Adding a fecundity-survival trade-off to a discrete population model with maturation delay

Although maturation delays of new individuals are frequently added in population models, researchers rarely account for the mortality of individuals between birth and maturity. Previous studies have added mortality of immature individuals to the maturation delay in discrete population models, finding that increasing the delay results in a decreased population level at equilibrium. Furthermore, increasing the maturation delay beyond a critical threshold ultimately leads to the extinction of the modeled species. Because maturation delays beyond one breeding cycle are often found in nature, clearly, maturation delays must also have a positive benefit leading to a trade-off. We explore such a trade-off by deriving a class of discrete single species models with a maturation delay and the realistic assumption of a reproductive trade-off. In this reproductive trade-off, although a prolonged maturation delay decreases the total probability of survival, the delay results in larger body sizes which we assume, based on some fish species, to result in higher fecundity. We examine two scenarios: density independent survival and cohort density dependent survival of immature individuals. For the mature and immature individuals, we consider two different, but popular, survival functions: the Beverton-Holt model and the Ricker model. Overall, we identify a positive maturation delay that maximizes the population size, which we refer to as "optimal maturation delay".