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

Some previously studied discrete delayed population models have included the often neglected mortality of immature individuals during the maturation delay, making them a more suitable representation of populations with delayed recruitment. For such models, it was found that increasing the delay decreases the equilibrium population size, eventually leading to extinction. However, since maturation delays beyond one breeding cycle are often found in nature, such delayed maturation must also have a benefit leading to a trade-off. In this work, we expand on these discrete delayed population models that incorporate mortality during the immature phase by including a trade-off with reproduction. We motivate such trade-off with the observation that a longer maturation allows the individuals to reach larger sizes, which has been shown to be positively correlated with egg production. We then distinguish between density independent survival and cohort density dependent survival of immature individuals. Prototypes for the survival include Beverton-Holt and Ricker survival. Across all models, we identify a critical delay threshold that indicates the extinction of the species and discuss the existence and stability of a positive equilibrium. For a class of models, we obtain an optimal delay that results in the largest equilibrium level, as well as a delay range that results in oscillatory dynamics for the Ricker survival function. Overall, our delay model sets up a useful phenomenological framework to test multiple combinations of trade-offs in parent survival, offspring survival, and reproductive investment.