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A stochastic two species competition model: nonequilibrium fluctuation and stability. (English) Zbl 1229.92080
Summary: The object of this paper is to study the stability behaviours of deterministic and stochastic versions of a two-species symmetric competition model. The logistic parameters of the competitive species are perturbed by colored noises or Ornstein-Uhlenbeck processes due to the random environment. The Fokker-Planck equation has been used to obtain probability density functions. We have also discussed the relationship between stability behaviours of this model in a deterministic environment and the corresponding model in a stochastic environment.

92D40 Ecology
60J70 Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.)
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
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[1] J. Roerdink and A. Weyland, “A generalized Fokker-Planck equation in the case of the Volterra model,” Bulletin of Mathematical Biology, vol. 43, no. 1, pp. 69-79, 1981. · Zbl 0449.92023
[2] R. M. Nisbet and W. S. C. Gurney, Modelling Fluctuating Populations, Wiley-Interscience, New York, NY, USA, 1982. · Zbl 0593.92013
[3] C. W. Gardiner, Handbook of Stochastic Methods, vol. 13 of Springer Series in Synergetics, Springer, Berlin, Germany, 1983. · Zbl 0515.60002
[4] R. Wang and Z. Zhang, “Exact stationary solutions of the Fokker-Planck equation for nonlinear oscillators under stochastic parametric and external excitations,” Nonlinearity, vol. 13, no. 3, pp. 907-920, 2000. · Zbl 0958.82039
[5] Z. Zhang, R. Wang, and K. Yasuda, “On joint stationary probability density function of nonlinear dynamic systems,” Acta Mechanica, vol. 130, no. 1-2, pp. 29-39, 1998. · Zbl 0917.60074
[6] R. Wang, K. Yasuda, and Z. Zhang, “A generalized analysis technique of the stationary FPK equation in nonlinear systems under Gaussian white noise excitations,” International Journal of Engineering Science, vol. 38, no. 12, pp. 1315-1330, 2000. · Zbl 1210.70024
[7] R. Wang, Y.-B. Duan, and Z. Zhang, “Resonance analysis of the finite-damping nonlinear vibration system under random disturbances,” European Journal of Mechanics. A. Solids, vol. 21, no. 6, pp. 1083-1088, 2002. · Zbl 1027.74032
[8] R. M. May, “Stability in randomly fluctuating versus deterministic environments,” The American Naturalist, vol. 107, pp. 621-650, 1973.
[9] G. P. Samanta, “A two-species competitive system under the influence of toxic substances,” Applied Mathematics and Computation, vol. 216, no. 1, pp. 291-299, 2010. · Zbl 1184.92058
[10] S. B. Hsu, S. P. Hubbell, and P. A. Waltman, “A contribution to the theory of competing predators,” Ecological Monographs, vol. 48, pp. 337-349, 1979.
[11] J. D. Murray, Mathematical Biology, vol. 19 of Biomathematics, Springer, New York, NY, USA, 2nd edition, 1993. · Zbl 0779.92001
[12] G. E. Uhlenbeck and I. S. Ornstein, “On the theory of Brownian motion,” in Selected Papers on Noise and Stochastic Process, N. Wax, Ed., Dover, New York, NY, USA, 1954. · Zbl 0059.11903
[13] G. Q. Cai and Y. K. Lin, “On exact stationary solutions of equivalent nonlinear stochastic systems,” International Journal of Non-Linear Mechanics, vol. 23, no. 4, pp. 315-325, 1988. · Zbl 0655.70027
[14] G. P. Samanta, “Influence of environmental noises on the Gomatam model of interacting species,” Ecological Modelling, vol. 91, no. 1-3, pp. 283-291, 1996.
[15] G. P. Samanta and Alakes Maiti, “Stochastic Gomatam model of interacting species: non-equilibrium fluctuation and stability,” Systems Analysis Modelling Simulation, vol. 43, no. 6, pp. 683-692, 2003. · Zbl 1062.92075
[16] A. Maiti and G. P. Samanta, “Deterministic and stochastic analysis of a prey-dependent predator-prey system,” International Journal of Mathematical Education in Science and Technology, vol. 36, no. 1, pp. 65-83, 2005. · Zbl 02363858
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