Measures of success in a class of evolutionary models with fixed population size and structure

We investigate a class of evolutionary models, encompassing many established models of well-mixed and spatially structured populations. Models in this class have fixed population size and structure. Evolution proceeds as a Markov chain, with birth and death probabilities dependent on the current population state. Starting from basic assumptions, we show how the asymptotic (long-term) behavior of the evolutionary process can be characterized by probability distributions over the set of possible states. We then define and compare three quantities characterizing evolutionary success: fixation probability, expected frequency, and expected change due to selection. We show that these quantities yield the same conditions for success in the limit of low mutation rate, but may disagree when mutation is present. As part of our analysis, we derive versions of the Price equation and the replicator equation that describe the asymptotic behavior of the entire evolutionary process, rather than the change from a single state. We illustrate our results using the frequency-dependent Moran process and the birth–death process on graphs as examples. Our broader aim is to spearhead a new approach to evolutionary theory, in which general principles of evolution are proven as mathematical theorems from axioms.     

Invasion and adaptive evolution for individual-based spatially structured populations

The interplay between space and evolution is an important issue in population dynamics, that is particularly crucial in the emergence of polymorphism and spatial patterns. Recently, biological studies suggest that invasion and evolution are closely related. Here, we model the interplay between space and evolution starting with an individual-based approach and show the important role of parameter scalings on clustering and invasion. We consider a stochastic discrete model with birth, death, competition, mutation and spatial diffusion, where all the parameters may depend both on the position and on the phenotypic trait of individuals. The spatial motion is driven by a reflected diffusion in a bounded domain. The interaction is modelled as a trait competition between individuals within a given spatial interaction range. First, we give an algorithmic construction of the process. Next, we obtain large population approximations, as weak solutions of nonlinear reaction–diffusion equations. As the spatial interaction range is fixed, the nonlinearity is nonlocal. Then, we make the interaction range decrease to zero and prove the convergence to spatially localized nonlinear reaction–diffusion equations. Finally, a discussion of three concrete examples is proposed, based on simulations of the microscopic individual-based model. These examples illustrate the strong effects of the spatial interaction range on the emergence of spatial and phenotypic diversity (clustering and polymorphism) and on the interplay between invasion and evolution. The simulations focus on the qualitative differences between local and nonlocal interactions.      

Spatial Moment Description of Birth–Death–Movement Processes Incorporating the Effects of Crowding and Obstacles

Birth–death–movement processes, modulated by interactions between individuals, are fundamental to many cell biology processes. A key feature of the movement of cells within in vivo environments is the interactions between motile cells and stationary obstacles. Here we propose a multi-species model of individual-level motility, proliferation and death. This model is a spatial birth–death–movement stochastic process, a class of individual-based model (IBM) that is amenable to mathematical analysis. We present the IBM in a general multi-species framework and then focus on the case of a population of motile, proliferative agents in an environment populated by stationary, non-proliferative obstacles. To analyse the IBM, we derive a system of spatial moment equations governing the evolution of the density of agents and the density of pairs of agents. This approach avoids making the usual mean-field assumption so that our models can be used to study the formation of spatial structure, such as clustering and aggregation, and to understand how spatial structure influences population-level outcomes. Overall the spatial moment model provides a reasonably accurate prediction of the system dynamics, including important effects such as how varying the properties of the obstacles leads to different spatial patterns in the population of agents.     

Evolutionary tracking is determined by differential selection on demographic rates and density dependence

Recent ecological forecasts predict that ~25% of species worldwide will go extinct by 2050. However, these estimates are primarily based on environmental changes alone and fail to incorporate important biological mechanisms such as genetic adaptation via evolution. Thus, environmental change can affect population dynamics in ways that classical frameworks can neither describe nor predict. Furthermore, often due to a lack of data, forecasting models commonly describe changes in population demography by summarizing changes in fecundity and survival concurrently with the intrinsic growth rate (r). This has been shown to be an oversimplification as the environment may impose selective pressure on specific demographic rates (birth and death) rather than directly on r (the difference between the birth and death rates). This differential pressure may alter population response to density, in each demographic rate, further diluting the information combined to produce r. Thus, when we consider the potential for persistence via adaptive evolution, populations with the same r can have different abilities to persist amidst environmental change. Therefore, we cannot adequately forecast population response to climate change without accounting for demography and selection on density dependence. Using a continuous‐time Markov chain model to describe the stochastic dynamics of the logistic model of population growth and allow for trait evolution via mutations arising during birth events, we find persistence via evolutionary tracking more likely when environmental change alters birth rather than the death rate. Furthermore, species that evolve responses to changes in the strength of density dependence due to environmental change are less vulnerable to extinction than species that undergo selection independent of population density. By incorporating these key demographic considerations into our predictive models, we can better understand how species will respond to climate change.     

