Dynamics analysis and numerical simulations of a stochastic non-autonomous predator–prey system with impulsive effects

Abstract In this paper, we propose a stochastic non-autonomous Lotka–Volterra predator–prey model with impulsive effects and investigate its stochastic dynamics. We first prove that the subsystem of the system has a unique periodic solution which is globally attractive. Furthermore, we obtain the threshold value in the mean which governs the stochastic persistence and the extinction of the prey–predator system. Our results show that the stochastic noises and impulsive perturbations have crucial effects on the persistence and extinction of each species. Finally, we use the different stochastic noises and impulsive effects parameters to provide a series of numerical simulations to illustrate the analytical results.

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

[2]  Ruiqing Shi,et al.  A predator-prey model with disease in the prey and two impulses for integrated pest management , 2009 .

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

[4]  Xinzhu Meng,et al.  Global dynamics analysis of a nonlinear impulsive stochastic chemostat system in a polluted environment , 2016 .

[5]  Ryszard Rudnicki,et al.  Influence of stochastic perturbation on prey-predator systems. , 2007, Mathematical biosciences.

[6]  Shujing Gao,et al.  Application of inequalities technique to dynamics analysis of a stochastic eco-epidemiology model , 2016 .

[7]  Xinzhu Meng,et al.  Evolutionary dynamics in a Lotka–Volterra competition model with impulsive periodic disturbance , 2016 .

[8]  Yingmin Jia,et al.  Stability analysis for stochastic differential equations with infinite Markovian switchings , 2016 .

[9]  Lansun Chen,et al.  A delayed epidemic model with stage-structure and pulses for pest management strategy , 2008 .

[10]  Weihai Zhang,et al.  Observer-based controller design for singular stochastic Markov jump systems with state dependent noise , 2016, Journal of Systems Science and Complexity.

[11]  Xinzhu Meng,et al.  Optimal harvesting control and dynamics of two-species stochastic model with delays , 2017 .

[12]  Juan J. Nieto,et al.  Seasonality and mixed vaccination strategy in an epidemic model with vertical transmission , 2011, Math. Comput. Simul..

[13]  Xianning Liu,et al.  Global dynamics of the periodic logistic system with periodic impulsive perturbations , 2004 .

[14]  Ryszard Rudnicki,et al.  Long-time behaviour of a stochastic prey–predator model , 2003 .

[15]  Qun Liu,et al.  Dynamics of stochastic predator-prey models with Holling II functional response , 2016, Commun. Nonlinear Sci. Numer. Simul..

[16]  Xinzhu Meng,et al.  Stability of a novel stochastic epidemic model with double epidemic hypothesis , 2010, Appl. Math. Comput..

[17]  D. O’Regan,et al.  Periodic solution for a non-autonomous Lotka–Volterra predator–prey model with random perturbation , 2015 .

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

[19]  Abdelghani Ouahab,et al.  Random fixed point theorems in partially ordered metric spaces , 2016 .

[20]  D. Jiang,et al.  Periodic solutions for a stochastic non-autonomous Holling–Tanner predator–prey system with impulses , 2016 .

[21]  Ke Wang,et al.  On a stochastic logistic equation with impulsive perturbations , 2012, Comput. Math. Appl..

[22]  J. Nieto,et al.  New results for the periodic boundary value problem for impulsive integro-differential equations , 2009 .

[23]  R. Khasminskii.  Stochastic Stability of Differential Equations , 1980 .

[24]  Tonghua Zhang,et al.  A Stage-Structured Predator-Prey SI Model with Disease in the Prey and Impulsive Effects , 2013 .

[25]  R. Sakthivel,et al.  Asymptotic stability of nonlinear impulsive stochastic differential equations , 2009 .

[26]  Ivanka M. Stamova,et al.  Asymptotic stability of competitive systems with delays and impulsive perturbations , 2007 .

[27]  J. Nieto,et al.  Existence of multiple solutions of some second order impulsive differential equations , 2016 .

[28]  Tongqian Zhang,et al.  A new predator-prey model with a profitless delay of digestion and impulsive perturbation on the prey , 2011, Appl. Math. Comput..

[29]  Ke Wang,et al.  Persistence, extinction and global asymptotical stability of a non-autonomous predator–prey model with random perturbation , 2012 .

[30]  Sanyi Tang,et al.  Global attractivity in a “food-limited” population model with impulsive effects☆ , 2004 .

[31]  Jitao Sun,et al.  Stability analysis of nonlinear stochastic differential delay systems under impulsive control , 2010 .

[32]  Tonghua Zhang,et al.  Dynamics analysis of a pest management prey–predator model by means of interval state monitoring and control , 2017 .

[33]  Xinzhu Meng,et al.  Dynamics of a novel nonlinear stochastic SIS epidemic model with double epidemic hypothesis , 2016 .

[34]  Zuxiong Li,et al.  Periodic solution of a chemostat model with variable yield and impulsive state feedback control , 2012 .