Mathematical theory of selection is developed within the frameworks of general models of inhomogeneous populations with continuous time. Methods that allow us to study the distribution dynamics under natural selection and to construct explicit solutions of the models are developed. All statistical characteristics of interest, such as the mean values of the fitness or any trait can be computed effectively, and the results depend in a crucial way on the initial distribution. The developed theory provides an effective method for solving selection systems; it reduces the initial complex model to a special system of ordinary differential equations (the escort system). Applications of the method to the Price equations are given; the solutions of some particular inhomogeneous Malthusian, Ricker and logistic-like models used but not solved in the literature are derived in explicit form.
A commentary on the mathematical model of the heterogeneous gypsy moth larva population by G. Dwyer and his co-authors
在这篇评论中，我利用异质群体数学理论的一般方法，指出[Dwyer et al（2000），Am Nat，156（2）：105--120]中数学模型分析中的遗漏，这导致[Elderd et al（2008）Am Nat，172（6）：829--842]中的结论，即由于新数据，必须用替代模型取代原始模型。我表明，更彻底的理解的基础模型允许抽搐的模型参数，以考虑到观察到的数据。
Classical approaches to analyzing dynamical systems, including bifurcation analysis, can provide invaluable insights into underlying structure of a mathematical model, and the spectrum of all possible dynamical behaviors. However, these models frequently fail to take into account population heterogeneity, which, while critically important to understanding and predicting the behavior of any evolving system, is a common simplification that is made in analysis of many mathematical models of ecological systems. Attempts to include population heterogeneity frequently result in expanding system dime…
Single species population models and discrete stochastic gene frequency models are two standards of mathematical biology important for the evolution of populations. An agent based model is presented which reproduces these models and then explores where these models agree and disagree under relaxed specifications. For the population models, the requirement of homogeneous mixing prevents prediction of extinctions due to local resource depletion. These models also suggest equilibrium based on attainment of constant population levels though underlying population characteristics may be nowhere clos…
We define a general class of models representing natural selection between two alleles. The population size and spatial structure are arbitrary, but fixed. Genetics can be haploid, diploid, or otherwise; reproduction can be asexual or sexual. Biological events (e.g.~births, deaths, mating, dispersal) depend in arbitrary fashion on the current population state. Our formalism is based on the idea of genetic sites. Each genetic site resides at a particular locus and houses a single allele. Each individual contains a number of sites equal to its ploidy (one for haploids, two for diploids, etc.). S…
Evolution occurs in populations of reproducing individuals. It is well known that population structure can affect evolutionary dynamics. Traditionally, natural selection is studied between mutants that differ in reproductive rate, but are subject to the same population structure. Here we study how natural selection acts on mutants that have the same reproductive rate, but experience different population structures. In our framework, mutation alters population structure, which is given by a graph that specifies the dispersal of offspring. Reproduction can be either genetic or cultural. Competin…
Some populations, such as red blood cells (RBCs), exhibit a pattern of population decline that is closer to linear rather than exponential, which has proven to be unexpectedly challenging to describe with a single simple mathematical model. Here we show that a sub-exponential model of population extinction can approximate very well the experimental curves of RBC extinction, and that one possible mechanism underlying sub-exponential decay is population heterogeneity with respect to death rates of individuals. We further show that a sub-exponential model of population decline can be derived with…
This chapter gives a synopsis of recent approaches to model and analyse the evolution of microbial populations under selection. The first part reviews two population genetic models of Lenski's long-term evolution experiment with Escherichia coli, where models aim at explaining the observed curve of the evolution of the mean fitness. The second part describes a model of a host-pathogen system where the population of pathogenes experiences balancing selection, migration, and mutation, as motivated by observations of the genetic diversity of HCMV (the human cytomegalovirus) across hosts.
