A Reconsideration of Galton's Problem (Using a Two-Sex Population)

The main purposes of this paper are to promote and expound the bisexual Galton–Watson branching process as a relevant model for the consideration of Francis Galton's problem regarding the extinction of surnames of “men of note.” A scheme for adapting the bisexual process to consider Galton's problem is introduced. A necessary and sufficient condition for the certain extinction of a male-induced property in a two-sex species is presented. An approach for calculating the extinction of a male-generated characteristic in the two-sex species is proposed. That approach is then used to find the probability of the extinction of surnames in a bisexual population for Alfred Lotka's data based on an United States Census. Finally, these results are then compared with the classic extinction probabilities (from Lotka) associated with the traditional Galton–Watson branching process using asexual reproduction.     

The Probability of Extinction of Infectious Salmon Anemia Virus in One and Two Patches

Single-type and multitype branching processes have been used to study the dynamics of a variety of stochastic birth–death type phenomena in biology and physics. Their use in epidemiology goes back to Whittle’s study of a susceptible–infected–recovered (SIR) model in the 1950s. In the case of an SIR model, the presence of only one infectious class allows for the use of single-type branching processes. Multitype branching processes allow for multiple infectious classes and have latterly been used to study metapopulation models of disease. In this article, we develop a continuous time Markov chain (CTMC) model of infectious salmon anemia virus in two patches, two CTMC models in one patch and companion multitype branching process (MTBP) models. The CTMC models are related to deterministic models which inform the choice of parameters. The probability of extinction is computed for the CTMC via numerical methods and approximated by the MTBP in the supercritical regime. The stochastic models are treated as toy models, and the parameter choices are made to highlight regions of the parameter space where CTMC and MTBP agree or disagree, without regard to biological significance. Partial extinction events are defined and their relevance discussed. A case is made for calculating the probability of such events, noting that MTBPs are not suitable for making these calculations.     

Optimization and Phenotype Allocation

We study the phenotype allocation problem for the stochastic evolution of a multitype population in a random environment. Our underlying model is a multitype Galton–Watson branching process in a random environment. In the multitype branching model, different types denote different phenotypes of offspring, and offspring distributions denote the allocation strategies. Two possible optimization targets are considered: the long-term growth rate of the population conditioned on nonextinction, and the extinction probability of the lineage. In a simple and biologically motivated case, we derive an explicit formula for the long-term growth rate using the random Perron–Frobenius theorem, and we give an approximation to the extinction probability by a method similar to that developed by Wilkinson. Then we obtain the optimal strategies that maximize the long-term growth rate or minimize the approximate extinction probability, respectively, in a numerical example. It turns out that different optimality criteria can lead to different strategies.     

Population extinction and quasi-stationary behavior in stochastic density-dependent structured models

Density-independent and density-dependent, stochastic and deterministic, discrete-time, structured models are formulated, analysed and numerically simulated. A special case of the deterministic, density-independent, structured model is the well-known Leslie age-structured model. The stochastic, density-independent model is a multitype branching process. A review of linear, density-independent models is given first, then nonlinear, density-dependent models are discussed. In the linear, density-independent structured models, transitions between states are independent of time and state. Population extinction is determined by the dominant eigenvalue λ of the transition matrix. If λ ≤ 1, then extinction occurs with probability one in the stochastic and deterministic models. However, if λ > 1, then the deterministic model has exponential growth, but in the stochastic model there is a positive probability of extinction which depends on the fixed point of the system of probability generating functions. The linear, density-independent, stochastic model is generalized to a nonlinear, density-dependent one. The dependence on state is in terms of a weighted total population size. It is shown for small initial population sizes that the density-dependent, stochastic model can be approximated by the density-independent, stochastic model and thus, the extinction behavior exhibited by the linear model occurs in the nonlinear model. In the deterministic models there is a unique stable equilibrium. Given the population does not go extinct, it is shown that the stochastic model has a quasi-stationary distribution with mean close to the stable equilibrium, provided the population size is sufficiently large. For small values of the population size, complete extinction can be observed in the simulations. However, the persistence time increases rapidly with the population size. This author received partial support by the National Science Foundation grant # DMS-9626417.     

