On periodic cohort solutions of a size-structured population model
The paper considers a size-structured population with \(n\) cohorts and fixed death rate. The growth rate of an individual depends on its size and on the amount of substrate. They reproduce, leaving \(k\) offspring, and then die, when they reach a fixed size. The substate has its own dynamics (depending on a bifurcation parameter \(\beta)\) with an equilibrium point in the absence of consumption. The model (quite general) is described by a set of ordinary differential equations plus initial conditions and jump conditions at the time instants of reproduction. Existence of nontrivial periodic \(n\)-cohort solutions is investigated. The case \(n=1\) is thoroughly studied and sufficient conditions are given for the nontrivial periodic one-cohort solution to be stable with respect to splitting into \(n\) cohorts. Existence of periodic solutions for \(n\) cohorts is shown. Moreover, the paper discusses the question whether \(n\) cohorts \((n\geq 2)\) with small size differences will tend to a periodic one-cohort solution as \(t\to\infty\). The basic method is to look for fixed points of the discrete-time map obtained by looking at the system at the times of reproduction.
scientific article; zbMATH DE number 1058826