Effects of Immigration on the Dynamics of Simple Population Models

Many simple population models exhibit the period doubling route to chaos as a single parameter, commonly the growth rate, is increased. Here we examine the effect of an immigration process on such models and explain why in the case of one-dimensional ("single-humped") maps, immigration often tends to suppress chaos and stabilise equilibrium behaviour or cyclical oscillations of long period. The conditions for which an increase of immigration "simplifies" population dynamics are examined.     

The Population and Evolutionary Dynamics of Homologous Gene Recombination in Bacteria

In bacteria, recombination is a rare event, not a part of the reproductive process. Nevertheless, recombination—broadly defined to include the acquisition of genes from external sources, i.e., horizontal gene transfer (HGT)—plays a central role as a source of variation for adaptive evolution in many species of bacteria. Much of niche expansion, resistance to antibiotics and other environmental stresses, virulence, and other characteristics that make bacteria interesting and problematic, is achieved through the expression of genes and genetic elements obtained from other populations of bacteria of the same and different species, as well as from eukaryotes and archaea. While recombination of homologous genes among members of the same species has played a central role in the development of the genetics and molecular biology of bacteria, the contribution of homologous gene recombination (HGR) to bacterial evolution is not at all clear. Also, not so clear are the selective pressures responsible for the evolution and maintenance of transformation, the only bacteria-encoded form of HGR. Using a semi-stochastic simulation of mutation, recombination, and selection within bacterial populations and competition between populations, we explore (1) the contribution of HGR to the rate of adaptive evolution in these populations and (2) the conditions under which HGR will provide a bacterial population a selective advantage over non-recombining or more slowly recombining populations. The results of our simulation indicate that, under broad conditions: (1) HGR occurring at rates in the range anticipated for bacteria like Streptococcus pneumoniae, Escherichia coli, Haemophilus influenzae, and Bacillus subtilis will accelerate the rate at which a population adapts to environmental conditions; (2) once established in a population, selection for this capacity to increase rates of adaptive evolution can maintain bacteria-encoded mechanisms of recombination and prevent invasion of non-recombining populations, even when recombination engenders a modest fitness cost; and (3) because of the density- and frequency-dependent nature of HGR in bacteria, this capacity to increase rates of adaptive evolution is not sufficient as a selective force to provide a recombining population a selective advantage when it is rare. Under realistic conditions, homologous gene recombination will increase the rate of adaptive evolution in bacterial populations and, once established, selection for higher rates of evolution will promote the maintenance of bacteria-encoded mechanisms for HGR. On the other hand, increasing rates of adaptive evolution by HGR is unlikely to be the sole or even a dominant selective pressure responsible for the original evolution of transformation.     

Local Replicator Dynamics: A Simple Link Between Deterministic and Stochastic Models of Evolutionary Game Theory

Classical replicator dynamics assumes that individuals play their games and adopt new strategies on a global level: Each player interacts with a representative sample of the population and if a strategy yields a payoff above the average, then it is expected to spread. In this article, we connect evolutionary models for infinite and finite populations: While the population itself is infinite, interactions and reproduction occurs in random groups of size N. Surprisingly, the resulting dynamics simplifies to the traditional replicator system with a slightly modified payoff matrix. The qualitative results, however, mirror the findings for finite populations, in which strategies are selected according to a probabilistic Moran process. In particular, we derive a one-third law that holds for any population size. In this way, we show that the deterministic replicator equation in an infinite population can be used to study the Moran process in a finite population and vice versa. We apply the results to three examples to shed light on the evolution of cooperation in the iterated prisoner’s dilemma, on risk aversion in coordination games and on the maintenance of dominated strategies.     

具年龄结构的种群在破碎化生境中的空间分布模型

随着人类和其他生物赖以生存的环境破碎化程度的加剧,许多以前是连续分布的物种,目前不得不在破碎化生境(斑块)中求生存,所以,种群在破碎化生境(斑块)中分布问题的研究对生物保护和生境重建意义重大．本文运用Leslie矩阵和Markov链建立了一个具年龄结构的种群在破碎化生境中随时间动态变化的分布模型,讨论了种群在该生境中持续存在以及灭绝的条件．     

Asymptotic Distribution of Density-Dependent Stage-Grouped Population Dynamics Models

Matrix models are widely used in biology to predict the temporal evolution of stage-structured populations. One issue related to matrix models that is often disregarded is the sampling variability. As the sample used to estimate the vital rates of the models are of finite size, a sampling error is attached to parameter estimation, which has in turn repercussions on all the predictions of the model. In this study, we address the question of building confidence bounds around the predictions of matrix models due to sampling variability. We focus on a density-dependent Usher model, the maximum likelihood estimator of parameters, and the predicted stationary stage vector. The asymptotic distribution of the stationary stage vector is specified, assuming that the parameters of the model remain in a set of the parameter space where the model admits one unique equilibrium point. Tests for density-dependence are also incidentally provided. The model is applied to a tropical rain forest in French Guiana.     