Fixation probabilities for the Moran process with three or more strategies: general and coupling results

We study fixation probabilities for the Moran stochastic process for the evolution of a population with three or more types of individuals and frequency-dependent fitnesses. Contrary to the case of populations with two types of individuals, in which fixation probabilities may be calculated by an exact formula, here we must solve a large system of linear equations. We first show that this system always has a unique solution. Other results are upper and lower bounds for the fixation probabilities obtained by coupling the Moran process with three strategies with birth–death processes with only two strategies. We also apply our bounds to the problem of evolution of cooperation in a population with three types of individuals already studied in a deterministic setting by Núñez Rodríguez and Neves (J Math Biol 73:1665–1690, 2016). We argue that cooperators will be fixated in the population with probability arbitrarily close to 1 for a large region of initial conditions and large enough population sizes.

     

On some Markov models of certain interacting populations

We give a stochastic foundation to the Volterra prey-predator population in the following case. We take Volterra's predator equations and let a free host birth and death process support the evolution of the predator population. The purpose of this article is to present a rigorous population sample path construction of this interacted predator process and study the properties of this interacted process. The constructions yields a strong Markov process. The existence of steady-state distribution for the interacted predator process means the existence of equilibrium population level. We find a necessary and sufficient condition for the existence of a steady-state distribution. Next we see that if the host process possesses a steady-state distribution, so does the interacted predator process and this distribution satisfies a difference equation. For special choices of the auto death and interaction parametersa andb of the predator, whenever the host process visits the particular statea ^*=a/b the predator takes rest (saturates) from its evolution. We find the probability of asymptotic saturating of the predator.     

Evolutionary dynamics in structured populations

Evolutionary dynamics shape the living world around us. At the centre of every evolutionary process is a population of reproducing individuals. The structure of that population affects evolutionary dynamics. The individuals can be molecules, cells, viruses, multicellular organisms or humans. Whenever the fitness of individuals depends on the relative abundance of phenotypes in the population, we are in the realm of evolutionary game theory. Evolutionary game theory is a general approach that can describe the competition of species in an ecosystem, the interaction between hosts and parasites, between viruses and cells, and also the spread of ideas and behaviours in the human population. In this perspective, we review the recent advances in evolutionary game dynamics with a particular emphasis on stochastic approaches in finite sized and structured populations. We give simple, fundamental laws that determine how natural selection chooses between competing strategies. We study the well-mixed population, evolutionary graph theory, games in phenotype space and evolutionary set theory. We apply these results to the evolution of cooperation. The mechanism that leads to the evolution of cooperation in these settings could be called ‘spatial selection’: cooperators prevail against defectors by clustering in physical or other spaces.     

Estimating individual fitness in the wild using capture–recapture data

The concept of Darwinian fitness is central in evolutionary ecology, and its estimation has motivated the development of several approaches. However, measuring individual fitness remains challenging in empirical case studies in the wild. Measuring fitness requires a continuous monitoring of individuals from birth to death, which is very difficult to get in part because individuals may or may not be controlled at each reproductive event and recovered at death. Imperfect detection hampers keeping track of mortality and reproductive events over the whole lifetime of individuals. We propose a new statistical approach to estimate individual fitness while accounting for imperfect detection. Based on hidden process modelling of longitudinal data on marked animals, we show that standard metrics to quantify fitness, namely lifetime reproductive success, individual growth rate and lifetime individual contribution to population growth, can be extended to cope with imperfect detection inherent to most monitoring programs in the wild. We illustrate our approach using data collected on individual roe deer in an intensively monitored population.     

Extinction times and moment closure in the stochastic logistic process

We investigate the statistics of extinction times for an isolated population, with an initially modest number M of individuals, whose dynamics are controlled by a stochastic logistic process (SLP). The coefficient of variation in the extinction time V is found to have a maximum value when the death and birth rates are close in value. For large habitat size K we find that Vmax is of order K1/4 / M1/2, which is much larger than unity so long as M is small compared to K1/2. We also present a study of the SLP using the moment closure approximation (MCA), and discuss the successes and failures of this method. Regarding the former, the MCA yields a steady-state distribution for the population when the death rate is low. Although not correct for the SLP model, the first three moments of this distribution coincide with those calculated exactly for an adjusted SLP in which extinction is forbidden. These exact calculations also pinpoint the breakdown of the MCA as the death rate is increased.     

