On circulant populations. I. The algebra of semelparity

We consider a class of nonlinear Leslie matrix models, describing the population dynamics of an age-structured semelparous species. Semelparous species are those whose individuals reproduce only once and die afterwards. Competitive interaction between individuals is modelled via a one-dimensional environmental quantity. Age classes are characterised by their impact on, and their sensitivity to, the environment. We do not restrict ourselves to some particular form of functional dependence and keep the model otherwise as general as possible. The system possesses a cyclic symmetry. Due to the symmetry it exhibits so-called vertical bifurcations, where a manifold filled with periodic orbits appears in the phase space for specific parameter combinations. This bifurcation serves as a switch between the main types of behaviour: coexistence of all year classes or a periodic regime with some year classes missing. In particular, the vertical bifurcation takes place when a certain circulant matrix is singular. We also analyse the local stability of the unique coexistence equilibrium state and derive a characteristic equation for it. The dynamics of populations with two and, especially, with three age classes are analysed in detail.