We consider a discrete time model of semelparous biennial population dynamics. Interactions between individuals are modelled with the aid of an "environmental" variable I. The impact on and the sensitivity to the environmental condition is age specific. The main result is that competitive exclusion between the year classes is possible as is their coexistence. For moderate values of the basic reproduction ratio R(0) there is a strict dichotomy: depending on the other parameters we either find competitive exclusion or coexistence. We characterize rather precisely the patterns of age specific impact and sensitivity that lead to either of these outcomes.

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.

Dedicated to Jim Cushing for all the inspiration, ever since Cortona 1979.We motivate and describe a class of nonlinear Leslie matrix models for semelparous populations, like cicadas and Pacific salmon. We then focus on a Cushing-inspired special case for which one can show rigorously that a heteroclinic boundary cycle exists and attracts nearby orbits for a certain range of parameters. Along the way we formulate some open problems concerning a carrying simplex and global behaviour.Keywords: Heteroclinic cycle; Leslie matrix model; Semelparous species What is a year class?A species is called semelparous if reproduction also amounts to signing one's own death sentence. When there is exactly one reproduction opportunity per year (usually in the spring), we take one year as a natural discrete time unit. For several species, in particular many cicada species, the period in between being born and going to reproduce is strictly fixed at, say, k years. The population then subdivides into subpopulations according to the year of birth modulo k (or, equivalently, the year of reproduction modulo k). Such a subpopulation is called a year class (or, also, a brood). Note that whereas the age class to which an individual belongs increases by one when a year has passed, the year class to which it belongs is fixed once and for all.Year classes only mate among themselves and hence are reproductively isolated. In particular, if a year class goes extinct, it remains extinct. We then say that the year class is "missing". The periodical insects [1] are those for which all but one year classes are missing. The prime example is the Magicicadas (the k ¼ 17 species had its most recent emergence in the North-East United States in 2004).