Figure 1: Visual representation of the various hypotheses (via simulated data), where yellow indicates high growth rates and blue low growth rates. A represents a truly asynchronous population cycle, where each population (line) cycles independently of its neighbours. B shows partial synchrony where the neighbouring populations’ cycle almost simultaneously, though they are not perfectly synchronised (decomposed into subsequent models).C is a perfectly synchronised population where each population cycles at precisely the same time (where \(r_{t}\) should best be represented as varying with time, model N2). Fshows a purely spatial pattern (where any perceived spatio-temporal pattern is merely spatial, model N3, as Sherratt and Smith 2008 suggested may be the case for the apparent snowshoe hare travelling wave). D is a single planar wave at a snapshot in time (Moss et al . 2000; Lambin et al . 1998, Bjørnstadet al . 2002, Berthier et al . 2014, model P). Erepresents either an expanding or contracting single radial travelling wave (radially expanding from a central location as suggested by Johnsonet al . 2006 [model RE] or contracting as suggested by Sherratt & Smith 2008, [model RC]). G shows two isolated planar waves separated by a physical feature, the Duero river (inferred from Sherratt & Smith 2008, model PF). H shows two radial waves separated by the same physical feature but may be either contracting or expanding (models RFE and RFC). J represents dual overlapping planar waves, which additively form a single overall pattern (model PD). K is either dual overlapping contracting or expanding radial waves, additively forming an overall pattern (models RDE and RDC). I represents the modelling approach, represented by the borders and arrows, used by the various parameterisations for each model, described in Table 1, to recreate a synchronised cycle in order to infer the form of partial asynchrony.