The joint allele frequency spectrum of multiple populations: a coalescent theory approach

The allele frequency spectrum is a series of statistics that describe genetic polymorphism, and is commonly used for inferring population genetic parameters and detecting natural selection. Population genetic theory on the allele frequency spectrum for a single population has been well studied using both coalescent theory and diffusion equations. Recently, the theory was extended to the joint allele frequency spectrum (JAFS) for three populations using diffusion equations and was shown to be very useful in inferring human demographic history. In this paper, I show that the JAFS can be analytically derived with coalescent theory for a basic model of two isolated populations and then extended to multiple populations and various complex scenarios, such as those involving population growth and bottleneck, migration, and positive selection. Simulation study is used to demonstrate the accuracy and applicability of the theoretical model. The coalescent theory-based approach for the JAFS can characterize the demographic history with comprehensive statistical models as the diffusion approach does, and in addition gains several novel advantages: the computational complexity of calculating the JAFS with coalescent theory is reduced, and thus it is feasible to analytically obtain the JAFS for multiple populations; the hitchhiking effect can be efficiently modeled in coalescent theory, enabling the development of methodologies for detecting selection via multi-population polymorphism data. As an alternative to the diffusion approximation approach, the coalescent theory for the JAFS also provides a foundation for population genetic inference with the advent of large-scale genomic polymorphism data.     

Optimal harvesting strategies for a metapopulation

We consider optimal strategies for harvesting a population that is composed of two local populations. The local populations are connected by the dispersal of juveniles, e.g. larvae, and together form a metapopulation. We model the metapopulation dynamics using coupled difference equations. Dynamic programming is used to determine policies for exploitation that are economically optimal. The metapopulation harvesting theory is applied to a hypothetical fishery and optimal strategies are compared to harvesting strategies that assume the metapopulation is composed either of single unconnected populations or of one well-mixed population. Local populations that have high per capita larval production should be more conservatively harvested than would be predicted using conventional theory. Recognizing the metapopulation structure of a stock and using the appropriate theory can significantly improve economic gains.     

Evolution to alternative levels of stable diversity leaves areas of niche space unexplored

One of the oldest and most persistent questions in ecology and evolution is whether natural communities tend to evolve toward saturation and maximal diversity. Robert MacArthur’s classical theory of niche packing and the theory of adaptive radiations both imply that populations will diversify and fully partition any available niche space. However, the saturation of natural populations is still very much an open area of debate and investigation. Additionally, recent evolutionary theory suggests the existence of alternative evolutionary stable states (ESSs), which implies that some stable communities may not be fully saturated. Using models with classical Lotka-Volterra ecological dynamics and three formulations of evolutionary dynamics (a model using adaptive dynamics, an individual-based model, and a partial differential equation model), we show that following an adaptive radiation, communities can often get stuck in low diversity states when limited by mutations of small phenotypic effect. These low diversity metastable states can also be maintained by limited resources and finite population sizes. When small mutations and finite populations are considered together, it is clear that despite the presence of higher-diversity stable states, natural populations are likely not fully saturating their environment and leaving potential niche space unfilled. Additionally, within-species variation can further reduce community diversity from levels predicted by models that assume species-level homogeneity.     

Numerical modelling of plasticity induced by transcranial magnetic stimulation

We use neural field theory and spike-timing dependent plasticity to make a simple but biophysically reasonable model of long-term plasticity changes in the cortex due to transcranial magnetic stimulation (TMS). We show how common TMS protocols can be captured and studied within existing neural field theory. Specifically, we look at repetitive TMS protocols such as theta burst stimulation and paired-pulse protocols. Continuous repetitive protocols result mostly in depression, but intermittent repetitive protocols in potentiation. A paired pulse protocol results in depression at short ( < ～ 10 ms) and long ( > ～ 100 ms) interstimulus intervals, but potentiation for mid-range intervals. The model is sensitive to the choice of neural populations that are driven by the TMS pulses, and to the parameters that describe plasticity, which may aid interpretation of the high variability in existing experimental results. Driving excitatory populations results in greater plasticity changes than driving inhibitory populations. Modelling also shows the merit in optimizing a TMS protocol based on an individual’s electroencephalogram. Moreover, the model can be used to make predictions about protocols that may lead to improvements in repetitive TMS outcomes.     

