A continuous time version and a generalization of a Markov-recapture model for trapping experiments

Wileyto et al. [E.P. Wileyto, W.J. Ewens, M.A. Mullen, Markov-recapture population estimates: a tool for improving interpretation of trapping experiments, Ecology 75 (1994) 1109] propose a four-state discrete time Markov process, which describes the structure of a marking-capture experiment as a method of population estimation. They propose this method primarily for estimation of closed insect populations. Their method provides a mark-recapture estimate from a single trap observation by allowing subjects to mark themselves. The estimate of the unknown population size is based on the assumption of a closed population and a simple Markov model in which the rates of marking, capture, and recapture are assumed to be equal. Using the one step transition probability matrix of their model, we illustrate how to go from an embedded discrete time Markov process to a continuous time Markov process assuming exponentially distributed holding times. We also compute the transition probabilities after time t for the continuous time case and compare the limiting behavior of the continuous and discrete time processes. Finally, we generalize their model by relaxing the assumption of equal per capita rates for marking, capture, and recapture. Other questions about how their results change when using a continuous time Markov process are examined.     

On a population genetic model by S. Karlin

Summary One of the basic models in population genetics by S. Karlin is the model for a population of haploid individuals with several types and mutation and fixed population size. Mathematically the model is a derived multitype branching process. In order to estimate the approach to homozygosity of the population one has to compute the eigenvalues of the finite transition matrix of the corresponding Markov chain. Karlin (1968) gave explicit expressions for these eigenvalues. The idea of his proof is to transform the transition matrix into triangular form. Starting from the same idea the paper establishes a somewhat simpler and more elaborate proof.     

High host density favors greater virulence: a model of parasite–host dynamics based on multi-type branching processes

We use a multitype continuous time Markov branching process model to describe the dynamics of the spread of parasites of two types that can mutate into each other in a common host population. While most mathematical models for the virulence of infectious diseases focus on the interplay between the dynamics of host populations and the optimal characteristics for the success of the pathogen, our model focuses on how pathogen characteristics may change at the start of an epidemic, before the density of susceptible hosts decline. We envisage animal husbandry situations where hosts are at very high density and epidemics are curtailed before host densities are much reduced. The use of three pathogen characteristics: lethality, transmissibility and mutability allows us to investigate the interplay of these in relation to host density. We provide some numerical illustrations and discuss the effects of the size of the enclosure containing the host population on the encounter rate in our model that plays the key role in determining what pathogen type will eventually prevail. We also present a multistage extension of the model to situations where there are several populations and parasites can be transmitted from one of them to another. We conclude that animal husbandry situations with high stock densities will lead to very rapid increases in virulence, where virulent strains are either more transmissible or favoured by mutation. Further the process is affected by the nature of the farm enclosures.     

Population processes under the influence of disasters occurring independently of population size

Markov branching processes and in particular birth-and-death processes are considered under the influence of disasters that arrive independently of the present population size. For these processes we derive an integral equation involving a shifted and rescaled argument. The main emphasis, however, is on the (random) probability of extinction. Its distribution density satisfies an equation which can be solved numerically at least up to a multiplicative constant. In an example it is also found by simulation.     

Evolutionary Branching in a Finite Population: Deterministic Branching vs. Stochastic Branching

Adaptive dynamics formalism demonstrates that, in a constant environment, a continuous trait may first converge to a singular point followed by spontaneous transition from a unimodal trait distribution into a bimodal one, which is called “evolutionary branching.” Most previous analyses of evolutionary branching have been conducted in an infinitely large population. Here, we study the effect of stochasticity caused by the finiteness of the population size on evolutionary branching. By analyzing the dynamics of trait variance, we obtain the condition for evolutionary branching as the one under which trait variance explodes. Genetic drift reduces the trait variance and causes stochastic fluctuation. In a very small population, evolutionary branching does not occur. In larger populations, evolutionary branching may occur, but it occurs in two different manners: in deterministic branching, branching occurs quickly when the population reaches the singular point, while in stochastic branching, the population stays at singularity for a period before branching out. The conditions for these cases and the mean branching-out times are calculated in terms of population size, mutational effects, and selection intensity and are confirmed by direct computer simulations of the individual-based model.     

A logistic branching process for population genetics

A logistic (regulated population size) branching process population genetic model is presented. It is a modification of both the Wright-Fisher and (unconstrained) branching process models, and shares several properties including the coalescent time and shape, and structure of the coalescent process with those models. An important feature of the model is that population size fluctuation and regulation are intrinsic to the model rather than externally imposed. A consequence of this model is that the fluctuation in population size enhances the prospects for fixation of a beneficial mutation with constant relative viability, which is contrary to a result for the Wright-Fisher model with fluctuating population size. Explanation of this result follows from distinguishing between expected and realized viabilities, in addition to the contrast between absolute and relative viabilities.     

