2011/03/08 by Clinton H. Durney, C. H. Durney, Sara Case +4 · 2 citations
Biochemistry, Genetics and Molecular Biology · Mathematics · Medicine · Physics and Astronomy · Social Sciences · #Biology #Competition (biology) #Ecology #Evolution and Genetic Dynamics #Evolutionary Game Theory and Cooperation #Evolutionary biology #Exponential function #Field (mathematics) #Function (biology) #Generalization #Mathematical analysis #Mathematical and Theoretical Epidemiology and Ecology Models #Mathematics #Physics #Population #Pure mathematics #Statistical physics #Statistics #Variety (cybernetics) #cond-mat.stat-mech #q-bio.PE
paper · pdf · doi:10.1103/physreve.83.051108
published as Phys. Rev. E 83, 051108 (2011) · 15 pages, 5 figures, submitted to Physical Review E
arxiv created 2011/03/08 · openalex publication_date 2011/05/09 · arxiv updated 2015/05/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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.