Percolation on random Johnson–Mehl tessellations and related models

Continuum line-of-sight percolation on Poisson–Voronoi tessellations

In this work, we study a new model for continuum line-of-sight percolation in a random environment driven by the Poisson–Voronoi tessellation in the d-dimensional Euclidean space. The edges (one-dimensional facets, or simply 1-facets) of this tessellation are the support of a Cox point process, while the vertices (zero-dimensional facets or simply 0-facets) are the support of a Bernoulli point process. Taking the superposition Z of these two processes, two points of Z are linked by an edge if and only if they are sufficiently close and located on the same edge (1-facet) of the supporting tessellation. We study the percolation of the random graph arising from this construction and prove that a 0–1 law, a subcritical phase, and a supercritical phase exist under general assumptions. Our proofs are based on a coarse-graining argument with some notion of stabilization and asymptotic essential connectedness to investigate continuum percolation for Cox point processes. We also give numerical estimates of the critical parameters of the model in the planar case, where our model is intended to represent telecommunications networks in a random environment with obstructive conditions for signal propagation.

The Role of Standing Variation in Geographic Convergent Adaptation

The extent to which populations experiencing shared selective pressures adapt through a shared genetic response is relevant to many questions in evolutionary biology. In this article, we explore how standing genetic variation contributes to convergent genetic responses in a geographically spread population. Geographically limited dispersal slows the spread of each selected allele, hence allowing other alleles to spread before any one comes to dominate the population. When selectively equivalent alleles meet, their progress is substantially slowed, dividing the species range into a random tessellation, which can be well understood by analogy to a Poisson process model of crystallization. In this framework, we derive the geographic scale over which an allele dominates and the proportion of adaptive alleles that arise from standing variation. Finally, we explore how negative pleiotropic effects of alleles can bias the subset of alleles that contribute to the species' adaptive response. We apply the results to the malaria-resistance glucose-6-phosphate dehydrogenase-deficiency alleles, where the large mutational target size makes it a likely candidate for adaptation from deleterious standing variation. Our results suggest that convergent adaptation may be common. Therefore, caution must be exercised when arguing that strongly geographically restricted alleles are the outcome of local adaptation. We close by discussing the implications of these results for ideas of species coherence and the nature of divergence between species.

Parallel adaptation: one or many waves of advance of an advantageous allele?

Models for detecting the effect of adaptation on population genomic diversity are often predicated on a single newly arisen mutation sweeping rapidly to fixation. However, a population can also adapt to a new environment by multiple mutations of similar phenotypic effect that arise in parallel, at the same locus or different loci. These mutations can each quickly reach intermediate frequency, preventing any single one from rapidly sweeping to fixation globally, leading to a "soft" sweep in the population. Here we study various models of parallel mutation in a continuous, geographically spread population adapting to a global selection pressure. The slow geographic spread of a selected allele due to limited dispersal can allow other selected alleles to arise and start to spread elsewhere in the species range. When these different selected alleles meet, their spread can slow dramatically and so initially form a geographic patchwork, a random tessellation, which could be mistaken for a signal of local adaptation. This spatial tessellation will dissipate over time due to mixing by migration, leaving a set of partial sweeps within the global population. We show that the spatial tessellation initially formed by mutational types is closely connected to Poisson process models of crystallization, which we extend. We find that the probability of parallel mutation and the spatial scale on which parallel mutation occurs are captured by a single compound parameter, a characteristic length, which reflects the expected distance a spreading allele travels before it encounters a different spreading allele. This characteristic length depends on the mutation rate, the dispersal parameter, the effective local density of individuals, and to a much lesser extent the strength of selection. While our knowledge of these parameters is poor, we argue that even in widely dispersing species, such parallel geographic sweeps may be surprisingly common. Thus, we predict that as more data become available, many more examples of intraspecies parallel adaptation will be uncovered.

Percolation thresholds on two-dimensional Voronoi networks and Delaunay triangulations

The site percolation threshold for the random Voronoi network is determined numerically, with the result pc=0.714 10+/-0.000,02 , using Monte Carlo simulation on periodic systems of up to 40,000 sites. The result is very close to the recent theoretical estimate pc approximately 0.7151 of Neher For the bond threshold on the Voronoi network, we find pc=0.666, 931+/-0.000,005 implying that, for its dual, the Delaunay triangulation pc=0.333 069+/-0.000 005 . These results rule out the conjecture by Hsu and Huang that the bond thresholds are 2/3 and 1/3, respectively, but support the conjecture of Wierman that, for fully triangulated lattices other than the regular triangular lattice, the bond threshold is less than 2 sin pi/18 approximately 0.3473 .