Bifurcation theory, adaptive dynamics and dynamic energy budget-structured populations of iteroparous species.

In this paper, we describe a technique to evaluate the evolutionary dynamics of the timing of spawning for iteroparous species. The life cycle of the species consists of three life stages, embryonic, juvenile and adult whereby the transitions of life stages (gametogenesis, birth and maturation) occur at species-specific sizes. The dynamics of the population is studied in a semi-chemostat environment where the inflowing food concentration is periodic (annual). A dynamic energy budget-based continuous-time model is used to describe the uptake of the food, storage in reserves and allocation of the energy to growth, maintenance, development (embryos, juveniles) and reproduction (adults). A discrete-event process is used for modelling reproduction. At a fixed spawning date of the year, the reproduction buffer is emptied and a new cohort is formed by eggs with a fixed size and energy content. The population consists of cohorts: for each year one consisting of individuals with the same age which die after their last reproduction event. The resulting mathematical model is a finite-dimensional set of ordinary differential equations with fixed 1-year periodic boundary conditions yielding a stroboscopic map. We will study the evolutionary development of the population using the adaptive dynamics approach. The trait is the timing of spawning. Pairwise and mutual invasibility plots are calculated using bifurcation analysis of the stroboscopic map. The evolutionary singular strategy value belonging to the evolutionary endpoint for the trait allows for an interpretation of the reproduction strategy of the population. In a case study, parameter values from the literature for the bivalve Macoma balthica are used.