Effective population size,genetic diversity,and coalescence time in subdivided populations

A formula for the effective population size for the finite island model of subdivided populations is derived. The formula indicates that the effective size can be substantially greater than the actual number of individuals in the entire population when the migration rate among subpopulations is small. It is shown that the mean nucleotide diversity, coalescence time, and heterozygosity for genes sampled from the entire population can be predicted fairly well from the theory for randomly mating populations if the effective population size for the finite island model is used.     

Network-based analysis of stochastic SIR epidemic models with random and proportionate mixing

In this paper, we outline the theory of epidemic percolation networks and their use in the analysis of stochastic susceptible-infectious-removed (SIR) epidemic models on undirected contact networks. We then show how the same theory can be used to analyze stochastic SIR models with random and proportionate mixing. The epidemic percolation networks for these models are purely directed because undirected edges disappear in the limit of a large population. In a series of simulations, we show that epidemic percolation networks accurately predict the mean outbreak size and probability and final size of an epidemic for a variety of epidemic models in homogeneous and heterogeneous populations. Finally, we show that epidemic percolation networks can be used to re-derive classical results from several different areas of infectious disease epidemiology. In an Appendix, we show that an epidemic percolation network can be defined for any time-homogeneous stochastic SIR model in a closed population and prove that the distribution of outbreak sizes given the infection of any given node in the SIR model is identical to the distribution of its out-component sizes in the corresponding probability space of epidemic percolation networks. We conclude that the theory of percolation on semi-directed networks provides a very general framework for the analysis of stochastic SIR models in closed populations.     

Spatial dynamics of adaptive sex ratios

According to Fisherian sex allocation theory, parents that can adjust their offspring sex ratio in response to skews in population sex ratio will maximize their fitness over parents lacking this ability. There is good evidence that adaptive sex ratio adjustment occurs in many natural populations, but deviations from theoretical predictions have also been observed. These anomalies may be more apparent than real. When the spatial dimension of sex ratio variation is ignored, then a mismatch between empirical data and theoretical predictions based on panmictic mating is to be expected. We illustrate this with data on human sex ratio variation in 21 preindustrial populations, and with a cellular automaton model built to obey Fisherian sex allocation rules. The results from the model generally match with the data. When information about the ambient sex ratio is limited, then the sex allocation decisions may appear locally maladaptive. In general, the results indicate that Fisher's sex-ratio theory may have greater explanatory power than previously thought.     

Persistence in fluctuating environments

Understanding under what conditions interacting populations, whether they be plants, animals, or viral particles, coexist is a question of theoretical and practical importance in population biology. Both biotic interactions and environmental fluctuations are key factors that can facilitate or disrupt coexistence. To better understand this interplay between these deterministic and stochastic forces, we develop a mathematical theory extending the nonlinear theory of permanence for deterministic systems to stochastic difference and differential equations. Our condition for coexistence requires that there is a fixed set of weights associated with the interacting populations and this weighted combination of populations’ invasion rates is positive for any (ergodic) stationary distribution associated with a subcollection of populations. Here, an invasion rate corresponds to an average per-capita growth rate along a stationary distribution. When this condition holds and there is sufficient noise in the system, we show that the populations approach a unique positive stationary distribution. Moreover, we show that our coexistence criterion is robust to small perturbations of the model functions. Using this theory, we illustrate that (i) environmental noise enhances or inhibits coexistence in communities with rock-paper-scissor dynamics depending on correlations between interspecific demographic rates, (ii) stochastic variation in mortality rates has no effect on the coexistence criteria for discrete-time Lotka–Volterra communities, and (iii) random forcing can promote genetic diversity in the presence of exploitative interactions.

One day is fine, the next is black.—The Clash     

Toxoplasma gondii and sex: essential or optional extra?

The evolutionary and biological significance of a female-biased sex ratio within apicomplexan parasites has been the subject of much discussion. It is proposed that the sex allocation theory, as applied to inbreeding populations, can explain the sex ratios observed for this diverse group of parasites. This is based on a mathematical model, which assumes that the majority of microgametes will succeed in fertilizing macrogametes. Is this a realistic assumption? It is possible, for different reasons, that the theory may not to be applicable to either malaria parasites or Toxoplasma gondii.     

