  Consider a large ecosystem (foodweb) with n species, where the abundances
follow a Lotka-Volterra system of coupled differential equations. We assume
that each species interacts with d other species and that their interaction
coefficients are independent random variables. This parameter d reflects the
connectance of the foodweb and the sparsity of its interactions especially if d
is much smaller that n. We address the question of feasibility of the foodweb,
that is the existence of an equilibrium solution of the Lotka-Volterra system
with no vanishing species. We establish that for a given range of d with an
extra condition on the sparsity structure, there exists an explicit threshold
depending on n and d and reflecting the strength of the interactions, which
guarantees the existence of a positive equilibrium as the number of species n
gets large. From a mathematical point of view, the study of feasibility is
equivalent to the existence of a positive solution (component-wise) to the
equilibrium linear equation. The analysis of such positive solutions
essentially relies on large random matrix theory for sparse matrices and
Gaussian concentration of measure. The stability of the equilibrium is
established. The results in this article extend to a sparse setting the results
obtained by Bizeul and Najim in Proc. AMS 2021.
