In this paper we develop a mathematical model in which any number of competing species
can coexist on four resources which regenerate according to an algebraic relationship.
We show that previous attempts to prove that n species cannot coexist on fewer
than n resources (the “competitive exclusion principle”) all make use of the
very restrictive assumption that the specific growth rates of all competing species are
linear functions of resource densities. When this restriction is relaxed, it becomes
possible to find situations in which n species can coexist on fewer than n
resources. On the basis of this and other observations we conclude that the competitive
exclusion principle should be considered to apply only to coexistence at fixed densities.