I have a semi-discrete model, tracking the number of insects that emerge from pupae, and I'm struggling a bit with the maths.
I have an emergence function, let's say $e(t)\sim N(\mu,\sigma^2)$, specifying that the adults emerge over time according to a normal distribution. So 50% emerge before $\mu$ and 50% after. If I integrate the normal distribution, from $-\infty$ to $\infty$, then all the adults emerge.
If I track the number of pupae, $P(t)$, starting from $P_0$, and the number of adults $A(t)$, then I can set up the system of equations:
$$ P(t_2) = P(t_1) - P(t_1) \int_{t_1}^{t_2} e(t) dt\\ A(t_2) = A(t_1) + P(t_1) \int_{t_1}^{t_2} e(t) dt $$
I have $e(t)$ multiply $P(t_1)$ so I can have events where some proportion of the pupae get killed at one or more points in time. However I still want the same proportion of remaining pupae emerge between any two points.
Nevertheless, even without killing some of the pupae this doesn't work, because I'm multiplying by $P(t_1)$ instead of $P_0$ (which obviously gets smaller over time) so not all the pupae emerge.
Anyone got any tips of how this should work?
Thanks