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

## Author

**Xue-Zhong He**

## Status

Research Report 97-1

Date: January 1997

## Abstract

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

Secondary:

## Content

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1997-1.dvi (111kB)

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