Description
The dynamics of ecological communities are intrinsically linked to ecological structures such as the network of species interactions or the phenotypic substructure of different species. In this talk, I will give two examples of minimal models that elucidate the interplay of these structures and ecological dynamics. In both cases, I will highlight open questions that (new) (numerical) algebraic methods could answer.
In the first part of my talk, I will analyse the effect of the network of competitive, mutualistic, and predator-prey interactions on stability of coexistence [1]. I will show that the possibility of stable coexistence in ecologies with Lotka-Volterra dynamics is determined completely by "irreducible ecologies", and I will explain how exhaustive analysis of all such interaction networks of N<6 species suggests that, strikingly, these irreducible ecologies form an exponentially small subset of all ecologies, as do the mathematically curious "impossible ecologies" in which stable coexistence is non-trivially impossible.
In the second part of my talk, I will introduce a minimal model of spatial structure in the competition of two species. One of these species switches, both stochastically and in response to the other species, to a phenotype resilient to competition [2]. In particular, I will ask: How does this phenotypic switching affect travelling waves by which one species invades the other? Combining exact and numerical results, I will reveal that, very surprisingly, phenotypic switching does not change the speed of these travelling waves.
[1] Meng, Horvát, Modes, and Haas, arXiv:2309.16261
[2] Gupta and Haas, in preparation
