Abstract: Intransitive loops of competition are akin to a game of Rock-Paper-Scissors (RPS) where there is not one dominant competitor (e.g. no dominant winning strategy in RPS). That is if Species A outcompetes Species B (A >B) and Species B outcompetes Species C (A >C), rather than Species A necessarily outcompeting Species C, we find in the intransitive case that Species C outcompetes Species A (C >A). Intransitive loops have been observed among a variety of organisms and provide a potential means of competitive coexistence fundamentally different than classically studied niche differences, in which all species impact themselves more than other species. A prevalent hypothesis about intransitive loop interactions is that loops including an even number of species are unstable whereas ones of odd length are stable. While this claim dates back to the work of Gilpin in the 1970s, there has not been a clear analytical proof of it in general. In particular, a more recent theoretical study supporting this assertion is based on the simplified case of zero-sum community dynamics, where the total size of the community is fixed. In this case, loops of odd length actually produce neutral cycles rather than a stable coexistence equilibrium point.
Here we instead use the Lotka-Volterra competition model to study intransitive loop dynamics. We employ an analytical approach applied in the 1970s by May & Leonard to the case of a 3-species intransitive loop. We extend this approach to communities with an arbitrary number of species interacting through a single intransitive loop that is “symmetric”, meaning that interaction strengths between pairs of interacting species in the loop are constant across interaction pairs. We show that for such symmetric loops, the coexistence equilibrium point of the Lotka-Volterra competition model with even-length loops is indeed always locally unstable, while for odd-length loops local stability is possible. We also use numerical analyses to show that these results are robust to small variation in competition strengths across species pairs. These results solidify a long-standing but unproved assertion that odd-length intransitive loops can lead to stable coexistence. However, we also find that the range of competition strengths allowing for this stable coexistence shrinks as the number of species increases. This points to the importance of further study of more complex interaction structures involving intransitive loops, to better understand them as a potential means by which competing species coexist in nature.