Determinism in a transient assemblage: the roles of dispersal and local competition

Both dispersal and local competitive ability may determine the outcome of competition among species that cannot coexist locally. I develop a spatially implicit model of two-species competition at a small spatial scale. The model predicts the relative fitness of two competitors based on local reproductive rates and regional dispersal rates in the context of the number, size, and extinction probability of habitat patches in the landscape. I test the predictions of this model experimentally using two genotypes of the bacteriophagous soil nematode Caenorhabditis elegans in patchy microcosms. One genotype has higher fecundity while the other is a better disperser. With such a fecundity-dispersal trade-off between competitors, the model predicts that relative fitness will be affected most by local population size when patches do not go extinct and by the number of patches when there is a high probability of patch extinction. The microcosm experiments support the model predictions. Both approaches suggest that competitive dominance in a patchily distributed transient assemblage will depend upon the architecture and predictability of the environment. These mechanisms, operating at a small scale with high spatial admixture, may be embedded in a larger metacommunity process.     

Some properties of a simple stochastic epidemic model of SIR type

We investigate the properties of a simple discrete time stochastic epidemic model. The model is Markovian of the SIR type in which the total population is constant and individuals meet a random number of other individuals at each time step. Individuals remain infectious for R time units, after which they become removed or immune. Individual transition probabilities from susceptible to diseased states are given in terms of the binomial distribution. An expression is given for the probability that any individuals beyond those initially infected become diseased. In the model with a finite recovery time R, simulations reveal large variability in both the total number of infected individuals and in the total duration of the epidemic, even when the variability in number of contacts per day is small. In the case of no recovery, R=infinity, a formal diffusion approximation is obtained for the number infected. The mean for the diffusion process can be approximated by a logistic which is more accurate for larger contact rates or faster developing epidemics. For finite R we then proceed mainly by simulation and investigate in the mean the effects of varying the parameters p (the probability of transmission), R, and the number of contacts per day per individual. A scale invariant property is noted for the size of an outbreak in relation to the total population size. Most notable are the existence of maxima in the duration of an epidemic as a function of R and the extremely large differences in the sizes of outbreaks which can occur for small changes in R. These findings have practical applications in controlling the size and duration of epidemics and hence reducing their human and economic costs.     

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.     

Stochastic models of tumor growth and the probability of elimination by cytotoxic cells

The probability of tumor extinction due to the action of cytotoxic cell populations is investigated by several one dimensional stochastic models of the population growth and elimination processes of a tumor. The several models are made necessary by the nonlinearity of the processes and the different parameter ranges explored. The deterministic form of the model is where γ₀, k′₆ and k ₁ are positive constants. The parameter of most import is which determines the stability of the T = 0 equilibrium. With an initial tumor size of one, a (linear) branching process is used to estimate the extinction probability. However, in the case λ = 0 when the linearization of the deterministic model gives no information (T = 0 is actually unstable) the branching model is unsatisfactory. This makes necessary the utilization of a density-dependent branching process to approximate the population. Through scaling a diffusion limit is reached which enables one to again compute the probability of extinction. For populations away from one a sequence of density-dependent jump Markov processes are approximated by a sequence of diffusion processes. In limiting cases, the estimates of extinction correspond to that computed from the original branching process. Table 1 summarizes the results.     

The spatial scale of population fluctuations and quasi-extinction risk

Using a spatially homogeneous population model with migration (random individual dispersal) and spatially autocorrelated environmental noise, we show how migration and local density regulation affect the spatial scale of fluctuations in the log of population sizes as well as the 1-yr differences in these. The difference between the squares of these two spatial scales of population fluctuations does not depend on the spatial scale of the noise but only on migration rate and strength of local density regulation. We also show how migration, local density regulation, and spatially correlated environmental noise affect the realized population process at a specific location. As the migration increases, the realized local density regulation and the expected population size increase, while the realized environmental noise decreases. This approach also enables us to analyze the dynamics of the total population size within quadrats of different sizes. The risk of local quasi extinction is strongly reduced by increasing quadrat size or migration rate, while an increase in environmental stochasticity or spatial correlation in the environmental noise increases the risk of quasi extinction.     

