Hopf bifurcation in pioneer-climax competing species models

Two-dimensional pioneer-climax models of competing species differential equations are introduced and examined. The per capita growth rates are functions of weighted densities of the individual species. Constant rate forcing is added representing stocking or harvesting. The ensuing stability properties of interior equilibria are studied. The primary interest is when stable periodic behavior of the populations results from a Hopf bifurcation. For linear and quadratic fitnesses, general formulas are calculated for determining the stability of the periodic orbit and for finding the Hopf bifurcation surface. Also strategies for restoring the system to a stable equilibrium are developed.