  We study the biodiversity problem for resource competition systems with
extinctions and self-limitation effects. Our main result establishes estimates
of biodiversity in terms of the fundamental parameters of the model. We also
prove the global stability of solutions for systems with extinctions and large
turnover rate. We show that when the extinction threshold is distinct from
zero, the large time dynamics of system is fundamentally non-predictable. In
the last part of the paper we obtain explicit analytical estimates of ecosystem
robustness with respect to variations of resource supply which support the
$R^*$ rule for a system with random parameters.