Space mediates coexistence of females and hermaphrodites

In gynodioecious populations of flowering plants females and hermaphrodites coexist. Gynodioecy is widespread and occurs in both asexual and sexual species but does not admit a satisfactory explanation from classical sex ratio theory. In sexual populations male fertility restoring genes have evolved to counter non-nuclear male sterility mutations. In pseudogamous asexual populations pollen retention and increased self-fertilization can make male sterility costly. Both of these mechanisms can promote coexistence. However, it remains unclear how either of these mechanisms could evolve if coexistence was not initially possible. In the absence of these adaptations non-spatial models predict that females either fail to invade hermaphrodite populations or else displace them until pollen shortage drives the population to extinction. We develop a pair approximation to a probabilistic cellular automata model in which females and hermaphrodites interact on a regular lattice. The model features independent pollination and colonization processes which take place on different timescales. The timescale separation is exploited to obtain, with perturbation methods, a more manageable aggregated pair approximation. We present both the mean field model which recreates the classical non-spatial predictions and the pair approximation, which strikingly predicts different invasion criteria and coexistence under a wide range of parameters. The pair approximation is shown to correspond well qualitatively with simulation behaviour.     

Morphologically-structured models of growing budding yeast populations

It has been well recognized that many key aspects of cell cycle regulation are encoded into the size distributions of growing budding yeast populations due to the tight coupling between cell growth and cell division present in this organism. Several attempts have been made to model the cell size distribution of growing yeast populations in order to obtain insight on the underlying control mechanisms, but most were based on the age structure of asymmetrically dividing populations. Here we propose a new framework that couples a morphologically-structured representation of the population with population balance theory to formulate a dynamic model for the size distribution of growing yeast populations. An advantage of the presented framework is that it allows derivation of simpler models that are directly identifiable from experiments. We show how such models can be derived from the general framework and demonstrate their utility in analyzing yeast population data. Finally, by employing a recently proposed numerical scheme, we proceed to integrate numerically the full distributed model to provide predictions of dynamics of the cell size structure of growing yeast populations.     

Joint inference of population assignment and demographic history

A new approach to assigning individuals to populations using genetic data is described. Most existing methods work by maximizing Hardy-Weinberg and linkage equilibrium within populations, neither of which will apply for many demographic histories. By including a demographic model, within a likelihood framework based on coalescent theory, we can jointly study demographic history and population assignment. Genealogies and population assignments are sampled from a posterior distribution using a general isolation-with-migration model for multiple populations. A measure of partition distance between assignments facilitates not only the summary of a posterior sample of assignments, but also the estimation of the posterior density for the demographic history. It is shown that joint estimates of assignment and demographic history are possible, including estimation of population phylogeny for samples from three populations. The new method is compared to results of a widely used assignment method, using simulated and published empirical data sets.     

Supporting Fisheries Management by Means of Complex Models: Can We Point out Isles of Robustness in a Sea of Uncertainty?

Ecosystems are usually complex, nonlinear and strongly influenced by poorly known environmental variables. Among these systems, marine ecosystems have high uncertainties: marine populations in general are known to exhibit large levels of natural variability and the intensity of fishing efforts can change rapidly. These uncertainties are a source of risks that threaten the sustainability of both fish populations and fishing fleets targeting them. Appropriate management measures have to be found in order to reduce these risks and decrease sensitivity to uncertainties. Methods have been developed within decision theory that aim at allowing decision making under severe uncertainty. One of these methods is the information-gap decision theory. The info-gap method has started to permeate ecological modelling, with recent applications to conservation. However, these practical applications have so far been restricted to simple models with analytical solutions. Here we implement a deterministic approach based on decision theory in a complex model of the Eastern English Channel. Using the ISIS-Fish modelling platform, we model populations of sole and plaice in this area. We test a wide range of values for ecosystem, fleet and management parameters. From these simulations, we identify management rules controlling fish harvesting that allow reaching management goals recommended by ICES (International Council for the Exploration of the Sea) working groups while providing the highest robustness to uncertainties on ecosystem parameters.     

Scale‐dependent role of demography and dispersal on the distribution of populations in heterogeneous landscapes

Both dispersal and local demographic processes determine a population's distribution among habitats of varying quality, yet most theory, experiments, and field studies have focused on the former. We use a generic model to show how both processes contribute to a population's distribution, and how the relative importance of each mechanism depends on scale. In contrast to studies only considering habitat‐dependent dispersal, we show that predictions of ideal free distribution (IFD) theory are relevant even at landscape scales, where the assumptions of IFD theory are violated. This is because scales that inhibit one process, promote the other's ability to drive populations to the IFD. Furthermore, because multiple processes can generate IFDs, the pattern alone does not specify a causal mechanism. This is important because populations with IFDs generated by dispersal or demography respond much differently to shifts in resource distributions.     

