Insect seasonality: circle map analysis of temperature-driven life cycles

Maintaining an adaptive seasonality, with life cycle events occurring at appropriate times of year and in synchrony with cohorts and ephemeral resources, is a basic ecological requisite for many cold-blooded organisms. There are many mechanisms for synchronizing developmental milestones, such as egg laying (oviposition), egg hatching, cocoon opening, and the emergence of adults. These are often irreversible, specific to particular life stages, and include diapause, an altered physiological state which can be reversed by some synchronizing environmental cue (e.g. photoperiod). However, many successful organisms display none of these mechanisms for maintaining adaptive seasonality. In this paper, we briefly review the mathematical relationship between environmental temperatures and developmental timing and discuss the consequences of viewing these models as circle maps from the cycle of yearly oviposition dates and temperatures to oviposition dates for subsequent generations. Of particular interest biologically are life cycles which are timed to complete in exactly 1 year, or univoltine cycles. Univoltinism, associated with reproductive success for many temperate species, is related to stable fixed points of the developmental circle map. Univoltine fixed points are stable and robust in broad temperature bands, but lose stability suddenly to maladaptive cycles at the edges of these bands. Adaptive seasonality may therefore break down with little warning with constantly increasing or decreasing temperature change, as in scenarios for global warming. These ideas are illustrated and explored in the context of Mountain Pine Beetle (Dendroctonus ponderosae Hopkins) occurring in the marginal thermal habitat of central Idaho's Rocky Mountains. Applications of these techniques have not been widely explored by the applied math community, but are likely to provide great insight into the response of biological systems to climate change.