LIKELIHOOD-BASED INFERENCE IN ISOLATION-BY-DISTANCE MODELS USING THE SPATIAL DISTRIBUTION OF LOW-FREQUENCY ALLELES

Estimating dispersal distances from population genetic data provides an important alternative to logistically taxing methods for directly observing dispersal. Although methods for estimating dispersal rates between a modest number of discrete demes are well developed, methods of inference applicable to "isolation-by-distance" models are much less established. Here, we present a method for estimating ρσ², the product of population density (ρ) and the variance of the dispersal displacement distribution (σ²). The method is based on the assumption that low-frequency alleles are identical by descent. Hence, the extent of geographic clustering of such alleles, relative to their frequency in the population, provides information about ρσ². We show that a novel likelihood-based method can infer this composite parameter with a modest bias in a lattice model of isolation-by-distance. For calculating the likelihood, we use an importance sampling approach to average over the unobserved intraallelic genealogies, where the intraallelic genealogies are modeled as a pure birth process. The approach also leads to a likelihood-ratio test of isotropy of dispersal, that is, whether dispersal distances on two axes are different. We test the performance of our methods using simulations of new mutations in a lattice model and illustrate its use with a dataset from Arabidopsis thaliana .     

High-throughput quantification of microbial birth and death dynamics using fluorescence microscopy

Background: Microbes live in dynamic environments where nutrient concentrations fluctuate. Quantifying fitness in terms of birth rate and death rate in a wide range of environments is critical for understanding microbial evolution and ecology. Methods: Here, using high-throughput time-lapse microscopy, we have quantified how Saccharomyces cerevisiae mutants incapable of synthesizing an essential metabolite (auxotrophs) grow or die in various concentrations of the required metabolite. We establish that cells normally expressing fluorescent proteins lose fluorescence upon death and that the total fluorescence in an imaging frame is proportional to the number of live cells even when cells form multiple layers. We validate our microscopy approach of measuring birth and death rates using flow cytometry, cell counting, and chemostat culturing. Results: For lysine-requiring cells, very low concentrations of lysine are not detectably consumed and do not support cell birth, but delay the onset of death phase and reduce the death rate compared to no lysine. In contrast, in low hypoxanthine, hypoxanthine-requiring cells can produce new cells, yet also die faster than in the absence of hypoxanthine. For both strains, birth rates under various metabolite concentrations are better described by the sigmoidal-shaped Moser model than the well-known Monod model, while death rates can vary with metabolite concentration and time. Conclusions: Our work reveals how time-lapse microscopy can be used to discover non-intuitive microbial birth and death dynamics and to quantify growth rates in many environments.     

TOWARDS A GENERAL THEORY OF GROUP SELECTION

The longstanding debate about the importance of group (multilevel) selection suffers from a lack of formal models that describe explicit selection events at multiple levels. Here, we describe a general class of models for two‐level evolutionary processes which include birth and death events at both levels. The models incorporate the state‐dependent rates at which these events occur. The models come in two closely related forms: (1) a continuous‐time Markov chain, and (2) a partial differential equation (PDE) derived from (1) by taking a limit. We argue that the mathematical structure of this PDE is the same for all models of two‐level population processes, regardless of the kinds of events featured in the model. The mathematical structure of the PDE allows for a simple and unambiguous way to distinguish between individual‐ and group‐level events in any two‐level population model. This distinction, in turn, suggests a new and intuitively appealing way to define group selection in terms of the effects of group‐level events. We illustrate our theory of group selection by applying it to models of the evolution of cooperation and the evolution of simple multicellular organisms, and then demonstrate that this kind of group selection is not mathematically equivalent to individual‐level (kin) selection.     

CAFE: a computational tool for the study of gene family evolution

SUMMARY: We present CAFE (Computational Analysis of gene Family Evolution), a tool for the statistical analysis of the evolution of the size of gene families. It uses a stochastic birth and death process to model the evolution of gene family sizes over a phylogeny. For a specified phylogenetic tree, and given the gene family sizes in the extant species, CAFE can estimate the global birth and death rate of gene families, infer the most likely gene family size at all internal nodes, identify gene families that have accelerated rates of gain and loss (quantified by a p-value) and identify which branches cause the p-value to be small for significant families. AVAILABILITY: Software is available from http://www.bio.indiana.edu/~hahnlab/Software.html     

Lévy flights in evolutionary ecology

We are interested in modeling Darwinian evolution resulting from the interplay of phenotypic variation and natural selection through ecological interactions. The population is modeled as a stochastic point process whose generator captures the probabilistic dynamics over continuous time of birth, mutation, and death, as influenced by each individual's trait values, and interactions between individuals. An offspring usually inherits the trait values of her progenitor, except when a random mutation causes the offspring to take an instantaneous mutation step at birth to new trait values. In the case we are interested in, the probability distribution of mutations has a heavy tail and belongs to the domain of attraction of a stable law and the corresponding diffusion admits jumps. This could be seen as an alternative to Gould and Eldredge's model of evolutionary punctuated equilibria. We investigate the large-population limit with allometric demographies: larger populations made up of smaller individuals which reproduce and die faster, as is typical for micro-organisms. We show that depending on the allometry coefficient the limit behavior of the population process can be approximated by nonlinear Lévy flights of different nature: either deterministic, in the form of non-local fractional reaction-diffusion equations, or stochastic, as nonlinear super-processes with the underlying reaction and a fractional diffusion operator. These approximation results demonstrate the existence of such non-trivial fractional objects; their uniqueness is also proved.     

