Saddles, arrows, and spirals: Deterministic trajectories in cyclic competition of four species
Durney, C. H.
Case, S. O.
Zia, R. K. P.
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Population dynamics in systems composed of cyclically competing species has been of increasing interest recently. Here we investigate a system with four or more species. Using mean field theory, we study in detail the trajectories in configuration space of the population fractions. We discover a variety of orbits, shaped like saddles, spirals, and straight lines. Many of their properties are found explicitly. Most remarkably, we identify a collective variable that evolves simply as an exponential: Q proportional to e(lambda t), where lambda is a function of the reaction rates. It provides information on the state of the system for late times (as well as for t -> -infinity). We discuss implications of these results for the evolution of a finite, stochastic system. A generalization to an arbitrary number of cyclically competing species yields valuable insights into universal properties of such systems.