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:

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