Judith R. Miller

FST and QST Under Neutrality

A commonly used test for natural selection has been to compare population differentiation for neutral molecular loci estimated by FST and for the additive genetic component of quantitative traits estimated by QST. Past analytical and empirical studies have led to the conclusion that when averaged over replicate evolutionary histories, QST = FST 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 QST and FST for a neutral quantitative trait determined by n unlinked diallelic loci with additive gene action. We studied analytical models of two scenarios. In one, a pair of demes has recently been formed through subdivision of a panmictic population; in the other, a pair of demes has been evolving in allopatry for a long time. A rigorous analysis of these two models showed that in general, it is not necessarily true that mean QST = FST (across evolutionary replicates) for a neutral, additive quantitative trait. In addition, we used finite-island model simulations to show there is a strong positive correlation between QST and the difference QST − FST because the evolutionary variance of QST is much larger than that of FST. If traits with relatively large QST values are preferentially sampled for study, the difference between QST and FST will also be large and positive because of this correlation. Many recent studies have used tests of the null hypothesis QST = FST to identify diversifying or uniform selection among subpopulations for quantitative traits. Our findings suggest that the distributions of QST and FST under the null hypothesis of neutrality will depend on species-specific biology such as the number of subpopulations and the history of subpopulation divergence. In addition, the manner in which researchers select quantitative traits for study may introduce bias into the tests. As a result, researchers must be cautious before concluding that selection is occurring when QST ≠ FST.

Nonexistence of singular pseudo-self-similar solutions of the Navier–Stokes system

Abstract. We show that there are no singular pseudo-self-similar solutions of the Navier-Stokes system with finite energy.

Long-Time Behavior of Scalar Viscous Shock Fronts in Two Dimensions

We prove nonlinear stability in L1 of planar shock front solutions to a viscous conservation law in two spatial dimensions and obtain an expression for the asymptotic form of small perturbations. The leading-order behavior is shown rigorously to be governed by an effective diffusion coefficient depending on forces transverse to the shock front. The proof is based on a spectral analysis of the linearized problem.

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. Beneficial mutations can also surf, as can deleterious mutations over finite time spans. We develop descriptive statistical models that quantify the relationship between the probability that a mutation will surf and demographic parameters for a cellular automaton model of surfing. We also provide a simple analytic model that performs well at predicting the probability of surfing for neutral and beneficial mutations in one dimension. The results suggest that factors – possibly including even abiotic factors – that promote invasion success may also increase the probability of surfing and associated adaptive genetic change, conditioned on such success.

Wave patterns, stability, and slow motions in inviscid and viscous hyperbolic equations with stiff reaction terms

We study the behavior of solutions to the inviscid ðA ¼ 0Þ and the viscous ðA > 0Þ hyperbolic conservation laws with stiff source terms ut þ f ðuÞx ¼� 1 e W 0 ðu Þþ eAuxx

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.

ORIGINAL ARTICLE: 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. Beneficial mutations can also surf, as can deleterious mutations over finite time spans. We develop descriptive statistical models that quantify the relationship between the probability that a mutation will surf and demographic parameters for a cellular automaton model of surfing. We also provide a simple analytic model that performs well at predicting the probability of surfing for neutral and beneficial mutations in one dimension. The results suggest that factors – possibly including even abiotic factors – that promote invasion success may also increase the probability of surfing and associated adaptive genetic change, conditioned on such success.

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. For two scenarios, described as “secondary contact” and “novel allele,” we use both deterministic and stochastic methods to describe the influence of gene flow between habitats, the strength of divergent selection, and the genetic distance between a marker and the QTL on the persistence of marker-QTL haplotypes. We find that for most reasonable values of selection on a locus (s ≤ 0.5) and migration (m > 1%) between differentially selected populations, haplotypes of typically spaced markers (5 cM) and QTL do not persist long enough (>100 generations) to provide accurate inference of the allelic state at the QTL.

DISPERSIVE EFFECTS IN A MODIFIED KURAMOTO–SIVASHINSKY EQUATION

ABSTRACT We study the limiting behavior of the Kuramoto–Sivashinsky/Korteweg-de Vries (KS/KdV) equation We show that in the appropriate sense, the solutions of KS/KdV tend to the solutions of the standard Korteweg-de Vries equation as . The proof relies, to a large extent, on precise estimates for oscillatory integrals that yield pointwise bounds on Greens functions.

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. Further, we provide numerical evidence of bistability and hysteresis for this system, modeling an initially confined population that colonizes new territory when some biotic or abiotic conditions change, and remains in its enlarged range even when conditions change back. Our analytical and numerical results indicate that the dynamics of this system are more complicated than previously recognized, and help make sense of evolutionary range dynamics predicted by other models that build upon it and sometimes challenge its predictions.