Numerical analysis and stability investigation of some system of reaction-diffusion equations

Šajā darbā tiek pētīts reakcijas–difūzijas tipa vienādojums, kurā reakcijas daļa ir Van der Pola tipa vienādojumu sistēma. Tiek analizēta vienādojumu sistēmas lineārā stabilitāte, kā arī meklēts sistēmas skaitliskais atrisinājums, izmantojot Čebiševa interpolācijas metodi.A reaction–diffusion type equation is being investigated in the Bachelor’s thesis, with Van der Pol type equation system as the reaction part. Main focus is on linear stability analysis of system of equations and also finding numerical solution for the system using Chebyshev interpolation method

Quantitative Human Paleogenetics:What can Ancient DNA Tell us About Complex Trait Evolution?

Genetic association data from national biobanks and large-scale association studies have provided new prospects for understanding the genetic evolution of complex traits and diseases in humans. In turn, genomes from ancient human archaeological remains are now easier than ever to obtain, and provide a direct window into changes in frequencies of trait-associated alleles in the past. This has generated a new wave of studies aiming to analyse the genetic component of traits in historic and prehistoric times using ancient DNA, and to determine whether any such traits were subject to natural selection. In humans, however, issues about the portability and robustness of complex trait inference across different populations are particularly concerning when predictions are extended to individuals that died thousands of years ago, and for which little, if any, phenotypic validation is possible. In this review, we discuss the advantages of incorporating ancient genomes into studies of trait-associated variants, the need for models that can better accommodate ancient genomes into quantitative genetic frameworks, and the existing limits to inferences about complex trait evolution, particularly with respect to past populations

Modelling the spatiotemporal spread of beneficial alleles using ancient genomes

Ancient genome sequencing technologies now provide the opportunity to study natural selection in unprecedented detail. Rather than making inferences from indirect footprints left by selection in present-day genomes, we can directly observe whether a given allele was present or absent in a particular region of the world at almost any period of human history within the last 10,000 years. Methods for studying selection using ancient genomes often rely on partitioning individuals into discrete time periods or regions of the world. However, a complete understanding of natural selection requires more nuanced statistical methods which can explicitly model allele frequency changes in a continuum across space and time. Here we introduce a method for inferring the spread of a beneficial allele across a landscape using two-dimensional partial differential equations. Unlike previous approaches, our framework can handle time-stamped ancient samples, as well as genotype likelihoods and pseudohaploid sequences from low-coverage genomes. We apply the method to a panel of published ancient West Eurasian genomes to produce dynamic maps showcasing the inferred spread of candidate beneficial alleles over time and space. We also provide estimates for the strength of selection and diffusion rate for each of these alleles. Finally, we highlight possible avenues of improvement for accurately tracing the spread of beneficial alleles in more complex scenarios