Some mathematical problems concerning the ecological principle of competitive exclusion

Abstract

The ecological principle of competitive exclusion asserts that two species cannot indefinitely occupy the same niche [Sj. Attempts have been made to state this principle as a mathematical theorem. A standard example is due to Volterra [14]. (See Example 2.1 below.) He constructed a model of two species competing for a single resource and showed that one of the species must go extinct. Although the concept of niche is rather vague, it is generally agreed that Volterra’s model is an example of two species competing for the same niche and is therefore an illustration of the principle of competitive exclusion. The generalizations of Volterra’s model have dealt largely with the introduction of more realistic assumptions about the interactions between species. These assumptions have led to some interesting mathematical problems, many of them unsolved. The purpose of this paper is to state these problems precisely, to give proofs of the known results, and to indicate the unanswered questions. The basic object of study is an idealized ecological community consisting of a certain number of species living together in an isolated geographical area. We shall assume that the community interactions are independent of both space and time and that each species is distributed uniformly over the region. The basic variables are the population densities of each of the species. The