Periodic travelling waves in cyclic populations: field studies and reaction–diffusion models

Abstract

Periodic travelling waves have been reported in a number of recent spatio-temporal field studies of populations undergoing multi-year cycles. Mathematical modelling has a major role to play in understanding these results and informing future empirical studies. We review the relevant field data and summarize the statistical methods used to detect periodic waves. We then discuss the mathematical theory of periodic travelling waves in oscillatory reactiondiffusion equations. We describe the notion of a wave family, and various ecologically relevant scenarios in which periodic travelling waves occur. We also discuss wave stability, including recent computational developments. Although we focus on oscillatory reactiondiffusion equations, a brief discussion of other types of model in which periodic travelling waves have been demonstrated is also included. We end by proposing 10 research challenges in this area, five mathematical and five empirical.

Long-term transients and complex dynamics of a stage-structured population with time delay and the Allee effect

“…The logistic equation, however, predicts only very simple dynamics, such as monotonic convergence to a steady state, and this is clearly at odds with the complexity of the dynamics observed in natural populations where sustained oscillations are much more typical (Ranta et al, 1995;Cushing et al, 2003;Turchin, 2003;Sherratt and Smith, 2008). This is of course hardly surprising, as the logistic equation obviously leaves many things out of focus; in particular, it assumes that offspring start reproducing the moment they are born and are born immediately after they are conceived.…”

Spatially mismatched trophic dynamics: cyclically outbreaking geometrids and their larval parasitoids

“…Furthermore, spatio-temporal dynamics compatible with trophic interactions, e.g. travelling waves, a form of spatial asynchrony likely to result from enemyvictim interactions (Sherratt & Smith 2008), have been reported to occur (e.g. Tenow et al 2007).…”

“…In that case, the spatial patterning of enemy-inflicted mortality should match with the pattern of victim population dynamics. However, Sherratt & Smith's (2008) review of spatiotemporal dynamics of cyclic populations due to trophic interactions found no examples of such studies. Here, we demonstrate how targeted field studies invoking spatial demographic data can be used to reject specific trophic mechanisms as decisive for the spatio-temporal dynamics of cyclic species, using geometrid moths in northern Fennoscandia as an example.…”

“…Recently, to potentially shed new light onto the decisive processes, the 'spatial dimension' of population cycles has been emphasized (for reviews, see Liebhold et al 2004;Sherratt & Smith 2008). A host of theoretical studies have deduced how various factors can bring about different spatial patterning of population cycles, from region-wide synchrony to complex patterns of small-scale asynchrony (e.g.…”

“…Spatial data on key demographic parameters, in addition to conventional population time series, would allow for stronger tests of hypotheses about processes (Bjørnstad et al 1999;Sherratt & Smith 2008). For instance, natural enemies have often been put forward as a decisive mechanism generating population cycles and distinct forms of spatial population asynchrony (e.g.…”