Understanding Coexistence Outcomes for Intransitive Competition Using Properties of Circulant Matrices

Abstract:

A persistent puzzle in community ecology is how so many competing species can coexist in nature despite a naive expectation that the best competitor for shared limiting resources should win. Intransitive interaction structures have been proposed to importantly influence competitive coexistence outcomes for ecological communities. This structure involves a loop of pairwise interactions in which each species dominates over the next if the two were isolated, but it contains no single dominant competitor for the entire system because the last species dominates the first. Intransitivity is distinctly different than “niche differentiation," the key mechanism of stable competitive coexistence that ecologists focus on, where interspecific competition is weaker than intraspecific competition. In contrast, intransitive structures require that the dominant competitor in each interacting pair has greater interspecific competitive effects on the other than it has on itself. Despite a clear difference in mechanism, so far, results have suggested that communities with intransitive competition can also lead to stable coexistence for loops of an odd number, but not for an even number, an idea we call the “even-odd” hypothesis. Existing literature, however, leaves many important questions open about the general tendency towards stable coexistence generated by intransitive interactions.

To answer some of these questions, we exploit the properties of circulant matrices. Both community interaction matrices and Jacobian matrices at the coexistence equilibrium take on this circulant structure under a Lotka-Volterra competition model with intransitive interactions of identical interaction strengths around the loop. We can understand coexistence outcomes for this system by analyzing the eigenvalues of these circulant matrices. We also carry out numerical eigenvalue analyses for non-circulant cases arising when interaction strengths vary. Overall, we provide a more general confirmation of the even-odd hypothesis for a single isolated intransitive loop interaction structure, but also elucidate the more complex story that arises in the contexts of additional community-wide interactions and multiple intransitive loops.