Some Discrete Competition Models and the Principle of Competitive Exclusion

Abstract

One of the fundamental tenets of ecology is the Competitive Exclusion Principle. According to this principle too much interspecific competition between two species results in the exclusion of one species. This Principle is supported by a wide variety of theoretical models, of which the Lotka/Volterra model based on differential equations is the most familiar. It is perhaps less well known that difference equations also played an important role in the historical development of the Competitive Exclusion Principle. The Leslie/Gower model was used in conjunction with influential competition experiments using species of Tribolium (flour beetles) carried out in the first half of the last century. This difference equation model exhibits the same dynamic scenarios as does the Lotka/Volterra model and also supports the Competitive Exclusion Principle. A recently developed competition for Tribolium species, however, exhibits a larger variety of dynamic scenarios and competitive outcomes, some of which seemingly stand in contradiction to the Principle. We discuss features of this model that differentiate it from the Leslie/Gower model. We give a simpler, lower dimensional “toy” model that illustrates some non-Lotka/Volterra dynamics. AMS Nos. 39A11, 92D40