Non-Markovian Rock-Paper-Scissors games

Abstract: There is mounting evidence that species interactions often involve long-term memory, with highly-varying waiting times between successive events and long-range temporal correlations. Accounting for memory undermines the common Markovian assumption, and dramatically impacts key ingredients of population dynamics including birth, foraging, predation, and competition processes. Here, we study a critical aspect of population dynamics, namely non-Markovian multi-species competition. This is done in the realm of the zero-sum rock-paper-scissors (zRPS) model that is broadly used in the life sciences to metaphorically describe cyclic competition between three interacting species. We develop a general non-Markovian formalism for multi-species dynamics, allowing us to determine the regions of the parameter space where each species dominates. In particular, when the dynamics are Markovian, the waiting times are exponentially distributed and the fate of the zRPS model in large well-mixed populations is encoded in a remarkably simple condition, often referred to as the ``law of the weakest'' (LOW), stating that the species with the lowest growth rate is the most likely to prevail. We show that the survival behavior and LOW of the zRPS model are critically affected by non-exponential waiting times, and especially, by their coefficient of variation. Our findings provide key insight into the influence of long waiting times on non-Markovian evolutionary processes.