Abstract: Biological systems can undergo sudden transitions when parameters change, making stability analysis of steady states essential. We develop an algebraic framework for studying the parameter spaces of ecological ODE models with polynomial or rational rates. Using tools from computational algebra, we identify boundaries separating stability regimes, and using routing functions, we compute the regions in their complement together with the stability behavior in each region. We demonstrate the method on the classical Levins-Culver competition-colonization model and a modern tripartite symbiosis model, where it uncovers rich and previously hidden stability structure.

Title: On strongly robust toric ideals: when Markov and Graver bases coincide.
Abstract: In the first part of the talk, I will give an introduction to Markov bases of toric ideals, which are one of the first connections between statistics and commutative algebra. In the second part of the talk, I will discuss recent work on strongly robust toric ideals. A toric ideal is called strongly robust when the Graver basis is a minimal system of generators. In the talk, I will explain how to build a strongly robust simplicial complex which determines the strongly robust property of toric ideals. I will then discuss our results on the strongly robust property in the case of monomial curves, as well as codimension 2 toric ideals and configurations in general position. This is joint work with A. Thoma and M. Vladoiu.