Population size dependence, competitive coexistence and habitat destruction

1. Spatial dynamics can lead to coexistence of competing species even with strong asymmetric competition under the assumption that the inferior competitor is a better colonizer given equal rates of extinction. Patterns of habitat fragmentation may alter competitive coexistence under this assumption.
2. Numerical models were developed to test for the previously ignored effect of population size on competitive exclusion and on extinction rates for coexistence of competing species. These models neglect spatial arrangement.
3. Cellular automata were developed to test the effect of population size on competitive coexistence of two species, given that the inferior competitor is a better colonizer. The cellular automata in the present study were stochastic in that they were based upon colonization and extinction probabilities rather than deterministic rules.
4. The effect of population size on competitive exclusion at the local scale was found to have little consequence for the coexistence of competitors at the metapopulation (or landscape) scale. In contrast, population size effects on extinction at the local scale led to much reduced landscape scale coexistence compared to simulations not including localized population size effects on extinction, especially in the cellular automata models. Spatially explicit dynamics of the cellular automata vs. deterministic rates of the numerical model resulted in decreased survival of both species. One important finding is that superior competitors that are widespread can become extinct before less common inferior competitors because of limited colonization.
5. These results suggest that population size–extinction relationships may play a large role in competitive coexistence. These results and differences are used in a model structure to help reconcile previous spatially explicit studies which provided apparently different results concerning coexistence of competing species.     

The Interaction between Selection,Demography and Selfing and How It Affects Population Viability

Population extinction due to the accumulation of deleterious mutations has only been considered to occur at small population sizes, large sexual populations being expected to efficiently purge these mutations. However, little is known about how the mutation load generated by segregating mutations affects population size and, eventually, population extinction. We propose a simple analytical model that takes into account both the demographic and genetic evolution of populations, linking population size, density dependence, the mutation load, and self-fertilisation. Analytical predictions were found to be relatively good predictors of population size and probability of population viability when verified using an explicit individual based stochastic model. We show that initially large populations do not always reach mutation-selection balance and can go extinct due to the accumulation of segregating deleterious mutations. Population survival depends not only on the relative fitness and demographic stochasticity, but also on the interaction between the two. When deleterious mutations are recessive, self-fertilisation affects viability non-monotonically and genomic cold-spots could favour the viability of outcrossing populations.     

Extinction, colonization, and species occupancy in tidepool fishes

Despite the increasing sophistication of ecological models with respect to the size and spatial arrangement of habitat, there is relatively little empirical documentation of how species dynamics change as a function of habitat size and the fraction of habitat occupied. In an assemblage of tidepool fishes, I used maximum-likelihood estimation to test whether models which included habitat size provided a better fit to empirical data on extinction and colonization probabilities than models that assumed constant probabilities over all habitats. I found species differences in how extinction and colonization probabilities scaled with habitat size (and hence local population size). However, there was little evidence for a relationship between extinction and colonization probabilities and the fraction of occupied tidepools, as assumed in simple metapopulation models. Instead, colonization and extinction were independent of the fraction of occupied tidepools, favoring a MacArthur-Wilson island-mainland model. When I incorporated declines in extinction probability with tidepool volume in a simple simulation model, I found that predicted occupancy could change greatly, especially when colonization was low. However, the predicted fraction of occupied patches in the simulation model changed little when I incorporated the range of values reported here for extinction and colonization and the rate at which they scale with habitat size. Quantifying extinction and colonization patterns of natural populations is fundamental to understanding how species are distributed spatially and whether metapopulation models of species occupancy provide explanatory power for field populations. Received: 14 March 1997 / Accepted: 21 September 1997     

A branching process, its application in biology: influence of demographic parameters on the social structure in mammal groups

Branching processes are widely used in biology. This theoretical tool is used in cell dynamics, epidemics and population dynamics. In population dynamics, branching processes are mainly used to access extinction probabilities of populations, groups or families, with the Galton-Watson branching process. Many mammal species live in socially-structured groups, and the smallest units of these groups are lineages (or families) of kin-related individuals. In many primate species, these lineages are matrilines, as females remain in their natal groups most of the time, whereas males generally disperse. Lineage parameters, such as numbers of matrilines, size of each matriline and average degree of relatedness, could strongly influence the genetic composition of groups. Evidence indicates that division along matrilines could induce substantial differentiation among fission groups. Here, we develop a novel mathematical model based on the branching process theory describing demographic dynamics of groups. The main result of this model is an explicit analytical expression of the joint distribution of numbers of lineages and sizes of socially-structured groups. We investigated the influence of parameters such as natality and mortality on the outcome of the process, including extinction probability. Finally, we discuss this theoretical result with respect to biological significance.     

