Using random matrix theory to determine the stability of the generalised Lotka-Volterra equations

The generalised Lotka-Volterra equations (GLVEs) are a widely-used modelling tool in theoretical ecology, and a prominent example of a disordered system. The GLVEs model the time evolution of species abundances in a large ecosystem. The interactions coefficients are drawn from a joint probability distribution and fixed throughout the dynamics. When we integrate the GLVEs forwards in time from some initial condition, some of the original pool of species will go extinct and some will survive. The stability of the resulting surviving community of species has been successfully deduced using Dynamic Mean-Field Theory (DMFT) previously. 



In this talk, we use DMFT to examine the statistics of the interactions between the species in the surviving community, which differ from those of the original pool of species. Using random matrix theory, we then show how the specifics of these interaction statistics affect the stability. In particular, we show that the full non-Gaussian nature of these statistics has to be taken into account in order to correctly predict stability. This is in contrast to the universality principle of random matrix theory, which suggests that the eigenvalues of a random matrix can be deduced from only the first and second moments and correlations of its elements. We discuss the implications of our findings both for random matrix theory and for theoretical ecology.



https://zoom.us/j/98286706234?pwd=bm1JUFVYcTJkaVl1VU55L0FiWDRIUT09