On the dynamics of a stochastic ratio-dependent predator–prey model with a specific functional response

In this paper, a new stochastic two-species predator–prey model which is ratio-dependent and a specific functional response is considered in, is proposed. The existence of a global positive solution to the model for any positive initial value is shown. Stochastically ultimate boundedness and uniform continuity are derived. Moreover, under some sufficient conditions, the stochastic permanence and extinction are established for the model. At last, numerical simulations are carried out to support our results.

[1]  X. Mao,et al.  Stochastic Differential Equations and Applications , 1998 .

[2]  Ke Wang,et al.  Persistence and extinction in stochastic non-autonomous logistic systems , 2011 .

[3]  Desmond J. Higham,et al.  An Algorithmic Introduction to Numerical Simulation of Stochastic Differential Equations , 2001, SIAM Rev..

[4]  Donald L. DeAngelis,et al.  A MODEL FOR TROPHIC INTERACTION , 1975 .

[5]  M. Hassell,et al.  New Inductive Population Model for Insect Parasites and its Bearing on Biological Control , 1969, Nature.

[6]  Attila Szolnoki,et al.  Coevolutionary Games - A Mini Review , 2009, Biosyst..

[7]  Yanli Zhou,et al.  Survival and Stationary Distribution in a Stochastic SIS Model , 2013 .

[8]  Philip H. Crowley,et al.  Functional Responses and Interference within and between Year Classes of a Dragonfly Population , 1989, Journal of the North American Benthological Society.

[9]  Shuenn-Ren Cheng,et al.  Stochastic Population Systems , 2009 .

[10]  Abdelilah Kaddar,et al.  On the dynamics of a delayed SIR epidemic model with a modified saturated incidence rate , 2009 .

[11]  Xinyu Song,et al.  Analysis of a stage-structured predator-prey model with Crowley-Martin function , 2011 .

[12]  Extinction and permanence in delayed stage-structure predator–prey system with impulsive effects , 2009 .

[13]  Xuerong Mao,et al.  Population dynamical behavior of non-autonomous Lotka-Volterra competitive system with random perturbation , 2009 .

[14]  Ke Wang,et al.  Analysis on a Stochastic Two-Species Ratio-Dependent Predator-Prey Model , 2015 .

[15]  Marija Krstić,et al.  Stochastically perturbed vector-borne disease models with direct transmission , 2012 .

[16]  Xueyong Zhou,et al.  GLOBAL STABILITY OF THE VIRAL DYNAMICS WITH CROWLEY-MARTIN FUNCTIONAL RESPONSE , 2011 .

[17]  X. Mao,et al.  Environmental Brownian noise suppresses explosions in population dynamics , 2002 .

[18]  M. Liu,et al.  Persistence and extinction of a stochastic single-specie model under regime switching in a polluted environment. , 2010, Journal of theoretical biology.

[19]  Yanqiu Li,et al.  Existence, uniqueness and global asymptotic stability of positive solutions of a predator–prey system with Holling II functional response with random perturbation , 2008 .

[20]  Ioannis Karatzas,et al.  Brownian Motion and Stochastic Calculus , 1987 .

[21]  Shouming Zhong,et al.  Asymptotic properties of a stochastic predator-prey model with Crowley-Martin functional response , 2013 .

[22]  Zhijun Liu,et al.  Analysis of a Leslie–Gower-type prey–predator model with periodic impulsive perturbations , 2009 .

[23]  Donald L. DeAngelis,et al.  A Model for Tropic Interaction , 1975 .

[24]  J. Beddington,et al.  Mutual Interference Between Parasites or Predators and its Effect on Searching Efficiency , 1975 .

[25]  Daqing Jiang,et al.  Analysis of a predator-prey model with modified Leslie-Gower and Holling-type II schemes with stochastic perturbation , 2009 .

[26]  Xuerong Mao,et al.  Stochastic Versions of the LaSalle Theorem , 1999 .

[27]  H. I. Freedman.  Deterministic mathematical models in population ecology , 1982 .