The Shapes of Clines and Wavefronts

Cline theory has a central place in speciation studies. Cline locations delimit taxon boundaries, cline widths scale with barrier strength, and the shapes of clines (smooth or stepped) suggest whether species barriers are mono- or polygenic. How cline shapes vary along chromosomes therefore forms part of the genome's species barrier landscape. Further, asymmetric moving clines (wave fronts) can mark adaptive introgression puncturing species barriers, potentially leading to their collapse or decay. Here we review the development of cline and wavefront models and relate this to the use of dispersal kernels in epidemiology and ecology. We contrast classical results to those for a thick-tailed kernel, showing how cline shape affects the speed of spatial processes, including the widening of neutral clines and the spatial coalescent. We critique current cline models used for inference (both spatial and genomic clines) and address Barton's question: Why (after decades of cline fitting) is there so little evidence of stepped clines? We suggest evidence is weak because stepped cline models are over-parameterised, while current genomic cline models are under-parameterised. We explore goldilocks cline parameterisations and discuss non-parametric approaches that may help resolve these issues. This broadens to a discussion of the future of, and alternatives to, cline fitting.