| Abstract: |
| Intransitive loops of competition are akin to a game of Rock-Paper-Scissors where there is not one dominant competitor. A prevalent hypothesis for intransitive loop interactions is that even-lengthed loops of species are unstable whereas odd-lengthed ones are stable. While this claim is common throughout the literature, there has not been a clear analytical proof of this result in general. Here we use the Lotka-Volterra competition model to study intransitive loop dynamics. We employ analytical approaches applied to the case of a 3-species intransitive loop in which the competition strengths are constant across species pairs in the loop and extend these approaches to arbitrarily many species. We also use numerical analyses to test the robustness of these results to variation in competition strengths across species pairs. We show analytically in the symmetric case that the coexistence equilibrium point of the Lotka-Volterra competition model with even-lengthed loops is indeed locally unstable, while for odd-lengthed loops it is locally stable. With numerical simulations, we can also understand the impacts of competition variability and correlation on the stability of such systems. All these results solidify a long-standing but unproved assertion that odd-lengthed intransitive loops can lead to stable coexistence. |
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