Is There Selection on RFLP Differences in Mitochondrial Dna?

Experimental populations of Drosophila simulans were established for the purpose of detecting the presence or absence of selection on a restriction fragment length polymorphism in mitochondrial DNA (mtDNA). It was then discovered that the founding strains differed with respect to the Rickettsia-mediated incompatibility system in this species, which is maternally transmitted together with the mtDNA differences. A population model was constructed using the known fitness effects of the incompatibility system, with the result that the population trajectories can be completely explained by the effects of the microorganism with no need to invoke selection on mtDNA. The strong conclusion is that in this case we can rule out the strong selection proposed by MacRae and Anderson to explain the "dramatic mtDNA changes" in their Drosophila pseudoobscura populations. The population theory used for the experiments is discussed in the context of natural populations. Estimated parameters include the possibility that with two populations, one with the organism and one without it, there may be no bias as to which will invade the other, which in turn suggests no global tendency for the infection to spread or decline.     

Does increasing mortality change the response of fish populations to environmental fluctuations?

Fluctuations of fish populations abundances are shaped by the interplay between population dynamics and the stochastic forcing of the environment. Age-structured populations behave as a filter of the environment. This filter is characterised by the species-specific life cycle and life-history traits. An increased mortality of mature individuals alters these characteristics and may therefore induce changes in the variability of populations. The response of a generic age-structured model was analysed to investigate the expected changes in the fluctuations of fish populations in response to decreased adult survival. These expectations were then tested on an extensive dataset. In accordance with theory, the analyses revealed that decreased adult survival and mean age of spawners were linked to an increase in the relative importance of short-term fluctuations. It suggests that intensive exploitation can lead to a change in the variability of fish populations, an issue of central interest from both conservation and management perspectives.     

Group beneficial norms can spread rapidly in a structured population

Group beneficial norms are common in human societies. The persistence of such norms is consistent with evolutionary game theory, but existing models do not provide a plausible explanation for why they are common. We show that when a model of imitation used to derive replicator dynamics in isolated populations is generalized to allow for population structure, group beneficial norms can spread rapidly under plausible conditions. We also show that this mechanism allows recombination of different group beneficial norms arising in different populations.     

Predictions of the somatic mutation and mortalization theories of cellular ageing are contrary to experimental observations

An accumulation of recessive lethal somatic mutations has often been proposed as a basis of cellular ageing. We have developed a mathematical model for the somatic mutation theory as applied to the finite in vitro lifespan of diploid fibroblast populations. Provided the mutation rate is sufficiently high, the model readily explains the cessation of proliferation of fibroblast cultures, but it predicts a much lower proportion of viable cells than is observed experimentally and also requires an unrealistically short cell division time. It is noted that the somatic mutation model is formally quite similar to the “mortalization” theory of Shall & Stein (1979), and that the mortalization theory is also incompatible with the same experimental data. We conclude that neither the somatic mutation theory nor the mortalization theory can explain the observed features of the growth of fibroblast populations in vitro. We discuss the possibility that deleterious mutations become important in the terminal stages of the lifespan, when they may accumulate as an indirect result of a general breakdown in information transfer between macromolecules.     

A covariance structure model for the admixture of binary genetic variation

I derive a covariance structure model for pairwise linkage disequilibrium (LD) between binary markers in a recently admixed population and use a generalized least-squares method to fit the model to two different data sets. Both linked and unlinked marker pairs are incorporated in the model. Under the model, a pairwise LD matrix is decomposed into two component matrices, one containing LD attributable to admixture, and another containing, in an aggregate form, LD specific to the populations forming the mixture. I use population genetics theory to show that the latter matrix has block-diagonal structure. For the data sets considered here, I show that the number of source populations can be determined by statistical inference on the canonical correlations of the sample LD matrix.     

The application of demographic models to anthropological data

There has recently been great interest in the demography of primitive human populations. This can often be dealt with successfully by the use of demographic theory and model life tables; however, small populations present problems not encountered in large national censuses. This paper discusses some of these problems and illustrates the determination of model life tables in anthropology with a particular population, the Yanomama Indians. The concern is for the methods to be used, and their applicability, and not with the specific population illustrating them. By applying theoretical models, one can derive information that is otherwise unavailable but is necessary in the important biomedical, social, and ecological understanding of primitive societies.