Feedback between Population and Evolutionary Dynamics Determines the Fate of Social Microbial Populations

The evolutionary spread of cheater strategies can destabilize populations engaging in social cooperative behaviors, thus demonstrating that evolutionary changes can have profound implications for population dynamics. At the same time, the relative fitness of cooperative traits often depends upon population density, thus leading to the potential for bi-directional coupling between population density and the evolution of a cooperative trait. Despite the potential importance of these eco-evolutionary feedback loops in social species, they have not yet been demonstrated experimentally and their ecological implications are poorly understood. Here, we demonstrate the presence of a strong feedback loop between population dynamics and the evolutionary dynamics of a social microbial gene, SUC2, in laboratory yeast populations whose cooperative growth is mediated by the SUC2 gene. We directly visualize eco-evolutionary trajectories of hundreds of populations over 50–100 generations, allowing us to characterize the phase space describing the interplay of evolution and ecology in this system. Small populations collapse despite continual evolution towards increased cooperative allele frequencies; large populations with a sufficient number of cooperators “spiral” to a stable state of coexistence between cooperator and cheater strategies. The presence of cheaters does not significantly affect the equilibrium population density, but it does reduce the resilience of the population as well as its ability to adapt to a rapidly deteriorating environment. Our results demonstrate the potential ecological importance of coupling between evolutionary dynamics and the population dynamics of cooperatively growing organisms, particularly in microbes. Our study suggests that this interaction may need to be considered in order to explain intraspecific variability in cooperative behaviors, and also that this feedback between evolution and ecology can critically affect the demographic fate of those species that rely on cooperation for their survival.     

Reduction of Nonautonomous Population Dynamics Models with Two Time Scales

The purpose of this work is reviewing some reduction results to deal with systems of nonautonomous ordinary differential equations with two time scales. They could be included among the so-called approximate aggregation methods. The existence of different time scales in a system, together with some long-term features, are used to build up a simpler system governed by a lesser number of state variables. The asymptotic behavior of the latter system is then used to describe the asymptotic behaviour of the former one. The reduction results are stated in two particular but important cases: periodic systems and asymptotically autonomous systems. The reduction results are illustrated with the help of simple spatial SIS epidemic models including either periodic or asymptotically autonomous terms.     

The Evolutionary Dynamics of Bluetongue Virus

Bluetongue virus (BTV) is a midge-borne member of the genus Orbivirus that causes an eponymous debilitating livestock disease of great agricultural impact and which has expanded into Europe in recent decades. Reassortment among the ten segments comprising the double-stranded (ds) RNA genome of BTV has played an important role in generating the epidemic strains of this virus in Europe. In this study, we investigated the dynamics of BTV genome segment evolution utilizing time-structured data sets of complete sequences from four segments, totalling 290 sequences largely sampled from ruminant hosts. Our analysis revealed that BTV genome segments generally evolve under strong purifying selection and at substitution rates that are generally lower (mean rates of ~0.5–7 × 10⁻⁴ nucleotide substitutions per site, per year) than vector-borne positive-sense viruses with single-strand (ss) RNA genomes. These also represent the most robust estimates of the nucleotide substitution rate in a dsRNA virus generated to date. Additionally, we determined that patterns of geographic structure and times to most recent common ancestor differ substantially between each segment, including a relatively recent origin for the diversity of segment 10 within the past millennium. Together, these findings demonstrate the effect of reassortment to decouple the evolutionary dynamics of BTV genome segments.     

Population Dynamics of Evolutionary Change: Demographic Parameters as Indicators of Fitness

I evaluated demographic parameters as indicators of fitness by calculating the net reproductive rate (R₀), exponential rate of change (r), lifetime reproductive success (LRS), and Malthusian parameter (m) for nine genotypes and four phenotypes (two alleles at each of two independent loci) of an age-structured population. The given starting conditions included age-specific survival rates of males and females and age-specific fecundity of females for each genotype (to simplify the problem I presumed no differences in survivorship or fecundity of genotypes with the same phenotype) and the same age structure for each genotype. The prevailing genotype had the greatestm, but it did not have the greatestr,R₀, or LRS, or even the greatest survivorship of either juveniles or adults, or the greatest fecundity. This result indicates thatmis the only correct measure of fitness (i.e., as a predictor of which genotype should prevail from among a group of genotypes) and that comparisons ofr,R₀, LRS, juvenile or adult survival rates, or fecundity may be misleading indicators of which genotype should prevail (i.e., be most “fit”) over time (i.e., be selected for).     

