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On Predator-Prey Systems and Small-Gain Theorems

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


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.
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Keywords monotone systems; feedback systems; Lotka- Volterra systems.; almost global stability

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


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Copyright Info: 2005, Patrick D. Leenheer, et al., licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (http://creativecommons.org/licenses/by/4.0)

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