Old and Young. Can they coexist?

Abstract

The object of the study is semelparous species, i.e. those whose individuals reproduce only once in their life and die afterwards. Examples are annual and biennial plants, salmon, cicada. If reproduction is restricted to a short time interval during the year (in the spring) and the duration of life is fixed to, say, k years, the population of semelparous species is subdivided into year classes according to the year of birth (modulo k). Since individuals in different year classes have different age, they can compete for a shared resource. As a result of the competition one or several year classes can go extinct. We have following questions: when we should expect to find coexistence of year classes (as, e.g., in salmon) and when competitive exclusion (as in the periodical insects, e.g. cicada)? When does one year class tune the environmental conditions such that other year classes are driven to extinction? And when, on the other hand, can a missing year class invade successfully? We construct a discrete-time k-dimensional nonlinear matrix model and investigate its dynamics. The main dynamical states are the "coexistence" equilibrium, i.e. a steady state with all year classes present, and k-periodic points on invariant coordinate axes and (hyper-)planes, which correspond to the situation of one or several year classes missing. The coexistence equilibrium is unique, we construct a characteristic equation for it and perform local bifurcation analysis for k=2 and 3 (biennial and triennial species). Also, in these cases, we analyse the local stability of the k-periodic points. In other words we derive criteria for coexistence or competitive exclusion in terms of parameters such that impacts of different age classes on the environment and sensitivities to the environment. One of the most interesting feature of the dynamics is the existence of so-called vertical bifurcations, i.e. highly degenerate bifurcations for which an invariant manifold filled with periodic points occurs in the phase space. These bifurcations can serve as a switch between coexistence and competitive exclusion, in other words the coexistence equilibrium becomes stable and the k-periodic points on the coordinate axes/planes become transversally unstable (or vice versa) for the same combinations of parameter values. Transversal (in)stability corresponds to directions transversal to the coordinate axes/planes. Conditions for a vertical bifurcation are when a certain circulant matrix becomes singular or a certain nonlinear circulant system becomes degenerate. We discuss also internal stability of the k-periodic points on the axes. They corresponds to the situation when only one year class is present (and the other are missing). We call it Single Year Class (SYC) behaviour. The k-periodic points are fixed points of the k-iterate of the original map restricted to the axes or a SYC-map. A SYC-map is a composition of several functions of similar type. We construct various analytical and numerical bifurcaion diagrams corresponding to local and global stability of these fixed points. In addition, we construct competition models of several age-structured populations to explain a well-known phenomenon of prime numbers in life cycle length of Magicicada. This species live 13 or 17 years and population consists of a single year class (a brood). Our explanation for this phenomenon is that the long-living cicada "do not want" to be in resonance with short-living (2-3 years) cicada.