How Space and Noise Influence the Coexistence of Species in Cyclic Competition

In Nature, organisms interact with a finite number of individuals in their neighbourhood, they move and ecosystems are therefore not aptly described by well-mixed models. In fact, populations often self-organise by forming spatial patterns whose origin is an intense subject of research. Recently, there has been an
increasing interest in investigating how fluctuations and nonlinearity influence the formation of biological patterns. In spite of the progress made, the biological significance of emerging patterns and mobility still remains unclear. Here, these questions will be investigated within the framework of the paradigmatic spatial rock-paper-scissors game motivated by the cyclic competition in microbial experiments [B. Kerr et al., Nature *418*, 171 (2002)].
In a series of works, Mobilia and co-workers considered a spatial rock-paper-scissors model with linear diffusion and established a diffusion threshold below which all species coexist and form spiralling patterns, while above the threshold only one species survives [see, e.g., Nature *448*, 1046 (2007)]. Recently, in a PhD thesis supervised by Mobilia and Rucklidge, a metapopulation modelling approach was developed to establish a rigorous connection between the
individual-based model and its macroscopic description [EPL *102*, 28012 (2013); Phys. Rev. E *90*, 032704 (2014)]. However, as yet no systematic studies have been carried out to understand how species coexistence is affected by the joint effect of nonlinear mobility and noise, and what is the evolutionary significance of the resulting patterns.
These and related questions will be investigated for a rock-paper-scissors model in which the agents' movement is directed (driven by nutrients). Within a metapopulation approach, this model will be studied at micro-level and then up-scaled to yield partial differential equations with nonlinear diffusion terms. One goal will be to understand under which circumstances coherent patterns emerge and how noise and directed nonlinear mobility affect their properties. We will also systematically investigate how the self-organisation into patterns influences the time during which species coexist before extinction.