Mathematical Biology Seminar Daniel Maes, University of Utah, Wednesday, September 10, 2025 12:30 pm in LCB 222 Coexistence Outcomes for Species Under Intransitive Competition 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 type of 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 (a Rock-Paper-Scissors dynamic). Thus far in the literature, analyses have suggested that communities with intransitive competition can 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--together with general Lotka-Volterra competition assumptions--to understand community coexistence outcomes. We can understand coexistence outcomes for such systems by analyzing the eigenvalues of circulant community matrices, and also carry out numerical eigenvalue analyses for non-circulant cases that retain intransitivity. 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.