Evol Ecol Res 11: 295-304 (2009) Full PDF if your library subscribes.
Diversification along environmental gradients in spatially structured populations
Jaroslav Ispolatov and Michael Doebeli
Department of Zoology and Department of Mathematics, University of British Columbia, Vancouver, British Columbia, Canada
Correspondence: M. Doebeli, Department of Zoology, University of British Columbia, 6270 University Boulevard, Vancouver, BC V6T 1Z4, Canada.
Questions: Does frequency-dependent and spatially localized competition in geographically structured populations generate phenotypic diversification in the form of multimodal phenotype distributions? Is such pattern formation an artifact of the boundary conditions assumed for numerical simulations? Do environmental gradients facilitate phenotypic diversification, and how does the slope of the environmental gradient influence this facilitation?
Mathematical methods: We solve numerically partial differential equations models for the dynamics of density distributions of phenotypically varying populations that occupy a continuous geographical area.
Key assumptions: We assume a linear spatial gradient in the optimal phenotype for the stabilizing component of selection (carrying capacity). To address the role of boundary conditions, we use two types of models. First, we use models in which the carrying capacity declines to 0 in both the phenotypic and the spatial dimensions as the corresponding coordinates become either large or small enough. Second, we use models with periodic boundary conditions in both the phenotypic and the geographical dimensions. For sexual populations, we assume assortative mating without Allee effects.
Conclusions: Multimodal pattern formation in phenotype space – that is, phenotypic diversification and speciation – is a robust outcome of models incorporating frequency-dependent and spatially localized competition in geographically structured populations. Also, spatial gradients in the optimal phenotype lead to a facilitation of diversification. Confirming earlier results from individual-based models, this facilitation appears to be most pronounced for gradients of intermediate slopes. The pattern formation observed in our models is not an artifact of boundary conditions.
Keywords: adaptive speciation, diversification, environmental gradients, partial differential equations, spatially structured populations.
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