Calculation of Malthusian fertility and mortality from birth and death rates
If every member of the population breeds, the mean number of children born from a single individual during their whole life is $\beta$ times the duration of their life. The mean life duration is $1/\mu$. Hence the total fertility rate in the homogenous model you suggest is $$\frac\beta\mu$$ If only some proportion $p$ of the population can breed, the total fertility rate of each breeding individual is $$\frac\beta{p\mu}$$