On the Genealogy of a Rare Allele

The gene genealogy is derived for a rare allele that is descended from a mutant ancestor that arose at a fixed time in the past. Following Thompson (1976,Amer. J. Human Genet.28, 442–452), the fractional linear branching process is used as a model of the demography of a rare allele. The model does not require the total population size to be constant or the mutant class to be neutral; so long as individuals in the class are selectively equivalent, the class as a whole may have a selective advantage, or disadvantage, relative to other alleles in the population. An exact result is given for the joint probability distribution of the coalescence times among a sample of alleles descended from the mutant. A method is described for rapidly simulating these coalescence times. The relationship between the genealogical structure of a discrete generation branching process and a continuous generation birth–death process is elucidated. The theory may be applied to the problem of estimating the ages of rare nonrecurrent mutations.     

Coalescent process with fluctuating population size and its effective size

We consider a Wright-Fisher model whose population size is a finite Markov chain. We introduce a sequence of two-dimensional discrete time Markov chains whose components describe the coalescent process and the fluctuation of population size. For the limiting process of the sequence of Markov chains, the relationship of the expectation of coalescence time to the harmonic and the arithmetic means of population sizes is shown, and the Laplace transform of the distribution of coalescence time is calculated. We define the coalescence effective population size (cEPS) by the expectation of coalescence time. We show that cEPS is strictly larger (resp. smaller) than the harmonic (resp. arithmetic) mean. As the population size fluctuates more quickly (resp. slowly), cEPS is closer to the harmonic (resp. arithmetic) mean. For the case of a two-valued Markov chain, we show the explicit expression of cEPS and its dependency on the sample size.     

The effective population size of some age-structured populations

It was shown in a previous paper that if generations are discrete, then the effective population size of a large population can be derived from the theory of multitype branching processes. It turns out to be proportional to the reciprocal of a term that appears in the denominator of expressions for survival probabilities when there is a supercritical positively regular branching process for which the dominant positive eigenvalue of the first moment matrix is slightly larger than 1. If there is an age-structured population with unchanging proportions among sexes and age groups, then the effective population size is shown to be also obtainable from the theory of multitype branching processes. The expression for this parameter has the same form as in the corresponding model for discrete generations, multiplied by an appropriate measure of the average length of a generation. Results are obtained for dioecious random mating populations, populations reproducing partly by selfing, and populations reproducing partly by full-sib mating.     

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.     

Consequences of fluctuating group size for the evolution of cooperation

Studies of cooperation have traditionally focused on discrete games such as the well-known prisoner’s dilemma, in which players choose between two pure strategies: cooperation and defection. Increasingly, however, cooperation is being studied in continuous games that feature a continuum of strategies determining the level of cooperative investment. For the continuous snowdrift game, it has been shown that a gradually evolving monomorphic population may undergo evolutionary branching, resulting in the emergence of a defector strategy that coexists with a cooperator strategy. This phenomenon has been dubbed the ‘tragedy of the commune’. Here we study the effects of fluctuating group size on the tragedy of the commune and derive analytical conditions for evolutionary branching. Our results show that the effects of fluctuating group size on evolutionary dynamics critically depend on the structure of payoff functions. For games with additively separable benefits and costs, fluctuations in group size make evolutionary branching less likely, and sufficiently large fluctuations in group size can always turn an evolutionary branching point into a locally evolutionarily stable strategy. For games with multiplicatively separable benefits and costs, fluctuations in group size can either prevent or induce the tragedy of the commune. For games with general interactions between benefits and costs, we derive a general classification scheme based on second derivatives of the payoff function, to elucidate when fluctuations in group size help or hinder cooperation.     

Statistical properties of estimators for continuous time Markov branching processes with varying and random environments

Asymptotic properties are established for estimators of time dependent intensities in Markov branching processes with varying and random environments. For the varying environment model, the estimators are shown to be uniformly strongly consistent on bounded intervals as the initial population size X₀ → ∞, and, when considered as empirical stochastic processes, to converge weakly to Gaussian processes with independent increments. For random environments, the estimators are shown to be asymptotically normal as t → ∞, where t is the time parameter.     

Linkage disequilibrium in two-locus,finite, random mating models without selection or mutation

Linkage disequilibrium is discussed for three two-locus, finite, random mating models in genetics, two models being Markov chains and the third a diffusion process. The expected value of the square of the disequilibrium function at any time is computed for the Markov chains. There is discussion of the relationships between the models and of the influence of finite population size on correlation between loci. It is suggested that there may be danger in assuming too simple a relationship between population size and degree of disequilibrium.     

Mutation-replication statistics of polymerase chain reactions.

