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