Reversing Period-Doubling Bifurcations in Models of Population Interactions Using Constant Stocking or Harvesting
This study considers a general class of two-dimensional, discrete population models where each per capita transition function (fitness) depends on a linear combination of the densities of the interacting populations. The fitness functions are either monotone decreasing functions (pioneer fitnesses) or one-humped functions (climax fitnesses). Conditions are derived which guarantee that an equilibrium loses stability through a period-doubling bifurcation with respect to the pioneer self-crowding parameter. A constant term which represents stocking or harvesting of the pioneer population is introduced into the system. Conditions are determined under which this stocking or harvesting will reverse the bifurcation and restabilize the equilibrium, and comparisons are made with the effects of density dependent stocking or harvesting. Examples illustrate the importance of the concavity of the pioneer fitness in determining whether stocking or harvesting has a stabilizing effect.