A stochastic model of tumor response to fractionated radiation: limit theorems and rate of convergence

The iterated birth and death Markov process is defined as an n-fold iteration of a birth and death Markov process describing kinetics of certain population combined with random killing of individuals in the population at moments tau 1,...,tau n with given survival probabilities s1,...,sn. A long-standing problem of computing the distribution of the number of clonogenic tumor cells surviving an arbitrary fractionated radiation schedule is solved within the framework of iterated birth and death Markov process. It is shown that, for any initial population size iota, the distribution of the size N of the population at moment t > or = tau n is generalized negative binomial, and an explicit computationally feasible formula for the latter is found. It is shown that if i --> infinity and sn --> 0 so that the product iota s1...sn tends to a finite positive limit, the distribution of random variable N converges to a probability distribution, which for t = tau n turns out to be Poisson. In the latter case, an estimate of the rate of convergence in the total variation metric similar to the classical Law of Rare Events is obtained.     

Extinction of populations by random influences

A unifying approach described by a random birth and death process which includes both environmental and demographic noise is introduced. It is shown that both of these noise sources play an essential role in extinction processes in general. The probability distribution of the lifetime of a population is determined and its dependence on the parameters of the model is discussed. Finally a population divided into subpopulations is modeled. The lifetime of this ensemble of subpopulations is compared to the lifetime of one large population.     

Season of birth affects short- and long-term survival

Recent research findings have highlighted the importance of early life conditions as risk factors for adult diseases and therefore determinants of subsequent survival. Given that individuals born during different seasons in seasonal environments experience different early-developmental conditions, an analysis of the effects of the season of birth on survival is considered an effective approach in clarifying the influence of early life conditions on survival in later life. In the present study, we analyzed the long-term effects of early developmental conditions in a historical population in which both nutritional levels and the burden of infectious diseases showed a seasonal variation. Using a semi-computerized linkage process, we were able to match birth and death data for 4,646 individuals born between 1634 and 1870 in the village of Es Mercadal (Minorca Island, Spain). To determine ecological differences associated with the season of birth, we first evaluated the association between season of birth and early life survival. This analysis helped us to determine seasonal variations in early life conditions such as infectious burden and nutritional levels. The season of birth had a significant effect on long-term survival in the birth cohort 1800-1870: summer births had a lower risk of death after age 15. We explain these results in terms of lower susceptibility to degenerative diseases in adult years due to superior in utero nutrition for summer births. These findings support the fetal origin hypothesis which states that the early life environment plays a key role in shaping the subsequent phenotype and risk of adult disease.     

A stochastic model of the dynamics of host-pathogen systems with mutation

In this paper a stochastic model of the dynamics of host-pathogen systems with mutation is constructed. In previous works deterministic models of host-pathogen systems with no mutation were considered. The evolution of the pathogen population in any generation of the host is formulated as a multidimensional birth and death process, while the evolution of genotypic frequencies in successive generations of the host is described by a solution of a nonlinear vector difference equation. A general solution of the differential equations of the multidimensional birth and death process is presented and expressions for the stationary distribution, whenever it exists, and the mean time to extinction, when absorbing states are present, are derived. Some answers to questions raised in the discussion of a previous paper (Mode, 1962) are also contained in this paper. The research reported in this paper was supported by the United States Atomic Energy Comission, Division of Biology and Medicine Project AT(45-1)-1729.     

Evolution of senescence: Influence of age-dependent death rates on the natural increase of a hypothetical population

A hypothetical population is characterized by functions of age which describe its longevity and its maternity rate. Solution of the renewal equation for the birth rate of the population yields a characteristic equation which, in contradiction to the results of previous studies, may have more than one real root. The largest real root of the characteristic equation is the rate of natural increase, r, of the population and is used as a measure of its selective advantage.The maternity rate is represented by a rising or falling exponential function of age. Longevity is represented by a series each term of which has the form of a gamma distribution function. As the number of terms increases, the mean longevity remains constant, but the function becomes progressively more rectangular in shape; the early death rate declines, while the death rate in old age increases. Unless the reproductive fraction is small, each such decrease in the youthful death rate more than compensates for the corresponding increase in old age and causes an increment in r which is interpreted as a step toward the evolution of senescence. Although the degree of change in r attendant upon a change in the age-dependency of the death rate is related to the initial value of the maternity function, it is not influenced by the age dependency of the maternity function.