Using stocking or harvesting to reverse period-doubling bifurcations in discrete population models*

Abstract

This study considers a general class of 2-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). Four sets of necessary inequality conditions are derived which guarantee generically that an equilibrium loses stability through a period-doubling bifurcation with respect to the pioneer self-crowding parameter. A stocking or harvesting term which is proportional to the pioneer density is introduced into the system. Conditions are determined under which this stocking or harvesting will reverse the bifurcation and restabilize the equilibrium. A numerical example illustrates how pioneer stocking can be used to reverse a period-doubling cascade and to maintain the system at any attracting cycle along the cascade.