A stochastic model of evolutionary dynamics with deterministic large-population asymptotics

An evolutionary birth-death process is proposed as a model of evolutionary dynamics. Agents residing in a continuous spatial environment X, play a game G, with a continuous strategy set S, against other agents in the environment. The agents’ positions and strategies continuously change in response to other agents and to random effects. Agents spawn asexually at rates that depend on their current fitness, and agents die at rates that depend on their local population density. Agents’ individual evolutionary trajectories in X and S are governed by a system of stochastic ODEs. When the number of agents is large and distributed in a smooth density on (X,S), the collective dynamics of the entire population is governed by a certain (deterministic) PDE, which we call a fitness-diffusion equation.     

Simulating natural selection as a culling mechanism on finite populations with the hawk-dove game

The behaviors of individuals and species are often explained in terms of evolutionary stable strategies (ESSs). The analysis of ESSs determines which, if any, combinations of behaviors cannot be invaded by alternative strategies. Two assumptions required to generate an ESS (i.e., an infinite population and payoffs described only on the average) do not hold under natural conditions. Previous experiments indicated that under more realistic conditions of finite populations and stochastic payoffs, populations may evolve in trajectories that are unrelated to an ESS, even in very simple games. The simulations offered here extend earlier research by employing truncation selection with random parental selection in a hawk-dove game. Payoffs are determined in pairwise contests using either the expected outcome, or the result of a random variable. In each case, however, the mean fraction of hawks over many generations and across many independent trials does not conform to the expected ESS. Implications of these results and philosophical underpinnings of ESS theory are offered.     

Evolution,games theory and polyhedra

The problem of finding an Evolutionary Stable Strategy (ESS) for an animal species is defined. It is shown how such strategies are a subset of the equilibrium solutions for a particular non-zero sum game. These equilibrium solutions are then shown to arise from the vertices of a particular convex polyhedron. A method of finding these equilibrium solutions through the vertices and then the ESS solutions is given. This is illustrated by a number of numerical examples taken from the literature. Finally an alternative approach based on solving a Linear Complementarity Problem is discussed.     

A rule-based approach to the modelling of bacterial ecosystems

This paper presents an approach to ecological/evolutionary modelling that is inspired by natural bacterial ecosystems and bacterial evolution. An individual-based artificial ecosystem model is proposed, which is designed to explore the evolvability of adaptive behavioural strategies in artificial bacteria represented by rule-based learning classifier systems. The proposed ecosystem model consists of a n-dimensional environmental grid, which can contain different types of artificial resources in arbitrary arrangements. The resources provide the energy that is necessary for the organisms to sustain life, and can trigger different types of behaviour in the organisms, such as movement towards nutrients and away from toxic substances, growth, and the controlled release of signalling resources. The balance between energy and material is modelled carefully to ensure that the ecosystem is dissipative. Those organisms that are able to efficiently exploit the available resources gradually accumulate enough energy to reproduce (by binary fission) and generate copies of themselves in the environment. Organisms are also able to produce their own resources, which can potentially be used as markers to send signals to other organisms (a behaviour known as quorum sensing). The complex relationships between stimuli and actions in the organisms are stochastically altered by means of mutations, thus enabling the organisms to adapt to their environment and maximise their lifespan and reproductive success. In this paper, the proposed bacterial ecosystem model is defined formally and its structure is discussed in detail. This is followed by results from simulation experiments that illustrate the model's operation and how it can be used in evolutionary modelling/computing scenarios.     

Game Dynamics with Learning and Evolution of Universal Grammar

We investigate a model of language evolution, based on population game dynamics with learning. First, we examine the case of two genetic variants of universal grammar (UG), the heart of the human language faculty, assuming each admits two possible grammars. The dynamics are driven by a communication game. We prove using dynamical systems techniques that if the payoff matrix obeys certain constraints, then the two UGs are stable against invasion by each other, that is, they are evolutionarily stable. Then, we prove a similar theorem for an arbitrary number of disjoint UGs. In both theorems, the constraints are independent of the learning process. Intuitively, if a mutation in UG results in grammars that are incompatible with the established languages, then the mutation will die out because mutants will be unable to communicate and therefore unable to realize any potential benefit of the mutation. An example for which these theorems do not apply shows that compatible mutations may or may not be able to invade, depending on the population's history and the learning process. These results suggest that the genetic history of language is constrained by the need for compatibility and that mutations in the language faculty may have died out or taken over due more to historical accident than to any straightforward notion of relative fitness. MSC 1991: 37N25 · 92D15 · 91F20     

