A Predator-Prey model in the chemostat with Holling Type II response function

Abstract

A model of predator-prey interaction in a chemostat with Holling Type II functional and numerical response functions ofthe Monod or Michaelis-Menten form is considered. It is proved that local asymptotic stability of the coexistence equilibriumimplies that it is globally asymptotically stable. It is also shown that when the coexistence equilibrium exists but is unstable,solutions converge to a unique, orbitally asymptotically stable periodic orbit. Thus the range of the dynamics of the chemostatpredator-prey model is the same as for the analogous classical Rosenzweig-MacArthur predator-prey model with Holling TypeII functional response. An extension that applies to other functional rsponses is also given.

A criterion for the existence of relaxation oscillations with applications to predator-prey systems and an epidemic model

“…If χpx 0 q " 0 and λpx 0 q ‰ 0, then by Theorem 2.1, for every small ą 0, system (48) has unique periodic orbit near γpx 0 q Ă Λ. Since Λ is invariant under (48), is also a periodic orbit for (1). If λpx 0 q ą 0, then is orbitally unstable for (48), and therefore is orbitally unstable for (1).…”

“…If λpx 0 q ă 0, then is locally orbitally asymptotically stable for (48). Since Λ is a hyperbolic attractor, it follows that is locally orbitally asymptotically stable for (1).…”

“…It can be verified that system (1) under assumption (2) has a unique positive equilibrium for all small ą 0. From the equations in (1),…”

“…that attracts all points in R 3 `. For system (1) with " 0, there is a continuous family of heteroclinic orbits on Λ as illustrated in Figure 1 (see equation (51) in Section 3 and its succeeding paragraph for the limiting system on Λ). Each heteroclinic orbit connects two points on the boundary of Λ, where x " 0.…”

“…Figure 1. System (1) with " 0 exhibits a family of heteroclinic orbits on Λ. The dynamics on the segment Λ X tx " 0u is governed by system (5).…”