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Volume 6,     Number 3,     Fall 1998

 

REVERSING PERIOD-DOUBLING BIFURCATIONS IN
MODELS OF POPULATION INTERACTIONS
USING CONSTANT STOCKING OR HARVESTING
JAMES F. SELGRADE AND JAMES H. ROBERDS
Dedicated to the memory of Geoffrey James Butler

Abstract. 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.

 

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