Locking life-cycles onto seasons: Circle-map models of population dynamics and local adaptation

We have formulated a model describing the timing of maturity and reproduction in briefly semelparous organisms whose development rate is primarily controlled by environmental factors. The model is expressed as a circle-map relating time of year at maturation in successive generations. The properties of this map enable us to determine the degree of synchrony to be expected between the life-cycles of members of a population exposed to a regular seasonal environment. We have proved that organisms with a life-history composed of a contiguous series of stages, all with development driven by the same seasonal function, cannot phase-lock their life-cycles to the seasons. However if the organism exhibits facultative diapause induced by a critical time/critical development mechanism of the type proposed by Norling (1984a,b,c) then it will always succeed in phase-locking to a perfectly periodic driving function. Within the context of this circle-map model we have examined population extinctions caused by attempting to over-winter in an inappropriate life-history stage, or by attempting to reproduce at a time of year when this is impossible. We have shown that the possibility of such extinctions limits both the shortness of the post-critical stage, and the lateness of the critical time. We have examined the fitness of persistent cohorts as a function of critical time and development. We find that if the post-critical stage is riskier than the pre-critical then natural selection favors a short post-critical stage and a late critical time; the limitation of this process being dependent on the proportion of the growing season over which successful reproduction is possible. We have determined the variation with life-cycle length (and hence latitude or altitude) of the maturation pattern corresponding to optimal life-history parameters. We find that for organisms which can mature only over a small part of the growing season the majority of any latitudinal gradient exhibits a unimodal maturation pattern. Organisms which can mature and reproduce over the majority of the growing season exhibit more complex patterns, but still exhibit substantial ranges of latitude over which unimodal or bimodal patterns are optimal.