  We consider the Generalized Lotka-Volterra system of equations with
all-to-all, random asymmetric interactions describing high-dimensional, very
diverse and well-mixed ecosystems. We analyze the multiple equilibria phase of
the model and compute its quenched complexity, i.e., the expected value of the
logarithm of the number of equilibria of the dynamical equations. We discuss
the resulting distribution of equilibria as a function of their diversity,
stability and average abundance. We obtain the quenched complexity by means of
the replicated Kac-Rice formalism, and compare the results with the same
quantity obtained within the annealed approximation, as well as with the
results of the cavity calculation and, in the limit of symmetric interactions,
of standard methods to compute the complexity developed in the context of
glasses.

Disordered Systems and Neural Networks;Statistical Mechanics