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

在自然界中，有机体与其周围有限数量的个体相互作用，它们在移动，因此生态系统不能很好地用混合模型来描述。事实上，种群经常通过形成空间模式进行自我组织，这种模式的起源是一个激烈的研究主题。最近，人们对研究涨落和非线性如何影响生物模式的形成越来越感兴趣。尽管取得了进展，但新出现的模式和流动性的生物学意义仍然不清楚。在一系列的工作中，Mobilia和他的同事考虑了一个线性扩散的空间岩石-布-剪刀模型，并建立了一个扩散阈值，低于这个阈值所有物种共存并形成螺旋状的模式，而高于这个阈值只有一个物种存活下来[见，例如，自然*448*，1046(2007)]。最近，在Mobilia和Rucklidge指导的一篇博士论文中，发展了一种集合种群建模方法，在基于个体的模型及其宏观描述之间建立了严格的联系[EPL*102*，28012(2013年)；Phys.修订本E*90*，032704(2014年)]。然而，到目前为止，还没有进行系统的研究来了解非线性迁移率和噪声的共同作用对物种共存的影响，以及由此产生的模式的进化意义是什么。这些问题和相关问题将被研究用于一个岩石-布-剪刀模型，在该模型中，代理人的运动是直接的(由营养物质驱动)。在集合种群方法中，这个模型将在微观层面上被研究，然后被放大以得到带有非线性扩散项的偏微分方程组。一个目标将是了解在什么情况下会出现相干模式，以及噪声和定向非线性迁移率如何影响它们的性质。我们还将系统地研究自组织模式如何影响物种灭绝前共存的时间。

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 anincreasing 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 theindividual-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.