Saddles, Arrows, and Spirals: Deterministic Trajectories in Cyclic Competition of Four Species
| Autor: | Durney, C. H., Case, S. O., Pleimling, M., Zia, R. K. P. |
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| Rok vydání: | 2011 |
| Předmět: | |
| Zdroj: | Phys. Rev. E 83, 051108 (2011) |
| Druh dokumentu: | Working Paper |
| DOI: | 10.1103/PhysRevE.83.051108 |
| Popis: | 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. Comment: 15 pages, 5 figures, submitted to Physical Review E |
| Databáze: | arXiv |
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