Lumped-density population models of pioneer-climax type and stability analysis of Hopf bifurcations

The effects of population density on the survival and growth of an individual species are modeled by assuming that the species per capita growth rate (i.e., fitness) is a function of a weighted total density. A species is called a pioneer population if it thrives at low density but its fitness decreases monotonically with increasing density. A species is called a climax population if its fitness increases up to a maximum value and then decreases as a function of its total density. Hopf bifurcations for deterministic models of the interaction of pioneer and climax populations are discussed. Stability properties for the Hopf invariant curves in the discrete models are compared to the stability properties of the Hopf periodic orbits in the corresponding continuous models. Examples illustrate that stability may be more common in the discrete models.