  We consider ecological difference equations of the form $X_{t+1}^i =X_t^i
A_i(X_t)$ where $X_t^i$ is a vector of densities corresponding to the
subpopulations of species $i$ (e.g. subpopulations of different ages or living
in different patches), $X_t=(X_t^1,X_t^2,\dots,X_t^m)$ is state of the entire
community, and $A_i(X_t)$ are matrices determining the update rule for species
$i$. These equations are permanent if they are dissipative and the extinction
set $\{X: \prod_i \|X^i\|=0\}$ is repelling. If permanence persists under
perturbations of the matrices $A_i(X)$, the equations are robustly permanent.
We provide sufficient and necessary conditions for robust permanence in terms
of Lyapunov exponents for invariant measures supported by the extinction set.
Applications to ecological and epidemiological models are given.
