title: Saddles, arrows, and spirals: deterministic trajectories in cyclic competition of four species.
abstract: 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 ∝ e(λt), where λ is a function of the reaction rates. It provides information on the state of the system for late times (as well as for t→-∞). 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.
authors: Department of Physics, Virginia Tech, Blacksburg, Virginia 24061-0435, USA.
authors: C H
authors: CH
authors: Durney
authors: S O
authors: SO
authors: Case
authors: M
authors: M
authors: Pleimling
authors: R K P
authors: RK
authors: Zia
mesh_terms: D005075:Biological Evolution; D053476:Extinction, Biological; D008962:Models, Theoretical; D011157:Population Dynamics; D013269:Stochastic Processes