  This paper is divided into two parts. The first part is devoted to the study
of a class of Approximate Message Passing (AMP) algorithms which are widely
used in the fields of statistical physics, machine learning, or communication
theory. The AMP algorithms studied in this part are those where the measurement
matrix has independent elements, up to the symmetry constraint when this matrix
is symmetric, with a variance profile that can be sparse. The AMP problem is
solved by adapting the approach of Bayati, Lelarge, and Montanari (2015) to
this matrix model. \\ The Lotka-Volterra (LV) model is the standard model for
studying the dynamical behavior of large dimensional ecological food chains.
The second part of this paper is focused on the study of the statistical
distribution of the globally stable equilibrium vector of a LV system in the
situation where the random symmetric interaction matrix among the living
species is sparse, and in the regime of large dimensions. This equilibrium
vector is the solution of a Linear Complementarity Problem, which distribution
is shown to be characterized through the AMP approach developed in the first
part. In the large dimensional regime, this distribution is close to a mixture
of a large number of truncated Gaussians.
