Graphical Representation and Stability Conditions of Predator-Prey Interactions

Abstract

The general nature of the predator-prey interaction has been depicted as a graph of predator versus prey densities from which conditions for stability of the interaction are predicted. An example of a three-species interaction is also presented. Variations of the graph are introduced, and it is shown that an otherwise unstable interaction may be stabilized by the presence of either an inviolable prey hiding place, or extremely low predation pressure at moderate predator and high prey densities, or another predator-limiting resource. Stability is always conferred when the predator is severely limited at its equilibrium density by one of its resources other than its supply of prey. Predators should tend to be limited at their equilibrium densities by more than one of their resources. When either of the two foregoing situations pertains, regular predator-prey oscillations should not be observable. The stability of the interaction close to equilibrium was found to depend exclusively, in the mathematically-continuous model, upon the slopes of two lines in the graph at equilibrium. Stability can be asymptotic rather than oscillatory in type. An equation for the period of oscillatory interactions is also advanced. The effects of Natural Selection on the isoclines, and thus the stability, is not clear-cut. Selection of the prey tends to stabilize the interaction; the opposite is true for selection on the predator.