Linda A. Buttel, Richard Durrett, and Simon A. Levin

Abstract. In models of competition in which space is treated as a continuum, and population size as continuous, there are no limits to the number of species that can coexist. For a finite number of sites, N, the results are different. The answer will, of course, depend on the model used to ask the question. In the Tilman-May-Nowak ordinary differential equation model, the number of species is asymptotically a constant times log N, with most species packed in at the upper end of the range of possible species. In contrast, for metapopulation models with discrete individuals and stochastic spatial systems with various competition neighborhoods, we find a traditional power-law species area relationship CN^a, with no species clumping. The exponent a is larger by a factor of 2 for spatially explicit models. In words, a spatial distribution of competitors allows for greater diversity than a metapopulation model.

Introduction. Two of the central problems in ecology are:

How can the large number of species on Earth can coexist?

and in the other direction:

What sets limits on diversity?

Almost 40 years ago, G.E. Hutchinson (1961) noted that the coexistence of hundreds of species of algae in lakes is not consistent with the

Competitive Exclusion Principle, which predicts that in homogeneous and equilibrial systems, the number of coexisting species cannot exceed the number of resources.

Tilman (1994) studied coexistence among a sequence of species in which the lower-numbered species are superior competitors. Letting p_i be the fraction of patches occupied by type i and taking the limit of an infinite number of patches, he arrived at the following dynamics:

dp_i/dt = b_i p_i ( 1 - p₁ - ... - p_i ) - d₁ p₁ - p₁ ( b₁ p₁ + ... + b_i-1 p_i-1 )

Here b_i and d_i are the birth and death rates for species i. The first term on the right-hand side represents births by type i onto sites that are vacant or occupied by inferior competitors. The second and third terms represent loss of sites of type i due to deaths or takeover by lower numbered species. When i = 1 this says

dp₁/dt = b₁ p₁ ( 1 - p₁) - d₁ p₁

so the fraction of space occupied by type 1 in equilibrium type is p₁ = (b ₁ - d ₁) / b₁. In general, the equation for each p_i only involves p₁ ... p _i so the equations can be solved recursively for the equilibrium frequencies.