Biology – Quantitative Biology – Populations and Evolution
Scientific paper
2011-03-08
Phys. Rev. E 83, 051108 (2011)
Biology
Quantitative Biology
Populations and Evolution
15 pages, 5 figures, submitted to Physical Review E
Scientific paper
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 which evolves simply as an exponential: $\mathcal{Q}% \propto e^{\lambda t}$, 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\rightarrow -\infty $). 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.
Case Sara O.
Durney Clinton H.
Pleimling Michel
Zia Royce K. P.
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