Background/Question/Methods Early experimental and modeling work in ecology led to the principle
of competitive exclusion : among several species competing for a
single common resource, only the best competitor survives, and, when
multiple resources are the subject of competition, the number of
coexisting species cannot exceed the number of resources. The
apparent contradiction between the competitive exclusion principle
and species diversity in natural communities has been a
long-standing enigma. Mathematical arguments in support of the
principle are based on the assumption that species in competition
approach an equilibrium.  In the case of competition for one or two
limiting resources in the setting of a chemostat, this assumption is
valid and leads to a rigorous mathematical proof. However, recent
numerical simulations of J. Huisman and F.J. Weissing suggest that
models with three or more essential (non-substitutable) limiting
resources can generate sustained oscillations or even chaotic
dynamics of species abundance,  under constant resource supply and
constant physical conditions. In particular, they show numerically
that periodic oscillations occur if three species compete for three
resources, chaotic oscillations occur if five species compete for
five resources, and up to nine species can seemingly be supported by
three recourses. The existence of an attracting limit cycle for the
case of three species competing for three resources plays a
fundamental role in generating coexistence of more species on three
essential resources.

Results/Conclusions

In this talk I will review the classical model of multispecies competition for multiple essential resources using Liebig'slaw of the minimum functional response. I will then focus on some special cases where mathematically rigorous results can be established. These include the case of 3 species competing for 3 resources and 4 species competing for 3 resources.