imilar way:
$$
 \overline{\overline{Af_{[t]}}}=
   \overline{\overline{Af_{[t-1]}}} * (1-\mu_{ad}) * \overline{\omega_{f}}\\
   +  \bigg( \overline{\phi} * \overline{E^{'}}+\overline{\nu f_{[t-1]}}\bigg)^{\top} * \bigg( \frac{\overline{\overline{\eta}}*\overline{Am_{[t-1]}}}{\sum{\overline{Am_{[t-1]}}}} \bigg)%\overline{\overline{Mf}}
$$
where we first compute the surviving female adults from one day to the next;
and then we calculate the mating composition of the female fraction emerging
from pupa stage. T