A diffusion model in population genetics with dynamic fitness

We analyze a degenerate diffusion equation with singular boundary data, modeling the evolution of a polygenic trait under selection and drift. The equation models the contributions of a large but finite number of loci (genes) to the trait and at the same time allows the population trait mean to vary in a way that affects the strength of selection at individual loci; in this respect it differs from other population-genetic models that have been rigorously analyzed. We present existence, uniqueness and stability results for solutions of the system. We also prove that the genetic variance in the system tends to zero in the long time limit, and relate the dynamics of the trait mean to the variance.     

A multiscale view on shape effects in the computational characterization of magnetorheological elastomers

This contribution highlights the influence of the shape of specimens of magnetorheological composites on their response under homogeneous magnetic fields. The analysis is performed in terms of two-scale computational characterization using a finite strain homogenization framework. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Non-parametric vs parametric forecasting in time series: A computational point of view

The finite sample behaviour of non-parametric predictors is considered for time series. Among other results, it is shown by simulation arguments that such predictors compare favourably with predictors based on parametric models in the spirit of the usual Box-Jenkins approach.     

Global well-posedness for the diffusion equation of population genetics

In this paper, we study the forward diffusion equation of population genetics. We prove the global existence of smooth solutions if the initial value is smooth. We also show that if the initial value is singular on the boundary, in a weighted Sobolev space, the diffusion equation exists a unique weak solution which is a probability density function. Moreover, we investigate the asymptotic behavior of the weak solution by the entropy method.     

Convergence to equilibrium in the classical model of population genetics

In the classical model of population genetics for continuous time (Fisher's equation) for n alleles a stationary point is called regular if the viability of each of the absent alleles is distinct from the average viability of the population. It is shown that if the ω-limit set of a trajectory contains a regular point, then it contains only one point.     

Models and people: An alternative view of the emergent properties of computational models

Computer models can help humans gain insight into the functioning of complex systems. Used for training, they can also help gain insight into the cognitive processes humans use to understand these systems. By influencing humans understanding (and consequent actions) computer models can thus generate an impact on both these actors and the very systems they are designed to simulate. When these systems also include humans, a number of self‐referential relations thus emerge which can lead to very complex dynamics. This is particularly true when we explicitly acknowledge and model the existence of multiple conflicting representations of reality among different individuals. Given the increasing availability of computational devices, the use of computer models to support individual and shared decision making could potentially have implications far wider than the ones often discussed within the Information and Communication Technologies community in terms of computational power and network communication. We discuss some theoretical implications and describe some initial numerical simulations. © 2015 Wiley Periodicals, Inc. Complexity 21: 202–213, 2016     

U.S. national population projections methods: A view from four forecasting traditions

Four traditions — judgmental, time series, demographic accounting, and explanatory — represent different approaches to national population projections. The U.S. Census Bureau's methodology for national population projections is mainly within the demographic accounting tradition incorporating selected aspects of the other three traditions. Several avenues for combining the best methods from each of the four forecasting traditions in future projections activities are outlined.     

Bifurcation and stability results for an equation arising in population genetics

Summary We consider a semilinear elliptic boundary value problem, which arises in population genetics. Although there is no obvious corresponding linearized problem, we establish, by using Implicit Function Theorem methods, necessary and sufficient conditions forbifurcation to occur from a branch of trivial solutions. The stability of the bifurcating solutions is also investigated.     

On a class of elliptic singular perturbations with applications in population genetics

With the maximum principle for differential equations asymptotic estimates are made for a class of linear elliptic singular perturbation problems with resonant turning point behaviour in some of the independent variables. The method is applied to the stationary solution of the Kolmogorov backward equation in population genetics.     

Existence and regularity for a nonlinear boundary flow problem of population genetics

In this paper we consider a heat equation with nonlinear boundary condition occurring in population genetics, the selection–migration problem for alleles in a region, considering flow of genes throughout the boundary. Such a problem determines a gradient flow in a convenient phase space and then the dynamics for large times depends heavily on the knowledge of the equilibrium solutions. We address the questions of the existence of a nontrivial equilibrium solution and its regularity.     

Diffusion approximation of the Wright-Fisher model of population genetics: Single-locus two alleles

We investigate an autoregressive diffusion approximation method applied to the Wright-Fisher model in population genetics by considering a Markov chain with Bernoulli distributed independent variables. The use of an autoregressive diffusion method and an averaged allelic frequency process lead to an Orn-stein-Uhlenbeck diffusion process with discrete time. The normalized averaged frequency process possesses independent allele frequency indicators with constant conditional variance at equilibrium. In a monoecious diploid population of size N with r generations, we consider the time to equilibrium of averaged allele frequency in a single-locus two allele pure sampling model.     

A computational model for interfaces

Decompositions of the plane into disjoint components separated by curves occur frequently. We describe a package of subroutines which provides facilities for defining, building, and modifying such decompositions and for efficiently solving various point and area location problems. Beyond the point that the specification of this package may be useful to others, we reach the broader conclusion that well-designed data structures and support routines allow the use of more conceptual or non-numerical portions of mathematics in the computational process, thereby extending greatly the potential scope of the use of computers in scientific problem solving. Ideas from conceptual mathematics, symbolic computation, and computer science can be utilized within the framework of scientific computing and have an important role to play in that area.