# 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.

**Synopsis:**

`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

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.`