. 39, 81–84. Google Scholar
[14]Khas'minskii, R. Z. (1960). Ergodic properties of recurrent diffusion processes and stabilization of the solution to the Cauchy problem for parabolic equations. Theory Prob. Appl. 5, 179–196. Google Scholar
[15]Kliemann, W. (1987). Recurrence and invariant measures for degenerate diffusions. Ann. Prob. 15, 690–707. Google Scholar
[16]Liu, M. and Wang, K. (2011). Global stability of a nonlinear stochastic predator–prey system with Beddington–DeAngelis functional response. Commun. Nonlinear Sci. Numer. Simul. 16, 1114–1121. Google Scholar
[17]Liu, Z., Shi, N., Jiang, D. and Ji, C. (2012). The asymptotic behavior of a stochastic predator–prey system with Holling II functional response. Abstr. Appl. Anal. 2012, 801812. Google Scholar
[18]Lv, J. and Wang, K. (2011). Asymptotic properties of a stochastic predator–prey system with Holling II functional response. Commun. Nonlinear Sci. Numer. Simul. 16, 4037–4048. CrossRefGoogle Scholar
[19]Mao, X., Sabanis, S. and Renshaw, E. (2003). Asymptotic behaviour of the stochastic Lotka–Volterra model. J. Math. Anal. Appl. 287, 141–156. Google Scholar
[20]Meyn, S. P. and Tweedie, R. L. (1993). Stability of Markovian processes. III. Foster–Lyapunov criteria for continuous-time processes. Adv. Appl. Prob. 25, 518–548. Google Scholar
[21]Rudnicki, R. (2003). Long-time behaviour of a stochastic prey–predator model. Stoch. Process. Appl. 108, 93–107. CrossRefGoogle Scholar
[22]Skorokhod, A. V. (1989). Asymptotic Methods in the Theory of Stochastic Differential Equations (Transl. Math. Monogr. 78). American Mathematical Society, Providence, RI. Google Scholar
[23]Stroock, D. W. and Varadhan, S. R. S. (1972). On the support of diffusion processes with applications to the strong maximum principle. In Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability, Vol. III, Probability Theory, University of California Press, pp. 333–359. Google Scholar
[24]Tuan, H. T., Dang, N. H. and Vu, V. K. (2012). Dynamics of a stochastic predator–prey model with Beddington DeAngelis functional response. SCIENTIA 22, 75–84. Google Scholar
[25]Zhang, X.-C., Sun, G.-Q. and Jin, Z. (2012). Spatial dynamics in a predator–prey model with Beddington–DeAngelis functional response. Physical Rev. E 85, 021924. Google Scholar
[26]Zhu, C. and Yin, G. (2009). On competitive Lotka–Volterra model in random environments. J. Math. Anal. Appl. 357, 154–170. Google Scholar