Mathematical investigation of changes in stability of dynamically evolving ecosystems

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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)