Dynamic of a stochastic predator-prey population. (English) Zbl 1238.92055

Yagi, Atsushi; Ton, Ta Viet

Appl. Math. Comput. 218, No. 7, 3100-3109 (2011).

Summary: A stochastic predator-prey model is studied. First we prove the existence, uniqueness and positivity of solutions. Then we show the upper bounds for the moments and the growth rate of the population. In some cases, the growth rate is negative and the population dies out rapidly. The paper ends with some reviews of the paper of B.G. Zhang and K. Gopalsamy, Stochastic Anal. Appl. 18, No. 2, 323–331 (2000; Zbl 0983.92023).

MSC:

92D40	Ecology
60H10	Stochastic ordinary differential equations (aspects of stochastic analysis)
34F05	Ordinary differential equations and systems with randomness

Keywords:

moment boundedness; asymptotic behaviour

Citations:

Zbl 0983.92023

References:

[1]	Arnold, L., Stochastic differential equations: theory and applications, (1972), Wiley New York
[2]	Beddington, J.R., Mutual interference between parasites or predators and its effect on searching efficiency, J. animal ecol., 44, 1, 331-340, (1975)
[3]	Cantrell, R.S.; Cosner, C., On the dynamics of predator – prey models with the bedding – deangelis functional response, J. math. anal. appl., 257, 206-222, (2001) · Zbl 0991.34046
[4]	DeAngelis, D.L.; Goldstein, R.A.; O’Neill, R.V., A model for trophic interaction, Ecology, 56, 881-892, (1975)
[5]	Fan, M.; Kuang, Y., Dynamics of a non-autonomous predator – prey system with the beddington – deangelis functional response, J. math. anal. appl., 295, 15-39, (2004) · Zbl 1051.34033
[6]	Friedman, A., Stochastic differential equations and their applications, (1976), Academic press New York
[7]	Hwang, T., Global analysis of the predator – prey system with beddington – deangelis functional response, J. math. anal. appl., 281, 395-401, (2003) · Zbl 1033.34052
[8]	Ioannis, K.; Steven, E.S., Brownian motion and stochastic calculus, (1991), Springer-Verlag New York · Zbl 0734.60060
[9]	X. Mao, Stochastic Differential Equations and Applications, Horwood, 1997. · Zbl 0892.60057
[10]	Mao, X.; Deng, F.; Luo, Q.; Pang, S., Noise suppresses or expresses exponential growth, Syst. control lett., 57, 262-270, (2008) · Zbl 1157.93515
[11]	Mao, X.; Hu, G.; Liu, M.; Song, M., Noise suppresses exponential growth under regime switching, J. math. anal. appl., 355, 783-795, (2009) · Zbl 1173.60024
[12]	Mao, X.; Marion, G.; Renshaw, E., Environmental Brownian noise suppresses explosions in population dynamics, Stoch. proc. appl., 97, 95-110, (2002) · Zbl 1058.60046
[13]	Zhang, B.G.; Gopalsamy, K., On the periodic solution of N-dimentional stochastic population models, Stoch. anal. appl., 18, 2, 323-331, (2000) · Zbl 0983.92023