An Impulsive Periodic Single-Species Logistic System with Diffusion

We study a single-species periodic logistic type dispersal system in a patchy environment with impulses. On the basis of inequality estimation technique, sufficient conditions of integrable form for the permanence and extinction of the system are obtained. By constructing an appropriate Lyapunov function, conditions for the existence of a unique globally attractively positive periodic solution are also established. Numerical examples are shown to verify the validity of our results and to further discuss the model.

[1]  Yoshiyuki Hino,et al.  Existence of periodic solutions and almost periodic solutions , 1991 .

[2]  Pierre Auger,et al.  Effect of predator density dependent dispersal of prey on stability of a predator-prey system. , 2007, Mathematical biosciences.

[3]  P ? ? ? ? ? ? ? % ? ? ? ? , 1991 .

[4]  Zhidong Teng,et al.  Permanence for a class of periodic time-dependent competitive system with delays and dispersal in a patchy-environment , 2007, Appl. Math. Comput..

[5]  Shouming Zhong,et al.  An impulsive periodic predator–prey system with Holling type III functional response and diffusion☆ , 2012 .

[6]  R. Stephenson.  A and V , 1962, The British journal of ophthalmology.

[7]  Lansun Chen,et al.  The dynamics of an impulsive delay predator-prey model with variable coefficients , 2008, Appl. Math. Comput..

[8]  Y. Takeuchi,et al.  Global stability of single-species diffusion volterra models with continuous time delays , 1987 .

[9]  Yasuhiro Takeuchi,et al.  Permanence of delayed population model with dispersal loss. , 2006, Mathematical biosciences.

[10]  Xiaoqin Wang,et al.  Dynamics of a periodic Watt-type predator-prey system with impulsive effect , 2009 .

[11]  Z Teng,et al.  The effect of dispersal on single-species nonautonomous dispersal models with delays , 2001, Journal of mathematical biology.

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

[13]  Ivanka M. Stamova,et al.  Asymptotic stability of an N-dimensional impulsive competitive system , 2007 .

[14]  Samir H. Saker,et al.  Existence of periodic solutions, global attractivity and oscillation of impulsive delay population model , 2007 .

[15]  H. I. Freedman,et al.  Global stability and predator dynamics in a model of prey dispersal in a patchy environment , 1989 .

[16]  M. U. Akhmeta,et al.  An impulsive ratio-dependent predator – prey system with diffusion , 2006 .

[17]  Zhidong Teng,et al.  Permanence and extinction of periodic predator-prey systems in a patchy environment with delay , 2003 .

[18]  Marat Akhmet,et al.  An impulsive ratio-dependent predator–prey system with diffusion , 2006 .

[19]  Y. Takeuchi,et al.  Predator-prey dynamics in models of prey dispersal in two-patch environments. , 1994, Mathematical biosciences.

[20]  Zhidong Teng,et al.  Permanence and global stability for nonautonomous N-species Lotka–Valterra competitive system with impulses , 2010 .

[21]  Z. A. Melzak.  Existence of periodic solutions , 1967 .

[22]  Lansun Chen,et al.  Persistence and global stability for two-species nonautonomous competition Lotka-Volterra patch-system with time delay , 1999 .

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