We have a population of people with different age $a$, time is indexed with $t$. There is a rate at which people die, $d(a, t)$. For simplicity, ignore births. I want to compute the evolution of the distribution of ages over time.
Let $m(a, t)$ denote the mass of people at age $a$ and point in time $t$. I will start with a discrete time approximation and let $\Delta$ go to zero. At each discrete point in time,
$$ m(a+\Delta, t+\Delta) = (1-P(a, t))m(a, t)$$
where $P(a, t) = exp(-d(a,t)\Delta)$ is the discrete time analogue of $d(a,t)$. As I'm going to let $\Delta\to 0$, I can approximate $P$ with $(1-\Delta d)$:
$$ m(a+\Delta, t+\Delta) = \Delta d(a,t)m(a, t)$$
Issue Already here I'm having trouble understanding what happened: $\Delta d(a,t)$ denotes the mass of people who died during $\Delta$ at age $a$ and time $t$. Shouldn't this be negatively affecting the mass of people alive? I was expecting something along the lines of
$$ m(a+\Delta, t+\Delta) = (1- \Delta d(a,t))m(a, t)$$