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Mathematical Biosciences and Engineering

2005, Volume 2, Issue 1: 25-42. doi: 10.3934/mbe.2005.2.25
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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
  • Received: 01 May 2003 Accepted: 29 June 2018 Published: 01 November 2004

  • MSC : 34C12, 92D40, 93D25.

  • 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

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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.


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