# Equivalence of Stock-Recruitment Functions and Parent-Progeny Relationships in Discrete-Time Multi-Stage Models

## TL;DR

This paper shows that traditional stock-recruitment models in fisheries management may be too simplistic and suggests using more detailed multi-stage models instead.

## Contribution

The study identifies mathematical conditions under which stock-recruitment functions align with multi-stage population models.

## Key findings

- Traditional stock-recruitment functions may not capture complex life-stage dynamics in fish populations.
- Specific mathematical conditions allow equivalence between stock-recruitment functions and multi-stage models.
- Multi-stage models could improve fisheries management by better representing population dynamics.

## Abstract

Understanding the relationship between adult fish populations (the "stock") and the number of new fish entering the population (the "recruits") is essential for effective fisheries management. Traditionally, this relationship is represented by a stock-recruitment (SR) function, which is a simplified mathematical model that directly links stock size to recruitment. However, fish populations pass through several life stages, each stage influenced by unique population dynamic factors. Current SR functions often overlook these complexities, assuming that recruitment depends solely on the adult population size. In this study, we use a multi-stage, age-structured discrete-time population dynamic model that accounts for all life stages and the transitions between them. We demonstrate that, in general, a closed-form, univariate SR function may not accurately represent the recruitment process when these life stages are considered. Instead, we identify specific mathematical conditions under which a SR function is equivalent to our multi-stage model. Our findings suggest a re-evaluation of conventional SR models, advocating for multi-stage approaches to support fisheries management decisions.