The Evolutionary and Epidemiological Dynamics of the Paramyxoviridae

Paramyxoviruses are responsible for considerable disease burden in human and wildlife populations: measles and mumps continue to affect the health of children worldwide, while canine distemper virus causes serious morbidity and mortality in a wide range of mammalian species. Although these viruses have been studied extensively at both the epidemiological and the phylogenetic scales, little has been done to integrate these two types of data. Using a Bayesian coalescent approach, we infer the evolutionary and epidemiological dynamics of measles, mumps and canine distemper viruses. Our analysis yielded data on viral substitution rates, the time to common ancestry, and elements of their demographic history. Estimates of rates of evolutionary change were similar to those observed in other RNA viruses, ranging from 6.585 to 11.350 × 10⁻⁴ nucleotide substitutions per site, per year. Strikingly, the mean Time to the Most Recent Common Ancestor (TMRCA) was both similar and very recent among the viruses studied, ranging from only 58 to 91 years (1908 to 1943). Worldwide, the paramyxoviruses studied here have maintained a relatively constant level of genetic diversity. However, detailed heterchronous samples illustrate more complex dynamics in some epidemic populations, and the relatively low levels of genetic diversity (population size) in all three viruses is likely to reflect the population bottlenecks that follow recurrent outbreaks.     

A Model of Oscillatory Protein Dynamics in Bacteria

Spatial oscillations of proteins in bacteria have recently attracted much attention. The cellular mechanism underlying these oscillations can be studied at molecular as well as at more macroscopic levels. We construct a minimal mathematical model with two proteins that is able to produce self-sustained regular pole-to-pole oscillations without having to take into account molecular details of the proteins and their interactions. The dynamics of the model is based solely on diffusion across the cell body and protein reactions at the poles, and is independent of stimuli coming from the environment. We solve the associated system of reaction-diffusion equations and perform a parameter scan to demonstrate robustness of the model for two possible sets of the reaction functions.     

Evolutionary Dynamics of HTLV-I

Using mathematical models to describe the in vivo dynamics of HTLV-I infection, an explanation is offered for the slow rate of evolution of HTLV-I relative to HIV-1. In agreement with experimental findings, it is assumed that cell activation is required for successful replication in T helper cells and that HTLV-I induces a significant degree of bystander activation. It is found that the rate of evolution of HTLV-I is limited by the restricted availability of activated uninfected T cells, both at high and low proviral loads. This limits the within-host sequence diversity of HTLV-I and may therefore account for the slow rate of evolution of the virus in the population. Specific differences in the in vivo dynamics of HTLV-I and HIV-1 are identified which may account for the discrepancy in the rate of evolution of these two retroviruses. Received: 7 September 1999 / Accepted: 6 December 1999     

The Genealogical Population Dynamics of HIV-1 in a Large Transmission Chain: Bridging within and among Host Evolutionary Rates

Transmission lies at the interface of human immunodeficiency virus type 1 (HIV-1) evolution within and among hosts and separates distinct selective pressures that impose differences in both the mode of diversification and the tempo of evolution. In the absence of comprehensive direct comparative analyses of the evolutionary processes at different biological scales, our understanding of how fast within-host HIV-1 evolutionary rates translate to lower rates at the between host level remains incomplete. Here, we address this by analyzing pol and env data from a large HIV-1 subtype C transmission chain for which both the timing and the direction is known for most transmission events. To this purpose, we develop a new transmission model in a Bayesian genealogical inference framework and demonstrate how to constrain the viral evolutionary history to be compatible with the transmission history while simultaneously inferring the within-host evolutionary and population dynamics. We show that accommodating a transmission bottleneck affords the best fit our data, but the sparse within-host HIV-1 sampling prevents accurate quantification of the concomitant loss in genetic diversity. We draw inference under the transmission model to estimate HIV-1 evolutionary rates among epidemiologically-related patients and demonstrate that they lie in between fast intra-host rates and lower rates among epidemiologically unrelated individuals infected with HIV subtype C. Using a new molecular clock approach, we quantify and find support for a lower evolutionary rate along branches that accommodate a transmission event or branches that represent the entire backbone of transmitted lineages in our transmission history. Finally, we recover the rate differences at the different biological scales for both synonymous and non-synonymous substitution rates, which is only compatible with the ‘store and retrieve’ hypothesis positing that viruses stored early in latently infected cells preferentially transmit or establish new infections upon reactivation.     

