Let $M$ be the real square matrix of a typed branching process, such that $M_{ij}$ is the expected value of offspring of type $j$ emanating from type $i$.

We know that if the first eigenvalue if $M$ is smaller than 1, then all types will be extinct, and if it is larger than 1, then with positive probability some types won't get extinct.

Is there some interpretation of the eigenvalue other than that? (its actual value.)

if it is larger than 1, then there are types that won't get extinct... is not accurate: rather, there is a positive probability that some types will not get extinct. – Did Apr 22 '13 at 7:57