  Random matrix theory successfully connects the structure of interactions of
large ecological communities to their ability to respond to perturbations. One
of the most debated aspects of this approach is the missing role of population
abundances. Despite being one of the most studied patterns in ecology, and one
of the most empirically accessible quantities, population abundances are always
neglected in random matrix approaches and their role in determining stability
is still not understood. Here, we tackle this question by explicitly including
population abundances in a random matrix framework. We obtain an analytical
formula that describes the spectrum of a large community matrix for arbitrary
feasible species abundance distributions. The emerging picture is remarkably
simple: while population abundances affect the rate to return to equilibrium
after a perturbation, the stability of large ecosystems is uniquely determined
by the interaction matrix. We confirm this result by showing that the
likelihood of having a feasible and unstable solution in the Lotka-Volterra
system of equations decreases exponentially with the number of species for
stable interaction matrices.