宿主-寄生物相互作用种群模型中的混沌与分形现象

自然界具有离散世代的宿主-寄生物相互作用的种群模型都以差分方程组来描述,由于宿主-寄生物相互关系具有不同的类型,模型的功能反应函数也具有多种形式.通过数值模拟试验,分析研究了聚集效应模型随参数变化时表现出的混沌动态行为以及吸引域的自相似分形属性.结果显示虽然在一定参数范围内种群数量显示严格的周期,但混沌动态是不可避免的,共存的多吸引子初值区域显示出自相似分形属性.说明混沌动态行为和分形属性是离散的种群互作用种群模型中必然出现的现象.     

The Evolutionary Dynamics of Human Influenza B Virus

Despite their close phylogenetic relationship, type A and B influenza viruses exhibit major epidemiological differences in humans, with the latter both less common and less often associated with severe disease. However, it is unclear what processes determine the evolutionary dynamics of influenza B virus, and how influenza viruses A and B interact at the evolutionary scale. To address these questions we inferred the phylogenetic history of human influenza B virus using complete genome sequences for which the date (day) of isolation was available. By comparing the phylogenetic patterns of all eight viral segments we determined the occurrence of segment reassortment over a 30-year sampling period. An analysis of rates of nucleotide substitution and selection pressures revealed sporadic occurrences of adaptive evolution, most notably in the viral hemagglutinin and compatible with the action of antigenic drift, yet lower rates of overall and nonsynonymous nucleotide substitution compared to influenza A virus. Overall, these results led us to propose a model in which evolutionary changes within and between the antigenically distinct 'Yam88' and 'Vic87' lineages of influenza B virus are the result of changes in herd immunity, with reassortment continuously generating novel genetic variation. Additionally, we suggest that the interaction with influenza A virus may be central in shaping the evolutionary dynamics of influenza B virus, facilitating the shift of dominance between the Vic87 and the Yam88 lineages.     

Immunobiological Outcomes of Repeated Chlamydial Infection from Two Models of Within-Host Population Dynamics

Background

Chlamydia trachomatis is a common human pathogen that mediates disease processes capable of inflicting serious complications on reproduction. Aggressive inflammatory immune responses are thought to not only direct a person''s level of immunity but also the potential for immunopathology. With human immunobiology being debated as a cause of prevailing epidemiological trends, we examined some fundamental issues regarding susceptibility to multiple chlamydial infections that could have implications for infection spread. We argue that, compared to less-frequent exposure, frequent exposure to chlamydia may well produce unique immunobiological characteristics that likely to have important clinical and epidemiological implications.

Methods and Results

As a novel tool for studying chlamydia, we applied principles of modeling within-host pathogen dynamics to enable an understanding of some fundamental characteristics of an individual''s immunobiology during multiple chlamydial infections. While the models were able to reproduce shorter-term infection kinetics of primary and secondary infections previously observed in animal models, it was also observed that longer periods between initial and second infection may increase an individual''s chlamydial load and lengthen their duration of infectiousness. The cessation of short-term repeated exposure did not allow for the formation of long-lasting immunity. However, frequent re-exposure non-intuitively linked the formation of protective immunity, persistent infection, and the potential for immunopathology.

Conclusions

Overall, these results provide interesting insights that should be verified with continued study. Nevertheless, these results appear to raise challenges for current evidence of the development of long-lasting immunity against chlamydia, and suggest the existence of a previously unidentified mechanism for the formation of persistent infection. The obvious next goal is to investigate the qualitative impact of these results on the spread of chlamydia.     

Synchronization and Oscillatory Dynamics in Heterogeneous, Mutually Inhibited Neurons

We study some mechanisms responsible for synchronous oscillations and loss of synchrony at physiologically relevant frequencies (10–200 Hz) in a network of heterogeneous inhibitory neurons. We focus on the factors that determine the level of synchrony and frequency of the network response, as well as the effects of mild heterogeneity on network dynamics. With mild heterogeneity, synchrony is never perfect and is relatively fragile. In addition, the effects of inhibition are more complex in mildly heterogeneous networks than in homogeneous ones. In the former, synchrony is broken in two distinct ways, depending on the ratio of the synaptic decay time to the period of repetitive action potentials (_s/T), where T can be determined either from the network or from a single, self-inhibiting neuron. With _s/T > 2, corresponding to large applied current, small synaptic strength or large synaptic decay time, the effects of inhibition are largely tonic and heterogeneous neurons spike relatively independently. With _s/T < 1, synchrony breaks when faster cells begin to suppress their less excitable neighbors; cells that fire remain nearly synchronous. We show numerically that the behavior of mildly heterogeneous networks can be related to the behavior of single, self-inhibiting cells, which can be studied analytically.