Mathematical investigation of changes in stability of dynamically evolving ecosystems

大型生态系统模型的局部稳定性问题已经得到了很多研究，其分析结果主要依赖于随机矩阵理论。从本质上讲，这种方法假设生态系统中物种的静态配置，并询问这样一个系统是否会稳定到物种丰度的小扰动。然而，众所周知，生态系统中物种的丰度和物种之间的相互作用会随着时间的推移而演变。这些进化力量可能会对生态系统的稳定性产生影响。在最近关于小（低维）生态系统演化的工作之后[1]，有可能将开发的技术扩展到通常属于随机矩阵理论范围的高维系统。通过结合这两种方法，打开了一个分析理解生态系统稳定性如何可能改变生态和进化尺度的潜力。正如所提到的，相互作用结构中的统计模式对生态系统稳定性的影响已经在生态时间尺度上进行了研究[2]，其中物种属性在时间上是固定的。然而，在进化系统中，物种会发生突变。这不仅导致物种特性的变化，而且还导致物种内变异的产生。这样做的一个后果是，动力系统的维数可以改变。因此，本研究的主要数学挑战之一将是修改标准的分析方法，使用随机矩阵理论来解释这种不寻常的动力学行为。在这样做的时候，我们的目标是弥合现有的方法之间的差距，到目前为止，一直限制在考虑高维系统的生态时间尺度，一方面，或低维系统的进化时间尺度上的其他。[1]“由突变相互作用驱动的种群规模变化和种群灭绝风险”HJ Park，Y Pichugin，W Huang，A Traulsen Physical Review E 99（2），022305（2019）[2]“微生物组的生态学：网络，竞争和稳定性”KZ Coyte，J Schluter，KR Foster Science 350（6261），663-666（2015）

Much research has been conducted on the local stability of large ecosystem models, with analytic results primarily reliant on random matrix theory. Essentially this approach assumes a static configuration of species in an ecosystem and asks whether such a system would be stable to small perturbations in species abundances. However, the abundance of, and interactions between, species in ecosystems are known to evolve over time. These evolutionary forces are likely to have an impact on ecosystem stability. Following recent work on the evolution of small (low-dimensional) ecosystems [1], there is scope to extend the techniques developed to the high dimensional systems that normally fall under the purview of random matrix theory. By combining these two approaches, one opens up the potential for an analytical understanding of how ecosystem stability might change over ecological and evolutionary scales. As addressed, the influence of statistical patterns in interaction structure on ecosystem stability has been studied at ecological time scales [2] in which species properties are fixed in time. However in evolving systems species are subject to mutation. This not only leads to changes in species properties, but also to the generation of within-species variation. A consequence of this is that the dimension of the dynamical system can change. One of the primary mathematical challenges of this study will therefore be to modify standard analytical approaches that use random matrix theory to account for this unusual dynamical behaviour. In doing so we aim to bridge the gap between existing approaches which have until now been constrained to considering high-dimensional systems at ecological timescales on the one hand, or low-dimensional systems at evolutionary timescales on the other. [1] "Population size changes and extinction risk of populations driven by mutant interactors" HJ Park, Y Pichugin, W Huang, A Traulsen Physical Review E 99 (2), 022305 (2019)[2] "The ecology of the microbiome: networks, competition, and stability" KZ Coyte, J Schluter, KR Foster Science 350 (6261), 663-666 (2015)