i'm currently looking at a problem from population dynamics. The assumption is that a colony of wild-type cells growth according to the "gompertz-function"

$f(t)=m^{1-\exp(-\lambda_0 t)}$

where $m$ is some given integer and $\lambda_0>0$. Mutants can be produced depending on the "size" of the wild-type population proportional to the mutation rate $\nu$. Mutants behave like a (supercritical) birth and death process, the associated generating function of this is denoted with $\phi(z,t)$.

In a paper by Dewanji et. al. the generating function $G(z,t)$ of the number of mutants at a fixed time is given by

$\log(G(z,t))=\int_0^t\nu f(s)[\phi(z,t-s)-1]ds$

Now, i'd like to take a limit $m\rightarrow\infty$, $\nu\rightarrow0$, such that, say, $m\nu=1$. But i have no idea how to tackle such a problem.

For the case when $f(t)=\exp(\lambda_0t)$, i do have an answer, but the method cannot be generalized. Does anyone have an idea how to tackle such a problem in general? It seems to me, it might not even possible to give an explicit answer? At least Mathematica failed and i don't see why/how i could exchange limit and integration.

Thanks in advance for any hints.

Best regards, Peter

PS: $\phi$ is the generating function of a negative binomial, but that might not be relevant.