On Predator-Prey Systems and Small-Gain Theorems

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2005, 2(1): 25-42. doi: 10.3934/mbe.2005.2.25

On Predator-Prey Systems and Small-Gain Theorems

Patrick D. Leenheer ^1, , David Angeli ^2, and Eduardo D. Sontag ^3,

1.	Department of Mathematics, University of Florida, Gainesville, FL 32611-8105, United States
2.	Dip. di Sistemi e Informatica, Universitá di Firenze, Via di S. Marta 3, 50139 Firenze, Italy
3.	Department of Mathematics, Rutgers University, New Brunswick, NJ 08903, United States

Received May 2003 Revised August 2004 Published November 2004

Abstract
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This paper deals with an almost global convergence result for Lotka-Volterra systems with predator-prey interactions. These systems can be written as (negative) feedback systems. The subsystems of the feedback loop are monotone control systems, possessing particular input-output properties. We use a small-gain theorem, adapted to a context of systems with multiple equilibrium points to obtain the desired almost global convergence result, which provides sufficient conditions to rule out oscillatory or more complicated behavior that is often observed in predator-prey systems.

Keywords: monotone systems, feedback systems, Lotka- Volterra systems., almost global stability.

Mathematics Subject Classification: 34C12, 92D40, 93D2.

Citation: Patrick D. Leenheer, David Angeli, Eduardo D. Sontag. On Predator-Prey Systems and Small-Gain Theorems. Mathematical Biosciences & Engineering, 2005, 2 (1) : 25-42. doi: 10.3934/mbe.2005.2.25

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