A stochastic predator-prey system with Watt-type functional response
Abstract
In this paper we consider a stochastic version of predator-prey systems with Watt-type functional response. We first prove the existence and uniqueness of the positive global solution by using the comparison theorem of stochastic equations. Then, we study the boundedness of moments of the solution. Furthermore, the growth rates, persistence and extinction of species are investigated.References
R. Arditi and L. R. Ginzburg, Coupling in predator-prey dynamics: Ratio-dependence, J. Theoret. Biol., vol. 139, pp. 311-326, 1989.
M. Bandyopadhyay and J. Chattopadhyay, Ratio-dependent predator-prey model: effect of environmental fluctuation and stability, Nonlinearity, vol. 18, no. 2, pp. 913-936, 2005.
J. R. Beddington, Mutual interference between parasites or predators and its effect on searching efficiency, J. Animal Ecol., vol. 44, pp. 331-340, 1975.
D. L. DeAngelis, R. A. Goldstein and R. V. O’Neill, A model for tropic interaction, Ecology, vol. 56, pp. 881-892, 1975.
N. T. Dung, On Delayed Logistic Equation Driven by Fractional Brownian Motion, J. Comput. Nonlinear Dynam, vol. 7, no. 3, 031005 (5 pages), 2012.
M. P. Hassell and C. C. Varley, New inductive population model for insect parasites and its bearing on biological control, Nature, vol. 223, pp. 1133-1137, 1969.
C. S. Holling, The components of predation as revealed by a study of small mammal predation of the European pine sawfly, Can. Entomologist, vol. 91, pp. 293-320, 1959.
N. Ikeda and S. Wantanabe, Stochastic Differential Equations and Diffusion Processes, North-Holland, Amsterdam, 1981.
C. Ji and D. Jiang, Dynamics of a stochastic density dependent predator-prey system with Beddington-DeAngelis functional response, J. Math. Anal. Appl., vol. 381, pp. 441-453, 2011.
R. Z. Khasminskiˇi and F. C. Klebaner, Long term behavior of solutions of the Lotka-Volterra system under small random perturbations, Ann. Appl. Probab., vol. 11, pp. 952-963, 2001.
P. E. Kloeden and E. Platen, Numerical solution of stochastic differential equations, Springer Verlag, 1992.
X. Lin, Y. Jiang and X. Wang, Existence of periodic solutions in predator-prey with Watt-type functional response and impulsive effects, Nonlinear Analysis, vol. 73, pp. 1684-1697, 2010.
P. S. Mandal and M. Banerjee, Stochastic persistence and stationary distribution in a Holling-Tanner type prey-predator model, Physica A, vol. 391, pp. 1216-1233, 2012.
X. Mao, Stochastic Differential Equations and Applications, Horwood, 1997.
X. Mao, G. Marion and E. Renshaw, Environmental Brownian noise suppresses explosions in population dynamics, Stochastic Process. Appl., vol. 97, no. 1, pp. 95-110, 2002.
R. M. May, Stability and Complexity in Model Ecosystems, Princeton University Press, New Jersey, 1973.
M. M´etivier, Semimartingales: A course on stochastic processes, Walter de Gruyter, Berlin, 1982.
J. D. Murray, Mathematical Biology: I. An Introduction, third ed., Springer, New York, 2002.
D. Revuz and M. Yor, Continuous martingales and Brownian motion, Third edition, Springer-Verlag, Berlin, 1999.
J. Szarski, Differential Inequalities, Warsaw, PWN, 1965.
W. Wang, X. Wang and Y. Lin, Complicated dynamics of a predator-prey system with Watt-type functional response and impulsive control strategy, Chaos, Solitons and Fractals, vol. 37, pp. 1427-1441, 2008.
X. Wang, W. Wang, Y. Lin and X. Lin, The dynamical complexity of an impulsive Watt-type prey-predator system, Chaos Solitons Fractals, vol. 40, no. 2, pp. 731-744, 2009.
K. E. F.Watt, A mathematical model for the effect of densities of attacked and attacking species on the number attacked, The Canadian Entomologist, vol. 91, pp 129-144, 1959.
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