The variability of the products of polymerase chain reactions, due to mutations and to incomplete replications, can have important clinical consequences. Sun (1995) and Weiss and von Haeseler (1995) modeled these errors by a branching process and introduced estimators of the mutation rate and of the efficiency of the reaction based, for example, on the empirical distribution of the mutations of a random sequence. This distribution involves a noncanonical branching Markov chain which, although easy to describe, is not analytically tractable except in the infinite-population limit. These authors for the infinite-target limit, and Wang et al. (2000) for finite targets, solved the infinite-population limit. In this paper, we provide bounds of the difference between the finite-target finite-population case and its finite-target infinite-population approximation. The bounds are explicit functions of the efficiency of the reaction, the mutation rate per site and per cycle, the size of the target, the number of cycles, and the size of the initial population. They concern every moment and, what might be more surprising, the histogram itself of the distributions. The bounds for the moments exhibit a phase transition at the value 1 - 1/N = 3/4 of the mutation rate per site and per cycle, where N = 4 is the number of letters in the encoding alphabet of DNA and RNA. Of course, in biological contexts, the mutation rates are much smaller than 3/4.     

When is sympatric speciation truly adaptive? An analysis of the joint evolution of resource utilization and assortative mating

The plausibility of sympatric speciation has long been debated among evolutionary ecologists. The process necessarily involves two key elements: the stable coexistence of at least two ecologically distinct types and the emergence of reproductive isolation. Recent theoretical studies within the theoretical framework of adaptive dynamics have shown how both these processes can be driven by natural selection. In the standard scenario, a population first evolves to an evolutionary branching point, next, disruptive selection promotes ecological diversification within the population, and, finally, the fitness disadvantage of intermediate types induces a selection pressure for assortative mating behaviour, which leads to reproductive isolation and full speciation. However, the full speciation process has been mostly studied through computer simulations and only analysed in part. Here I present a complete analysis of the whole speciation process by allowing for the simultaneous evolution of the branching ecological trait as well as a continuous trait controlling mating behaviour. I show how the joint evolution can be understood in terms of a gradient landscape, where the plausibility of different evolutionary paths can be evaluated graphically. I find sympatric speciation unlikely for scenarios with a continuous, unimodal, distribution of resources. Rather, ecological settings where the fitness inferiority of intermediate types is preserved during the ecological branching are more likely to provide opportunity for adaptive, sympatric speciation. Such scenarios include speciation due to predator avoidance or specialization on discrete resources. Electronic supplementary material  The online version of this article (doi:) contains supplementary material, which is available to authorized users.     

A Markov Process of Gene Frequency Change in a Geographically Structured Population

A Markov process (chain) of gene frequency change is derived for a geographically-structured model of a population. The population consists of colonies which are connected by migration. Selection operates in each colony independently. It is shown that there exists a stochastic clock that transforms the originally complicated process of gene frequency change to a random walk which is independent of the geographical structure of the population. The time parameter is a local random time that is dependent on the sample path. In fact, if the alleles are selectively neutral, the time parameter is exactly equal to the sum of the average local genetic variation appearing in the population, and otherwise they are approximately equal. The Kolmogorov forward and backward equations of the process are obtained. As a limit of large population size, a diffusion process is derived. The transition probabilities of the Markov chain and of the diffusion process are obtained explicitly. Certain quantities of biological interest are shown to be independent of the population structure. The quantities are the fixation probability of a mutant, the sum of the average local genetic variation and the variation summed over the generations in which the gene frequency in the whole population assumes a specified value.     

Restricted transition probabilities and their applications to some problems in the dynamics of biological populations

In this paper a theory of a class of restricted transition probabilities is developed and applied to a problem in the dynamics of biological populations under the assumption that the underlying stochastic process is a continuous time parameter Markov chain with stationary transition probabilities. The paper is divided into three parts. Part one contains sufficient background from the theory of Markov processes to define restricted transition probabilities in a rigorous manner. In addition, some basic concepts in the theory of stochastic processes are interpreted from the biological point of view. Part two is concerned with the problem of finding representations for restricted transition probabilities. Finally, in part three the theory of restricted transition probabilities is applied to the problem of finding and analyzing some properties of the distribution function of the maximum size attained by the population in a finite time interval for a rather wide class of Markov processes. Some other applications of restricted transition probabilities to other problems in the dynamics of biological populations are also suggested. These applications will be discussed more fully in a companion paper. The research reported in this paper was supported by the United States Atomic Energy Commission, Division of Biology and Medicine Project AT(45-1)-1729.     

Which States Matter? An Application of an Intelligent Discretization Method to Solve a Continuous POMDP in Conservation Biology

When managing populations of threatened species, conservation managers seek to make the best conservation decisions to avoid extinction. Making the best decision is difficult because the true population size and the effects of management are uncertain. Managers must allocate limited resources between actively protecting the species and monitoring. Resources spent on monitoring reduce expenditure on management that could be used to directly improve species persistence. However monitoring may prevent sub-optimal management actions being taken as a result of observation error. Partially observable Markov decision processes (POMDPs) can optimize management for populations with partial detectability, but the solution methods can only be applied when there are few discrete states. We use the Continuous U-Tree (CU-Tree) algorithm to discretely represent a continuous state space by using only the states that are necessary to maintain an optimal management policy. We exploit the compact discretization created by CU-Tree to solve a POMDP on the original continuous state space. We apply our method to a population of sea otters and explore the trade-off between allocating resources to management and monitoring. We show that accurately discovering the population size is less important than management for the long term survival of our otter population.