Stability and bifurcation in a non-kolmogorov type prey-predator system with time delay

Keywords--Prey-predator system, Time delay, Stability, Hopf bifurcation, Periodic solutions. 1. I N T R O D U C T I O N After the seminal models of Volterra and Lotka in the mid-1920s, predator-prey systems with delay have been studied extensively in recent years. There have been many results on stability, oscillation, persistence, and existence of positive periodic solutions, see for example, [1-5] and the references. It is well known that the dynamics of delayed systems depend not only on the parameters describing the models but also on the time delays from the feedback. Thus, both continuous delay and discrete delay prey-predator models are considered. The first distributed delay predator-prey model was proposed by Volterra [6], then Brelot [7] modified it. The model has the form, I ; : l :~ (t) = X (t) rl -a11x (t) -a12 (t -T) y (S) d8 , (1.1) ( t ) = y ( t ) r 2 + a21 a (t s ) x ( s ) d~ a 2 ~ v ( t ) . o~ Further detailed study on stability and Hopf bifurcation of system (1.1) can be found in [8]. However, the delayed predator-prey systems which have been studied are most of the Kolmogrovtype. As far as we know, there have few results about non-Kolmogorov type delayed prey-predator The research was supported by the National Natural Sciences Foundation of P.R. China (No. 10371074). 0895-7177/05/$ see front matter @ 2005 Elsevier Ltd. All rights reserved. Typeset by .A.AdS-~/X doi:10.1016/j.mcm.2004.02.038