We study the effect of biological confounders on the model selection problem between Kingman coalescents with population growth, and Xi-coalescents involving simultaneous multiple mergers. We use a low dimensional, computationally tractable summary statistic, dubbed the singleton-tail statistic, to carry out approximate likelihood ratio tests between these model classes. The singleton-tail statistic has been shown to distinguish between them with high power in the simple setting of neutrally evolving, panmictic populations without recombination. We extend this work by showing that cryptic reco…
In this work, two mathematical models for malaria under resistance are presented. More precisely, the first model shows the interaction between humans and mosquitoes inside a patch under infection of malaria when the human population is resistant to antimalarial drug and mosquitoes population is resistant to insecticides. For the second model, human-mosquitoes population movements in two patches is analyzed under the same malaria transmission dynamic established in one patch. For a single patch, existence and stability conditions for the equilibrium solutions in terms of the local basic reprod…
Effects of Periodic Migration on Selection in Subdivided Populations
Sharif University of Technology（谢里夫理工大学）；University of Minnesota（明尼苏达大学）；ICTP – The Abdus Salam International Centre for Theoretical Physics（国际理论物理中心（阿卜杜斯·萨拉姆国际理论物理中心））
Population genetics models typically consider a fixed population size and a unique selection coefficient. However, population dynamics inherently generate noise in numbers of individuals and selection acts on various components of the individuals' fitness. In plant species with seed banks, the size of both the above- and below-ground compartments present noise depending on seed production and the state of the seed bank. We investigate if this noise has consequences on 1)~the rate of genetic drift, and 2)~the efficacy of selection. We consider four variants of two-allele Moran-type models defin…
The diversity of T-cell receptor (TCR) repertoires is achieved by a combination of two intrinsically stochastic steps: random receptor generation by VDJ recombination, and selection based on the recognition of random self-peptides presented on the major histocompatibility complex. These processes lead to a large receptor variability within and between individuals. However, the characterization of the variability is hampered by the limited size of the sampled repertoires. We introduce a new software tool SONIA to facilitate inference of individual-specific computational models for the generatio…
The interaction between aphids, ants and ladybirds has been investigated from an ecological point of view since many decades, while there are no attempts to describe it from a mathematical point of view. This paper introduces a new mathematical model to describe the within-season population dynamics in an ecological patch of a system composed by aphids, ants and ladybirds, through a set of four differential equations. The proposed model is based on the Kindlmann and Dixon set of differential equations, focused on the prediction of the aphids-ladybirds population densities, that share a prey-pr…
Finite populations with frequency-dependent selection: a genealogical approach
常规模种群的演化模型经常使用Moran模型、Wright-Fisher模型或它们的扩散极限来研究。当进化是中性的，通过金曼的合并给出的随机系谱是用来理解这种模型的基本属性。在这里，我们解决了使用系谱的角度与弱频率依赖选择模型，即N s =：α是小的，和s是一个适合的个人和N的适应度优势是人口规模。当计算固定概率时，这导致了由Reynset（2003）提出的方法，他讨论了如何使用Kingman的合并来进行弱选择，或者扩展了Neuhauser和Krone（1997）和Neuhauser（1999）的祖先选择图。作为应用，我们重新推导了演化博弈论的三分之一定律（Nowak et al.，2004年）。此外，我们还给出了线性频率依赖下两个随机抽样个体的系谱距离的近似分布。
Many mathematical frameworks of evolutionary game dynamics assume that the total population size is constant and that selection affects only the relative frequency of strategies. Here, we consider evolutionary game dynamics in an extended Wright-Fisher process with variable population size. In such a scenario, it is possible that the entire population becomes extinct. Survival of the population may depend on which strategy prevails in the game dynamics. Studying cooperative dilemmas, it is a natural feature of such a model that cooperators enable survival, while defectors drive extinction. Alt…
The fixation probability of a single mutant invading a population of residents is among the most widely-studied quantities in evolutionary dynamics. Amplifiers of natural selection are population structures that increase the fixation probability of advantageous mutants, compared to well-mixed populations. Extensive studies have shown that many amplifiers exist for the Birth-death Moran process, some of them substantially increasing the fixation probability or even guaranteeing fixation in the limit of large population size. On the other hand, no amplifiers are known for the death-Birth Moran p…
Genetic data are often used to infer demographic history and changes or detect genes under selection. Inferential methods are commonly based on models making various strong assumptions: demography and population structures are supposed \textit{a priori} known, the evolution of the genetic composition of a population does not affect demography nor population structure, and there is no selection nor interaction between and within genetic strains. In this paper, we present a stochastic birth-death model with competitive interactions and asexual reproduction. We develop an inferential procedure fo…
We investigate how initial and boundary conditions influence the competition dynamics and outcome in dispersal-structured populations. The study is carried out through numerical modeling of the heterogeneous Brownian bugs model, in which individuals are characterized by diversified diffusion coefficients and compete for resources within a finite interaction range. We observed that if instead of being distributed randomly across the domain the population is initially localized within a small region, the dynamics and the stationary state of the system differ significantly. The outcome of the nat…
The human adaptive immune response is known to weaken in advanced age, resulting in increased severity of pathogen-born illness, poor vaccine efficacy, and a higher prevalence of cancer in the elderly. Age-related erosion of the T-cell compartment has been implicated as a likely cause, but the underlying mechanisms driving this immunosenescence have not been quantitatively modeled and systematically analyzed. T-cell receptor diversity, or the extent of pathogen-derived antigen responsiveness of the T-cell pool, is known to diminish with age, but inherent experimental difficulties preclude accu…
While Neutral Theory famously describes the number of discrete genetic differences in populations, we consider the number of genetic backgrounds under which such differences are observed - setting limits to the generalizability of their effects. This allow us to determine which population structures and diversity rates have maximal effect generalization across (1) environmental and (2) genetic variation, and to demonstrate that they correspond asymptotically to those of populations under (1) natural selection and (2) drift. At the same time, these results suggest distinct limits to the predict…
