  We discuss a resource-competition model, which takes the MacArthur's model as
a platform, to unveil interesting connections with glassy features and jamming
in high dimension. This model presents two qualitatively different phases: a
"shielded" phase, where a collective and self-sustained behavior emerges, and a
"vulnerable" phase, where a small perturbation can destabilize the system and
contribute to population extinction. We first present our perspective based on
a strong similarity with continuous constraint satisfaction problems in their
convex regime. Then, we discuss the stability in terms of the computation of
the leading eigenvalue of the Hessian matrix of the free energy in the replica
space. This computation allows us to efficiently distinguish between the two
aforementioned phases and to relate high-dimensional critical ecosystems to
glassy phenomena in the low-temperature regime.