Evolution of complex probability distributions in enzyme cascades

Unusual probability distribution profiles, including transient multi-peak distributions, have been observed in computer simulations of cell signaling dynamics. The emergence of these complex distributions cannot be explained using either deterministic chemical kinetics or simple Gaussian noise approximation. To develop physical insights into the origin of complex distributions in stochastic cell signaling, we compared our approximate analytical solutions of signaling dynamics with the exact numerical simulations. Our results are based on studying signaling in 2-step and 3-step enzyme amplification cascades that are among the most common building blocks of cellular protein signaling networks. We have found that while the multi-peak distributions are typically transient, and eventually evolve into single peak distributions, in certain cases these distributions may be stable in the limit of long times. We also have shown that introducing positive feedback loops results in diminution of the probability distribution complexity.     

The Evolution and Consequences of Sex-Specific Reproductive Variance

Natural selection favors alleles that increase the number of offspring produced by their carriers. But in a world that is inherently uncertain within generations, selection also favors alleles that reduce the variance in the number of offspring produced. If previous studies have established this principle, they have largely ignored fundamental aspects of sexual reproduction and therefore how selection on sex-specific reproductive variance operates. To study the evolution and consequences of sex-specific reproductive variance, we present a population-genetic model of phenotypic evolution in a dioecious population that incorporates previously neglected components of reproductive variance. First, we derive the probability of fixation for mutations that affect male and/or female reproductive phenotypes under sex-specific selection. We find that even in the simplest scenarios, the direction of selection is altered when reproductive variance is taken into account. In particular, previously unaccounted for covariances between the reproductive outputs of different individuals are expected to play a significant role in determining the direction of selection. Then, the probability of fixation is used to develop a stochastic model of joint male and female phenotypic evolution. We find that sex-specific reproductive variance can be responsible for changes in the course of long-term evolution. Finally, the model is applied to an example of parental-care evolution. Overall, our model allows for the evolutionary analysis of social traits in finite and dioecious populations, where interactions can occur within and between sexes under a realistic scenario of reproduction.     

Evolution of cannibalistic traits: scenarios derived from adaptive dynamics

The evolution of cannibalistic traits in consumer populations is studied in this paper with the approach of adaptive dynamics theory. The model is kept at its minimum complexity by eliminating some environmental characteristics, like heterogeneity and seasonalities, and by hiding the size-structure of the population. Evolutionary dynamics are identified through numerical bifurcation analysis, applied both to the ecological (resident-mutant) model and to the canonical equation of adaptive dynamics. The result is a rich catalog of evolutionary scenarios involving evolutionary stable strategies and branching points both in the monomorphic and dimorphic dynamics. The possibility of evolutionary extinction of highly cannibalistic populations is also ascertained. This allows one to explain why cannibalism can be a transient stage of evolution.     

On the stochastic model for estimation of mutational distance between homologous proteins

Summary A set of simple equations is derived which gives the relationship between the observed amino acid differences per 100 codons and the evolutionary distance per 100 codons using Holmquist's stochastic model of molecular evolution.Contribution No. 910 from the National Institute of Genetics, Mishima, Shizuoka-ken 411 Japan.     

Population size and rate of evolution

Summary It is suggested that in evolution there is much substitution of nearly neutral mutations, for which the selection intensity varies from time to time or from region to region. Since the variance among the selection coefficients of new mutants decreases when the environment becomes uniform, the probability of a mutant being advantageous to the species as a whole increases in more uniform environment (Fig. 1).Therefore the rate of gene substitution increases in smaller populations, as smaller populations are likely to be distributed over less varied environments.The adequacy of the model was discussed in relation with the following facts or plausible postulates. 1. A large number of amino acid substitutions during a period corresponding to the formation of new species. 2. Rapid evolution at the phenotypic level of populations having a small size. 3. Many extinctions and expansions of the species in the past.Contribution No. 871 from the National Institute of Genetics, Mishima, Shizuokaken 411 Japan.     

