Competition, Trait Variance Dynamics, and the Evolution of a Species' Range.

Geographic ranges of communities of species evolve in response to environmental, ecological, and evolutionary forces. Understanding the effects of these forces on species' range dynamics is a major goal of spatial ecology. Previous mathematical models have jointly captured the dynamic changes in species' population distributions and the selective evolution of fitness-related phenotypic traits in the presence of an environmental gradient.

Invasion waves and pinning in the Kirkpatrick-Barton model of evolutionary range dynamics.

The Kirkpatrick-Barton model, well known to invasion biologists, is a pair of reaction-diffusion equations for the joint evolution of population density and the mean of a quantitative trait as functions of space and time. Here we prove the existence of two classes of coherent structures, namely "bounded trait mean differential" traveling waves and localized stationary solutions, using geometric singular perturbation theory. We also give numerical examples of these (when they appear to be stable) and of "unbounded trait mean differential" solutions.

Range limits in spatially explicit models of quantitative traits.

In an influential paper, Kirkpatrick and Barton (Am Nat 150:1-23 1997) presented a system of diffusive partial differential equations modeling the joint evolution of population density and the mean of a quantitative trait when the trait optimum varies over a continuous spatial domain. We present a stability theorem for steady states of a simplified version of the system, originally studied in Kirkpatrick and Barton (Am Nat 150:1-23 1997). We also present a derivation of the system.

Survival of mutations arising during invasions.

When a neutral mutation arises in an invading population, it quickly either dies out or 'surfs', i.e. it comes to occupy almost all the habitat available at its time of origin.

F(ST) and Q(ST) under neutrality.

A commonly used test for natural selection has been to compare population differentiation for neutral molecular loci estimated by F(ST) and for the additive genetic component of quantitative traits estimated by Q(ST). Past analytical and empirical studies have led to the conclusion that when averaged over replicate evolutionary histories, Q(ST) = F(ST) under neutrality. We used analytical and simulation techniques to study the impact of stochastic fluctuation among replicate outcomes of an evolutionary process, or the evolutionary variance, of Q(ST) and F(ST) for a neutral quantitative trait determined by n unlinked diallelic loci with additive gene action.

Dynamics of Fst for the island model.

F(st) is a measure of genetic differentiation in a subdivided population. Sewall Wright observed that F(st)=1/1+2Nm in a haploid diallelic infinite island model, where N is the effective population size of each deme and m is the migration rate. In demonstrating this result, Wright relied on the infinite size of the population.

Steady state of homozygosity and Gst for the island model.

We examine homozygosity and G(st) for a subdivided population governed by the finite island model. Assuming an infinite allele model and strong mutation we show that the steady state distributions of G(st) and homozygosity have asymptotic expansions in the mutation rate. We use this observation to derive asymptotic expansions for various moments of homozygosity and to derive rigorous formulas for the mean and variance of G(st).

Durability of marker-quantitative trait loci haplotypes in structured populations.

Given the relative ease of identifying genetic markers linked to QTL (compared to finding the loci themselves), it is natural to ask whether linked markers can be used to address questions concerning the contemporary dynamics and recent history of the QTL. In particular, can a marker allele found associated with a QTL allele in a QTL mapping study be used to track population dynamics or the history of the QTL allele? For this strategy to succeed, the marker-QTL haplotype must persist in the face of recombination over the relevant time frame. Here we investigate the dynamics of marker-QTL haplotype frequencies under recombination, population structure, and divergent selection to assess the potential utility of linked markers for a population genetic study of QTL.

Comparing relative rates of pollen and seed gene flow in the island model using nuclear and organelle measures of population structure.

We describe a method for comparing nuclear and organelle population differentiation (F(ST)) in seed plants to test the hypothesis that pollen and seed gene flow rates are equal. Wright's infinite island model is used, with arbitrary levels of self-fertilization and biparental organelle inheritance. The comparison can also be applied to gene flow in animals.