On the cyclic replicator equation and the dynamics of semelparous populations

A semelparous organism reproduces only once in its life and dies thereafter. If there is only one opportunity for reproduction per year, and all individuals born in a certain year reproduce $k$ years later, then the population can be divided into year classes according to the year of birth modulo $k$. The dynamics is described by a discrete-time nonlinear Leslie matrix model, where the nonlinearity enters through the density dependent fertility and mortality rates. When the reproduction ratio is close to one, the full-life-cycle-map can be approximated by the solution of a differential equation of Lotka–Volterra type, which inherits the cyclic symmetry that is present in the full-life-cycle-map. The Lotka–Volterra equation can next be reduced to the replicator equation on the $(k-1)$-dimensional simplex. In this paper we classify the repertoire of dynamical behavior for $k=2,3$ and derive an almost complete picture for $k=4$, with some open problems identified. We pay special attention to the single year class (SYC) state (all but one year class are absent), multiple year class patterns (with several but not all year classes present), heteroclinic cycles, and periodic orbits.