Journal of Applied Probability

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

Nguyen Huu Du, Dang Hai Nguyen, and G. George Yin

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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.

Article information

Source
J. Appl. Probab. Volume 53, Number 1 (2016), 187-202.

Dates
First available in Project Euclid: 8 March 2016

Permanent link to this document
https://projecteuclid.org/euclid.jap/1457470568

Mathematical Reviews number (MathSciNet)
MR3471956

Zentralblatt MATH identifier
1338.34091

Subjects
Primary: 34C12: Monotone systems 60H10: Stochastic ordinary differential equations [See also 34F05] 92D25: Population dynamics (general)

Keywords
Ergodicity extinction permanence predator-prey Beddington-DeAngelis functional response stationary distribution

Citation

Du, Nguyen Huu; Nguyen, Dang Hai; Yin, G. George. Conditions for permanence and ergodicity of certain stochastic predator-prey models. J. Appl. Probab. 53 (2016), no. 1, 187--202. https://projecteuclid.org/euclid.jap/1457470568.


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References

  • Beddington, J. R. (1975). Mutual interference between parasites or predators and its effect on searching efficiency. J. Animal Ecol. 44, 331–340.
  • Bellet, L. R. (2006). Ergodic properties of Markov processes. In Open Quantum Systems II, Springer, Berlin, pp. 1–39.
  • DeAngelis, D. L., Goldstein, R. A. and O'Neill, R. V. (1975). A model for tropic interaction. Ecology 56, 881–892.
  • 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.
  • Hofbauer, J. and Sigmund, K. (1998). Evolutionary Games and Population Dynamics. Cambridge University Press.
  • 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.
  • Ikeda, N. and Watanabe, S. (1989). Stochastic Differential Equations and Diffusion Processes, 2nd edn. North-Holland, Amsterdam.
  • 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.
  • 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.
  • 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.
  • Jurdjevic, V. (1997). Geometric Control Theory. (Camb. Stud. Adv. Math. 52). Cambridge University Press.
  • Ichihara, K. and Kunita, H. (1974). A classification of the second order degenerate elliptic operators and its probabilistic characterization. Z. Wahrscheinlichkeitsth. 30, 235–254.
  • 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.
  • Khas$^\prime$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.
  • Kliemann, W. (1987). Recurrence and invariant measures for degenerate diffusions. Ann. Prob. 15, 690–707.
  • 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.
  • 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.
  • 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.
  • Mao, X., Sabanis, S. and Renshaw, E. (2003). Asymptotic behaviour of the stochastic Lotka–Volterra model. J. Math. Anal. Appl. 287, 141–156.
  • 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.
  • Rudnicki, R. (2003). Long-time behaviour of a stochastic prey–predator model. Stoch. Process. Appl. 108, 93–107.
  • Skorokhod, A. V. (1989). Asymptotic Methods in the Theory of Stochastic Differential Equations (Transl. Math. Monogr. 78). American Mathematical Society, Providence, RI.
  • 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.
  • 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.
  • 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.
  • Zhu, C. and Yin, G. (2009). On competitive Lotka–Volterra model in random environments. J. Math. Anal. Appl. 357, 154–170.