Traveling waves and spatial patterns from dispersal on homogeneous and gradient habitats

Models of dispersal for response to global climate change can refer to theory on traveling waves developed for invasions. The relevance of this theory may depend on its density- versus habitat-dependence, but it has not considered environmental gradients nor related feedbacks. An agent based simulation model (ABM) for a plant species or vegetation type is developed here, focusing on properties of traveling waves. The ABM simulated no-tailed, thin-tailed, and fat-tailed dispersal kernels (random uniform, Gaussian, and Cauchy kernels, respectively). Probabilities were based on either population density or habitat, and for the latter as homogeneous or on a gradient, and if a gradient positive feedback from populations to the habitat was added to represent process at ecotones such as a positive feedback switch or the stress gradient hypothesis (SGH) feedback. The uniform and Gaussian kernels produced traveling waves with constant speed, but the fat-tailed Cauchy kernel produced traveling waves that accelerated and also flattened as the low-probability portion of wave accelerated fastest. On a gradient, when populations approached the limit of habitability the waves steepened and stopped traveling. The effect of SGH feedback on traveling wave slopes depends on the dispersal kernel. With strong facilitation on the gradient, greater variety of spatial patterns were able to develop among the replicate simulations of the fat-tailed kernel when it flattened to a greater spatial extent of low probability presence. In addition to the heterogeneity of the habitat and its modification by facilitation, the details of dispersal can also affect the feedback between spatial pattern and the establishment process that will affect the rate of advance of a species or an ecotone into new territory.