Abstract
A model of predator-prey interaction in a chemostat with Holling Type II functional and numerical response functions of the Monod or Michaelis-Menten form is considered. It is proved that local asymptotic stability of the coexistence equilibrium implies 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 chemostat predator-prey model is the same as for the analogous classical Rosenzweig-MacArthur predator-prey model with Holling Type II functional response. An extension that applies to other functional rsponses is also given.
Highlights
In this paper, we analyze a predator-prey model in the chemostat with Holling Type II predator response function of Monod form and prey response function of mass action form
In the special case that the prey grow logistically in the absence of predators and the predator response function is of Holling Type I form, Hsu [21] proved that the coexistence equilibrium is globally asymptotically stable whenever it exists
It was shown that whenever the coexistence equilibrium is locally asymptotically stable, it is globally asymptotically stable, and whenever the coexistence equilibrium is unstable, there is a unique, orbitally asymptotically stable periodic orbit
Summary
We analyze a predator-prey model in the chemostat with Holling Type II predator response function of Monod form and prey response function of mass action form. We determine the global dynamics of a predator-prey model in the chemostat with predator response function of Holling Type II (Monod form) and prey response function of mass action form. In the special case that the prey grow logistically in the absence of predators and the predator response function is of Holling Type I form, Hsu [21] proved that the coexistence equilibrium is globally asymptotically stable whenever it exists. In the model considered in this paper, we assume instead that the prey response function is Holling Type I and the predator response function is Holling Type II (of Monod form) In this case, we prove that the dynamics are more complicated. The bifurcation diagrams were done using XPPAUT [12] and the simulations were done using Matlab [32]
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