Degenerate Lyapunov functionals and a prey-predator model with discrete delays


Xue-Zhong He


Research Report 97-1
Date: January 1997


The dynamics of the classical Lotka-Volterra prey-predator equation with discrete delays
x'(t) = x(t)[ r_1 - x(t-tau_1) - ay(t-tau_2) ]
y'(t) = y(t)[ - r_2 + bx(t-tau_3) ]
is concerned in this paper. We first show that, in some circumstances, the positive equilibrium of the model is locally asymptotically stable for small delays and the Hopf bifurcation occurs for large delays.

To study the stability of the positive equilibrium of the general system, we then introduce the concept of degenerate Lyapunov functionals. By constructing suitable degenerate Lyapunov functionals, we obtain some sufficient conditions on both local and global stabilities of the positive equilibrium. As a corollary, we show that small delays do not change the stability of the system. Furthermore, some explicit estimates on the delays are given.

Key phrases

Stability. degenerate Lyapunov functionals. prey-predator. delays. Hopf bifurcation.

AMS Subject Classification (1991)

Primary: 34D05, 34K20, 92D25


The paper is available in the following forms:
TeX dvi format:
1997-1.dvi.gz (38kB) or 1997-1.dvi (111kB)

PostScript: (89kB) or (298kB)

To minimize network load, please choose the smaller gzipped .gz form if and only if your browser client supports it.

Sydney Mathematics and Statistics