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IMA Journal of Applied Mathematics Advance Access published online on March 5, 2009
IMA Journal of Applied Mathematics, doi:10.1093/imamat/hxp007
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© The Author 2009. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

Permanence and extinction of an impulsive delay competitive Lotka–Volterra model with periodic coefficients

Zhijun Liu{dagger}

College of Mathematics and Information Science, Shaanxi Normal University, Xi'an, Shaanxi 710062, People's republic of China and Department of Mathematics, Hubei Institute for Nationalities, Enshi, Hubei 445000, People's republic of China

Jianhua Wu{ddagger}

College of Mathematics and Information Science, Shaanxi Normal University, Xi'an, Shaanxi 710062, People's republic of China

Ronghua Tan§

Department of Mathematics, Hubei Institute for Nationalities, Enshi, Hubei 445000, People's republic of China

{dagger} Email: zjliu1008{at}gmail.com

{ddagger} Email: jianhuaw{at}snnu.edu.cn

§ Email: ronghua_tan{at}hotmail.com

Received on May 16, 2008; Accepted on January 18, 2009

In this paper, a periodic competitive system with delays and pulses is proposed. By using the comparison theorem for impulsive differential equations and the property of globally asymptotic stability of a periodic single-species growth population model with impulsive perturbations, sufficient conditions for permanence and extinction of the above system are derived, respectively. Our main results show that under appropriate conditions, the permanence and extinction of system are irrespective of the size of delays, however, impulsive perturbations play an important role and have effects on the permanence and extinction of system.

Keywords: permanence; extinction; competitive system; delay; pulse.


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