带有正负系数的一阶震荡中立型微分方程

本文建立了中立型微分方程d/dt[x（t）-R（t）x（t-r）]＋P（t）x（t-r）-Q（t）x（t-δ）=0震荡的一个充分条件．     

Assessing the functional implications of soil biodiversity in ecosystems

Soil communities are among the most species-rich components of terrestrial ecosystems. A major challenge for soil ecologists is to formulate feasible research strategies that will preserve and capitalize on the biodiversity resources of the soil. This article considers the role of soil organism diversity by concentrating on: (i) the relationship between soil biodiversity and ecosystem function; (ii) what issues need to be explored; (iii) studies carried out in the Ecotron controlled environment facility; and (iv) how stable isotope techniques can improve our understanding of the relationship between soil biodiversity and ecosystem function. It is advocated that: (i) the objective of any soil biodiversity study should always be the generation of general concepts, rather than local, system-specific observations; and (ii) any empirical study can be properly interpreted only within a quantitative ecological framework.     

逃逸速率对分子马达定向运动的影响

在考虑逃逸速率的情况下采用主方程方法研究分子马达,计算了逃逸速率对分子马达的漂移速率、扩散系数、停留时间以及分子马达在轨道上移动的距离．通过计算发现,逃逸速率影响分子马达的定向运动．     

Evolutionary graph theory: breaking the symmetry between interaction and replacement

We study evolutionary dynamics in a population whose structure is given by two graphs: the interaction graph determines who plays with whom in an evolutionary game; the replacement graph specifies the geometry of evolutionary competition and updating. First, we calculate the fixation probabilities of frequency dependent selection between two strategies or phenotypes. We consider three different update mechanisms: birth-death, death-birth and imitation. Then, as a particular example, we explore the evolution of cooperation. Suppose the interaction graph is a regular graph of degree h, the replacement graph is a regular graph of degree g and the overlap between the two graphs is a regular graph of degree l. We show that cooperation is favored by natural selection if b/c>hg/l. Here, b and c denote the benefit and cost of the altruistic act. This result holds for death-birth updating, weak-selection and large population size. Note that the optimum population structure for cooperators is given by maximum overlap between the interaction and the replacement graph (g=h=l), which means that the two graphs are identical. We also prove that a modified replicator equation can describe how the expected values of the frequencies of an arbitrary number of strategies change on replacement and interaction graphs: the two graphs induce a transformation of the payoff matrix.     

Evolution probabilities and phylogenetic distance of dinucleotides

We develop here an analytical evolution model based on a dinucleotide mutation matrix 16 x 16 with six substitution parameters associated with the three types of substitutions in the two dinucleotide sites. It generalizes the previous models based on the nucleotide mutation matrices 4 x 4. It determines at some time t the exact occurrence probabilities of dinucleotides mutating randomly according to these six substitution parameters. Furthermore, several properties and two applications of this model allow to derive 16 evolutionary analytical solutions of dinucleotides and also a dinucleotide phylogenetic distance. Finally, based on this mathematical model, the SED (Stochastic Evolution of Dinucleotides) web server has been developed for deriving evolutionary analytical solutions of dinucleotides.     

A simple quantitative model of the molecular clock

Summary We present the ideas, and their motivation, at the basis of a simple model of nucleic acid evolution: thestationary Markov process, or Markov clock. After a brief review of its relevant mathematical properties, the Markov clock is applied to nucleotide sequences from mitochondrial and nuclear genes of different species. Particular emphasis is given to the necessity of carrying out a correct statistical analysis, which allows us to check quantitatively the applicability of our model. We find evidence that the Markov clock ticks in many different processes, and that its limitations can be understood in terms of a simple idea that we call the base-drift hypothesis. This hypothesis correlates the deviations from the stationarity of the Markov process to the evolutionary distanced _AB ^(P) of two species A and B, relative to the processP. We conclude by discussing the implications of our findings for future work.     

Natural selection and the origin and maintenance of standard genetic marker systems

Natural selection has always been assumed to be the major force of evolution, but its presence has been difficult to demonstrate. A review of the evidence for selective differences among genotypes for most human genetic polymorphisms indicates there is little of a direct nature. Indirect theoretical evidence, however, seems to support a major role for natural selection, and it does not seem to support the hypothesis that most amino acid substitutions within the human species are neutral. Among small isolates, most of the gene frequency differences are most likely due to genetic drift or the founder effect, and the principal counterbalancing force is gene flow or migration. But genetic differences among the major human subdivisions do not seem to be due to the same interacting forces. One reason for the inability to detect selection has been an oversimplified view of its operation, which assigns genotypes a constant fitness in every generation. Many recent theoretical developments of more complicated kinds of selection may lead to a resolution of the problem and suggest better interpretations of the enormous amount of data on human genetic variation that is rapidly accumulating.