# Calculation of Malthusian fertility and mortality from birth and death rates

In a simple Malthusian model, the population $p$ grows according to $p = p_0e^{rt}$, where $r = \beta - \mu$, $\beta$ is the "fertility" parameter and $\mu$ the "mortality" parameter. In the case of $\beta$, the closest thing to this parameter is the Total Fertility Rate. Is there a meaningful way to convert from one to the other? An easy formula might be something along the lines of $$\beta \approx \frac{1}{2T}\text{TFR},$$ where $T$ is the reproductive lifetime, but this doesn't account for gender imbalances or a more nuanced way of treating the instantaneous nature of the parameter.

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 $$\frac\beta{p\mu}$$