Modelling Biodiversity and Pattern Formation in Communities Exhibiting Cyclic Dominance
Understanding the mechanisms allowing the maintenance of biodiversity is a central issue in theoretical biology and ecology. Evolutionary game theory, where the success of one species depends on what the others are doing, provides a promising framework to theoretically investigate co-evolving populations. Recent experiments on microbial populations have shown that the existence of local and cyclic interactions promotes the long-term coexistence of all species and the formation of spatial patterns. In this context, rock-paper-scissors games - in which rock crushes scissors, scissors cut paper, and paper wraps rock - have emerged as a fruitful metaphor for non-hierarchical co-evolutionary dynamics.
Here, I shall discuss stochastic evolutionary models where three species cyclically dominate each other according to "rock-paper-scissors" interactions. After an overview of recent inspiring experimental results concerning biodiversity in some microbial communities, I will describe the mathematical modelling of the noisy nonlinear dynamics in the absence of spatial structure [Phys. Rev. E 74, 051907 (2006)]. I shall then report on recent findings concerning the combined influence of spatial degrees of freedom (local dispersal) and stochasticity on the population's coexistence [Nature 448, 1046 (2007)]. In particular, I will show that there is a subtle interplay between the individuals' mobility and local interactions which leads to the loss of biodiversity above a certain mobility threshold. Below that critical value, all species coexist and self-organise in fascinating moving patterns (entanglement of spirals waves). It has been elucidated that those kaleidoscopic spatio-temporal structures stem from an interplay between the deterministic dynamics and internal noise [Phys. Rev. Lett. 99, 238105 (2007), J. Theor. Biol. 254, 368 (2008)]. I will outline how the theory of front propagation allows to derive analytical expressions for the velocity and the wavelength of the rotating spiral waves, and thus to determine the state diagram of the spatially-extended system that is characterised by a uniform phase (only one species survives) and another one (bio-diverse) where all species coexist.