Student Probability Seminar
Stability and transience in complex systems via random matrices
Speaker: Guillaume Dubach, CIMS
Location: Warren Weaver Hall 1314
Date: Monday, May 6, 2019, 12:30 p.m.
One important area of research and debate in ecology is the so-called complexity vs. stability issue. It was started by a now celebrated article by Robert May (1972), which suggested that a large complex system driven by generic non-linear ordinary differential equations would be unlikely to exhibit a point of stable equilibrium if its variables are 'too well connected' and 'too random' at the same time; this surprising theoretical fact can be directly applied to population dynamics, where the system is typically driven by a system of Lotka-Volterra equations, and many other settings.
Following a review article by Allesina and Tang, we will see how May's result can be recovered and precisely quantified using eigenvalue statistics from non-Hermitian random matrix theory. This analysis has been refined very recently by Jacek Grela, who also studied the existence of stable non-transient trajectories. In this context, transient means that the trajectory goes away from the equilibrium before converging to it; understanding transient behavior implies not only eigenvalues, but eigenvectors of non-Hermitian random matrices.