Mathematical Preconditions for Existence of the Stock-Recruitment Function

Abstract

For marine species, several life stages link parents to their progeny (recruits), through a process referred to as recruitment. Current stock recruitment (SR) functions encapsulate this multi-stage relationship in a closed-form mathematical expression, which explicitly relates the biomass of parents, to the number of recruits. This functional relationship (between parents and recruits) is required for management purposes. No study, however, has investigated the conditions that validate the existence of the SR functions when all life history stages are incorporated into a model. In this study, we represent the processes leading to recruitment by a stage-and age-structured discrete-time population dynamic model. We show that in general, a SR function does not correctly represent the parent-progeny relationship. A valid relationship must incorporate information across the complete life cycle over several time periods. For populations with simple life cycle history, a functional relationship may result, which is not necessarily consistent with current SR functions. We present a brief discussion of the relevance of our results to effective management of fisheries.