  We consider a generic nonlinear extension of May's 1972 model by including
all higher-order terms in the expansion around the chosen fixed point (placed
at the origin) with random Gaussian coefficients. The ensuing analysis reveals
that as long as the origin remains stable, it is surrounded by a "resilience
gap": there are no other fixed points within a radius r*>0 and the system is
therefore expected to be resilient to a typical initial displacement small in
comparison to r*. The radius r* is shown to vanish at the same threshold where
the origin loses local stability, revealing a mechanism by which systems close
to the tipping point become less resilient. We also find that beyond the
resilience radius the number of fixed points in a ball surrounding the original
point of equilibrium grows exponentially with N, making systems dynamics highly
sensitive to far enough displacements from the origin.
