This paper is concerned with a stochastic predator-prey system with Beddington-DeAngelis functional response and time delay. Firstly, we show that this system has a unique positive solution as this is essential in any population dynamics model. Secondly, the validity of the stochastic system is guaranteed by stochastic ultimate boundedness of the analyzed solution. Finally, by constructing suitable Lyapunov functions, the asymptotic moment estimation of the solution was given. These properties of the solution can provide theoretical support for biological resource management.
References
[1]
Holling, C.S. (1959) The Components of Predation as Revealed by a Study of Small Mammal Predation of the European Pine Sawfly. The Canadian Entomologist, 91, 293-320. https://doi.org/10.4039/Ent91293-5
[2]
Holling, C.S. (1959) Some Characteristics of Simple Types of Predation and Parasitism. The Canadian Entomologist, 91, 385-395. https://doi.org/10.4039/Ent91385-7
[3]
Crowley, P.H. and Martin, E.K. (1989) Functional Response and Interference within and between Year Classes of a Dragonfly Population. Journal of the North American Benthological Society, 8, 211-221. https://doi.org/10.2307/1467324
[4]
Hassell, M.P. and Varley, C.C. (1969) New Inductive Population Model for Insect Parasites and Its Bearing on Biological Control. Nature, 223, 1133-1137. https://doi.org/10.1038/2231133a0
[5]
Beddington, J.R. (1975) Mutual Interference between Parasites or Predators and Its Effect on Searching Efficiency. Journal of Animal Ecology, 44, 331-341. https://doi.org/10.2307/3866
[6]
DeAngelis, D.L., Goldsten, R.A. and Neill, R. (1975) A Model for Tropic Interaction. Ecology, 56, 881-892. https://doi.org/10.2307/1936298
[7]
Jost, C. and Arditi, R. (2001) From Pattern to Process: Identifying Predator-Prey Interactions. Population Ecology, 43, 229-243. https://doi.org/10.1007/s10144-001-8187-3
[8]
Skalski, G.T. and Gilliam, J.F. (2001) Functional Responses with Predator Interference: Viable Alternatives to the Holling Type II Model. Ecology, 82, 3083-3092. https://doi.org/10.1890/0012-9658(2001)082[3083:FRWPIV]2.0.CO;2
[9]
Gard, T.C. (1984) Persistence in Stochastic Food Web Models. Bulletin of Mathematical Biology, 46, 357-370.
[10]
Gard, T.C. (1988) Introduction to Stochastic Differential Equations. Dekker, New York.
[11]
Mandal, P. and Bnerjee, M. (2012) Stochastic Persistence and Stationary Distribution in a Holling-Tanner Type Prey-Predator Model. Physica A, 391, 1216-1233. https://doi.org/10.1016/j.physa.2011.10.019
[12]
Liu, M. and Wang, K. (2012) Persistence, Extinction and Global Asymptotical Stability of a Non-Autonomous Predator-Prey Model with Random Perturbation. Applied Mathematical Modelling, 36, 5344-5353. https://doi.org/10.1016/j.apm.2011.12.057
[13]
Fan, M. and Kuang, Y. (2008) Dynamics of a Nonautonomous Predator-Prey System with the Beddington-DeAngelis Functional Response. Journal of Mathematical Analysis and Applications, 295, 15-39. https://doi.org/10.1016/j.jmaa.2004.02.038
[14]
Liu, S., Beretta, E. and Breda, D. (2010) Predator-Prey Model of Beddington-DeAngelis Type with Maturation and Gestation Delays. Nonlinear Analysis: Real World Applications, 11, 4072-4091. https://doi.org/10.1016/j.nonrwa.2010.03.013
[15]
Liu, M. and Wang, K. (2012) Global Asymptotic Stability of a Stochastic Lotka-Volterra Model with Infinite Delays, Commun. Communications in Nonlinear Science and Numerical Simulation, 17, 3115-3123. https://doi.org/10.1016/j.cnsns.2011.09.021
[16]
Geritz, S. and Gyllenberg, M. (2012) A Mechanistic Derivation of the DeAngelis-Beddington Functional Response. Journal of Theoretical Biology, 314, 106-108. https://doi.org/10.1016/j.jtbi.2012.08.030
[17]
Liu, M. and Bai, C. (2014) Global Asymptotic Stability of a Stochastic Delayed Predator-Prey System with Beddington-DeAngelis Functional Response. Applied Mathematics and Computation, 226, 581-588. https://doi.org/10.1016/j.amc.2013.10.052
Hu, Y., Mohammed, S.E. and Yan, F. (2004) Discrete-Time Approximations of Stochastic Delay Equations: The Milstein Scheme. The Annals of Probability, 32, 265-314. https://doi.org/10.1214/aop/1078415836
