Mathematical analysis of a population model with an age–weight structured two-stage life history: asymptotic behavior of solutions

In this paper, we propose a model for the dynamics of a physiologically structured population of individuals whose life cycle is divided into two stages: the first stage is structured by the weight, while the second one is structured by the age, the exit from the first stage occurring when a threshold weight is attained. The model originates in a complex one dealing with a fish population and covers a large class of situations encompassing two-stage life histories with a different structuring variable for each state, one of its key features being that the maturation process is determined in terms of a weight threshold to be reached by individuals in the first stage. Mathematically, the model is based on the classical Lotka–MacKendrick linear model, which is reduced to a delayed renewal equation including a constant delay that can be viewed as the time spent by individuals in the first stage to reach the weight threshold. The influence of the growth rate and the maturation threshold on the long-term behavior of solutions is analyzed using Laplace transform methods.