Fixation probability for a beneficial allele and a mutant strategy in a linear game under weak selection in a finite island model

The effect of population structure on the probability of fixation of a newly introduced mutant under weak selection is studied using a coalescent approach. Wright's island model in a framework of a finite number of demes is assumed and two selection regimes are considered: a beneficial allele model and a linear game among offspring. A first-order approximation of the fixation probability for a single mutant with respect to the intensity of selection is deduced. The approximation requires the calculation of expected coalescence times, under neutrality, for lineages starting from two or three sampled individuals. The results are obtained in a general setting without assumptions on the number of demes, the deme size or the migration rate, which allows for simultaneous coalescence or migration events in the genealogy of the sampled individuals. Comparisons are made with limit cases as the deme size or the number of demes goes to infinity or the migration rate goes to zero for which a diffusion approximation approach is possible. Conditions for selection to favor a mutant strategy replacing a resident strategy in the context of a linear game in a finite island population are addressed.     

The Markovian binary tree applied to demography

We apply matrix analytic methods and branching processes theory to a comparison of female populations in different countries. We show how the same mathematical model allows us to determine characteristics about individual women, such as the distribution of her lifetime, the time until her first and her last daughter, and the number of daughters, as well as to analyze properties of the whole female family generated by a first woman, such as the extinction probability of the family, the distributions of the time until extinction, of the family size at any given time and of the total progeny.     

A Stochastic Tick-Borne Disease Model: Exploring the Probability of Pathogen Persistence

We formulate and analyse a stochastic epidemic model for the transmission dynamics of a tick-borne disease in a single population using a continuous-time Markov chain approach. The stochastic model is based on an existing deterministic metapopulation tick-borne disease model. We compare the disease dynamics of the deterministic and stochastic models in order to determine the effect of randomness in tick-borne disease dynamics. The probability of disease extinction and that of a major outbreak are computed and approximated using the multitype Galton–Watson branching process and numerical simulations, respectively. Analytical and numerical results show some significant differences in model predictions between the stochastic and deterministic models. In particular, we find that a disease outbreak is more likely if the disease is introduced by infected deer as opposed to infected ticks. These insights demonstrate the importance of host movement in the expansion of tick-borne diseases into new geographic areas.     

A finite locus effect diffusion model for the evolution of a quantitative trait

A diffusion model is constructed for the joint distribution of absolute locus effect sizes and allele frequencies for loci contributing to an additive quantitative trait under selection in a haploid, panmictic population. The model is designed to approximate a discrete model exactly in the limit as both population size and the number of loci affecting the trait tend to infinity. For the case when all loci have the same absolute effect size, formal multiple-timescale asymptotics are used to predict the long-time response of the population trait mean to selection. For the case where loci can take on either of two distinct effect sizes, not necessarily with equal probability, numerical solutions of the system indicate that response to selection of a quantitative trait is insensitive to the variability of the distribution of effect sizes when mutation is negligible.     

Probability of a Disease Outbreak in Stochastic Multipatch Epidemic Models

Environmental heterogeneity, spatial connectivity, and movement of individuals play important roles in the spread of infectious diseases. To account for environmental differences that impact disease transmission, the spatial region is divided into patches according to risk of infection. A system of ordinary differential equations modeling spatial spread of disease among multiple patches is used to formulate two new stochastic models, a continuous-time Markov chain, and a system of stochastic differential equations. An estimate for the probability of disease extinction is computed by approximating the Markov chain model with a multitype branching process. Numerical examples illustrate some differences between the stochastic models and the deterministic model, important for prevention of disease outbreaks that depend on the location of infectious individuals, the risk of infection, and the movement of individuals.     

Establishment Probability in Fluctuating Environments: A Branching Process Model

We study the establishment probability of invaders in stochastically fluctuating environments and the related issue of extinction probability of small populations in such environments, by means of an inhomogeneous branching process model. In the model it is assumed that individuals reproduce asexually during discrete reproduction periods. Within each period, individuals have (independent) Poisson distributed numbers of offspring. The expected numbers of offspring per individual are independently identically distributed over the periods. It is shown that the establishment probability of an invader varies over the reproduction periods according to a stable distribution. We give a method for simulating the establishment probabilities and approximations for the expected establishment probability. Furthermore, we show that, due to the stochasticity of the establishment success over different periods, the expected success of sequential invasions is larger then that of simultaneous invasions and we study the effects of environmental fluctuations on the extinction probability of small populations and metapopulations. The results can easily be generalized to other offspring distributions than the Poisson.