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A stochastic competing-species model and ergodicity

Part of: Stochastic processes Stochastic analysis

Published online by Cambridge University Press:  14 July 2016

Zengjing Chen  and
Reg Kulperger
Zengjing Chen*
Affiliation:
Shandong University
Reg Kulperger*
Affiliation:
The University of Western Ontario
*
∗Postal address: Department of Mathematics, Shandong University, Jinan, 250100, P. R. China.
∗∗Postal address: Department of Statistical and Actuarial Science, The University of Western Ontario, London, Ontario N6A 5B7, Canada. Email address: kulperger@uwo.ca
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Abstract

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We consider a classic competing-species model with the rates changed to include Gaussian white noise. We show that if the noise is not too large, then the stochastic version is ergodic. An explicit relation between the noise and the original competing-species parameters gives a sufficient condition for ergodicity.

Keywords

Competing speciesstochasticItô processBrownian motionergodic

MSC classification

Primary: 60H10: Stochastic ordinary differential equations
Secondary: 60G99: None of the above, but in this section
Type
Research Papers
Information
Journal of Applied Probability , Volume 42 , Issue 3 , September 2005 , pp. 738 - 753
Copyright
© Applied Probability Trust 2005 

References

Bhattacharya, R. N. (1978). Criteria for recurrence and existence of invariant measures for multidimensional diffusions. Ann. Prob. 6, 541–553.CrossRefGoogle Scholar
Chen, Z. and Kulperger, R. (2003). A stochastic prey predator process and damping. In preparation.Google Scholar
Chessa, S. and Fujita, Y.H. (2002). The stochastic equation of predator-prey population dynamics. Boll. Unione Mat. Ital. Sez. B. Artic. Ric. Mat. 5, 789–804 (in Italian).Google Scholar
Friedman, A. (1973). Wandering out to infinity of diffusion processes. Trans. Am. Math. Soc. 184, 185–203.Google Scholar
Gard, T. C. (2000). Transient effects of stochastic multi-population models. In Electron. J. Differential Equat., Conf. 05 (Proc. Conf. Nonlinear Differential Equations, Coral Gables, FL, 1999), eds Cantrell, S. and Cosner, C., Texas State University, pp. 81–90.Google Scholar
Gard, T. and Kanna, D. (1976). On a stochastic differential equation modeling of prey–predator evolution. J. Appl. Prob. 13, 429–443.CrossRefGoogle Scholar
Karlin, S. and Taylor, H. (1981). A Second Course in Stochastic Processes. Academic Press, New York.Google Scholar
Khasminskii, R. Z. and Klebaner, F. C. (2001). Long term behavior of solutions of the Lotka–Volterra system under small random perturbations. Ann. Appl. Prob. 11, 952–963.Google Scholar
King, A. et al. (1996). Weakly dissipative predator–prey systems. Bull. Math. Biology 58, 835–859.Google Scholar
Mangel, M. and Ludwig, D. (1977). Probability of extinction in a stochastic competition. SIAM J. Appl. Math. 33, 256–266.CrossRefGoogle Scholar
Manthey, R. and Maslowski, B. (2002). A random continuous model for two interacting populations. Appl. Math. Optimization 45, 213–236.Google Scholar
Mao, X., Marion, G. and Renshaw, E. (2002). Environmental Brownian noise suppresses explosions in population dynamics. Stoch. Process. Appl. 97, 95–110.Google Scholar
Renshaw, E. (1991). Modelling Biological Populations in Space and Time. Cambridge University Press.CrossRefGoogle Scholar
Rudnicki, R. (2003). Long-time behaviour of a stochastic prey–predator model. Stoch. Process. Appl. 108, 93–107.Google Scholar
Spagnola, B. and La Barbera, A. (2002). Role of the noise on the transient dynamics of an ecosystem of interacting species. Physica A 315, 114–124.CrossRefGoogle Scholar