Speaker: Diego Rojas La Luz (University of Wisconsin-Madison)
Unveiling Surprising Connections Between the Classical Theory of Reaction Networks and Generalized Lotka-Volterra Systems
In this talk we explore the relationship between Reaction Networks and Population Dynamics, with a specific focus on Generalized Lotka-Volterra systems (also called Kolmogorov systems). Surprisingly, we find strong analogies between classical Mass Action Kinetics results (like the Horn-Jackson theorem and the deficiency-zero theorem) and new counterparts for Generalized Lotka-Volterra (GLV) systems, hinting at a deep connection, where previously none was known. Notably, in the GLV setting, we can prove that “complex-balanced” equilibria (properly defined) are globally attractive (which corresponds to the “global attractor conjecture" in the Reaction Networks setting). As an example, we show how to apply this new theory to characterize global stability for a large class of cooperative GLV systems with Higher Order Interactions. A remarkable feature of the theory is that it allows us to prove global stability without restrictions of the dimension or degree of the ODE. We also extend our results to analyze the case where the parameters are allowed to freely change over time (non-autonomous case), called variable-k systems, an area not yet fully explored in the context of GLV systems. We especially focus on versions of the Persistence Conjecture and Permanence Conjecture for GLV systems. This is joint work with Polly Yu and Gheorghe Craciun.