The dynamics of an impulsive delay predator-prey model with variable coefficients

Abstract In this paper, we formulate a new robust two-species nonautonomous predator–prey model with multi-delays and impulsive effects and perform a systematic mathematical and ecological study. Our results in this paper indicate that under the appropriate linear periodic impulsive perturbations, the system is permanent and has a unique positive globally attractive semi-trivial periodic solution. By using the Brouwer fixed point theorem, we prove that if the periodic system is permanent, then there is at least one positive periodic solution of the system. We show that the conditions for global attractivity of the positive semi-trivial periodic solution and permanence of the population of the model depend on time delay, so, we call it “profitless”. In this paper, the main feature is that we introduce multi-delays and impulses into the predator–prey model, exhibit a new modeling method which is applied to investigate multi-species impulsive multi-delays differential equations.

[1]  Shujing Gao,et al.  Pulse vaccination strategy in a delayed sir epidemic model with vertical transmission , 2006 .

[2]  V. Lakshmikantham,et al.  Theory of Impulsive Differential Equations , 1989, Series in Modern Applied Mathematics.

[3]  Bing Liu,et al.  The dynamical behaviors of a Lotka–Volterra predator–prey model concerning integrated pest management ☆ , 2005 .

[4]  Fengde Chen.  Permanence of periodic Holling type predator-prey system with stage structure for prey , 2006, Appl. Math. Comput..

[5]  Zhidong Teng,et al.  Dynamic behaviors of the periodic Lotka–Volterra competing system with impulsive perturbations , 2007 .

[6]  R. Gaines,et al.  Coincidence Degree and Nonlinear Differential Equations , 1977 .

[7]  K. Gopalsamy.  Stability and Oscillations in Delay Differential Equations of Population Dynamics , 1992 .

[8]  Yang Kuang,et al.  Periodic Solutions of Periodic Delay Lotka–Volterra Equations and Systems☆ , 2001 .

[9]  Xinyu Song,et al.  Dynamic behaviors of the periodic predator–prey model with modified Leslie-Gower Holling-type II schemes and impulsive effect , 2008 .

[10]  Lansun Chen,et al.  Existence of positive periodic solution of an impulsive delay Logistic model , 2007, Appl. Math. Comput..

[11]  Wanbiao Ma,et al.  Necessary and Sufficient Conditions for Permanence and Global Stability of a Lotka–Volterra System with Two Delays , 1999 .

[12]  Lansun Chen,et al.  Profitless delays for a nonautonomous Lotka-Volterra predator-prey almost periodic system with dispersion , 2007, Appl. Math. Comput..

[13]  Guilie Luo,et al.  Positive periodic solution for a two-species ratio-dependent predator–prey system with time delay and impulse , 2007 .

[14]  Hai-Feng Huo.  Existence of positive periodic solutions of a neutral delay Lotka-Volterra system with impulses , 2004 .

[15]  Meili Li,et al.  Periodic solutions for impulsive delay differential equations in the control model of plankton allelopathy , 2007 .

[16]  Xianning Liu,et al.  Complex dynamics of Holling type II Lotka–Volterra predator–prey system with impulsive perturbations on the predator ☆ , 2003 .

[17]  Jianjun Jiao,et al.  The dynamics of an age structured predator–prey model with disturbing pulse and time delays ☆ , 2008 .