Derivation and dynamics of discrete population models with distributed delay in reproduction

We introduce a class of discrete single species models with distributed delay in the reproductive process and a cohort dependent survival function that accounts for survival pressure during that delay period. These delay recurrences track the mature population for species in which individuals reach maturity after at least $\tau$ and at most $\tau +{\tau }_M$ breeding cycles. Under realistic model assumptions, we prove the existence of a critical delay threshold, ${\tilde{\tau }}_c$. For given delay kernel length ${\tau }_M$, if each individual takes at least ${\tilde{\tau }}_c$ time units to reach maturity, then the population is predicted to go extinct. We show that the positive equilibrium is decreasing in both $\tau$ and ${\tau }_M$. In the case of a constant reproductive rate, we provide an equation to determine ${\tilde{\tau }}_c$ for fixed ${\tau }_M$, and similarly, provide a lower bound on the kernel length, ${\tilde{\tau }}_M$ for fixed $\tau$ such that the population goes extinct if ${\tau }_M\ge {\tilde{\tau }}_M$. We compare these critical thresholds for different maturation distributions and show that if all else is the same, to avoid extinction it is best if all individuals in the population have the shortest delay possible. We apply the model derivation to a Beverton–Holt model and discuss its global dynamics. For this model with kernels that share the same mean delay, we show that populations with the largest variance in the time required to reach maturity have higher population levels and lower chances of extinction.