Stochastic Birth Death Research Articles

Replicator equations induced by microscopic processes in nonoverlapping population playing bimatrix games

This paper is concerned with exploring the microscopic basis for the discrete versions of the standard replicator equation and the adjusted replicator equation. To this end, we introduce frequency-dependent selection-as a result of competition fashioned by game-theoretic consideration-into the Wright-Fisher process, a stochastic birth-death process. The process is further considered to be active in a generation-wise nonoverlapping finite population where individuals play a two-strategy bimatrix population game. Subsequently, connections among the corresponding master equation, the Fokker-Planck equation, and the Langevin equation are exploited to arrive at the deterministic discrete replicator maps in the limit of infinite population size.

Dynamics of a birth–death process based on combinatorial innovation

A feature of human creativity is the ability to take a subset of existing items (e.g. objects, ideas, or techniques) and combine them in various ways to give rise to new items, which, in turn, fuel further growth. Occasionally, some of these items may also disappear (extinction). We model this process by a simple stochastic birth–death model, with non-linear combinatorial terms in the growth coefficients to capture the propensity of subsets of items to give rise to new items. In its simplest form, this model involves just two parameters (P, α). This process exhibits a characteristic ‘hockey-stick’ behaviour: a long period of relatively little growth followed by a relatively sudden ‘explosive’ increase. We provide exact expressions for the mean and variance of this time to explosion and compare the results with simulations. We then generalise our results to allow for more general parameter assignments, and consider possible applications to data involving human productivity and creativity.

Species-level phylogeny, fruit evolution and diversification history of Geranium (Geraniaceae)

The cosmopolitan genus Geranium L. (Geraniaceae) consists of c. 350 species distributed in temperate habitats worldwide, with most of its diversity concentrated in the Mediterranean region. Unlike other genera in Geraniaceae, the species of Geranium present contrasting seed discharge syndromes, i.e. the ‘Erodium-type’ (ET), the ‘carpel-projection type’ (CP), the ‘seed-ejection type’ (SE), and the ‘inoperative type’ (IT), which have been used to delimit major groups within the genus. However, phylogenetic relationships within Geranium are unknown and so is the evolution of the different seed discharge mechanisms. Here, we used a calibrated multispecies coalescent approach to infer the species-level phylogeny and divergence times of the genus based on chloroplast (rbcL, trnL-trnF) and nuclear (ITS) DNA sequences. Our sampling represents most of the morphological variation described in the genus. We reconstruct the evolution of the seed discharge mechanism using ancestral state reconstruction (ASR) techniques on the multispecies coalescent tree, and assess the association between fruit type evolution and species diversification using stochastic birth-death and trait-dependent diversification models. Finally, we reconstruct the early biogeographic history of the genus using discrete and continuous biogeographic analyses of species distribution centroids, including fossil evidence and tip dates. Our results show that fruit type is homoplasious and that the classification based on fruit type in Geranium is artificial. The taxonomy and putative apomorphic characters for Geranium are discussed. ASR of the fruit characters suggests that ET may represent the ancestral state in Geranium and from which CP originated twice, IT presumably once, and SE twice. The independent appearance of the SE syndrome is in both cases associated with increases in diversification rates in the genus. The biogeographic analysis centers the origin and early 10Ma diversification of Geranium on the Mediterranean region. The evolution of seed discharge mechanism about 5Ma might have allowed the species of Geranium to increase in geographic range and to ultimately, diversify.

Collective dynamics of stem cell populations

This year marks the 50th anniversary of the publication of a landmark paper in PNAS in which Till et al. proposed a remarkable model of stem cell proliferation (1). Their idea, based on assessment of colony-forming statistics in light of a mathematical model of a stochastic birth–death process, was that individual stem cell dynamics are inherently random. This surprising proposal quickly ignited a heated, and long-running, debate over stochastic and instructive models of stem cell behavior (2). In the intervening half century, many models of stem cell dynamics have been proposed, yet the mechanisms by which stem cell numbers and activity are regulated are still not completely understood. In PNAS, Lei et al. contribute an original idea to the ongoing discussion (3). Drawing on notions from evolutionary theory, they propose a general mathematical framework that views regulation of stem cell population activity as an optimization problem, which achieves best solution when there is cross-talk between genetic and epigenetic feedback mechanisms.

Functional responses and predator–prey models: a critique of ratio dependence

Arditi and Ginzburg (2012) propose ordinary differential equations (ODEs) with ratio-dependent functional responses as the new null model for predation, based on their earlier work on ratio-dependent food chains and a number of functional response measurements. Here, I discuss some of their claims, arguing for a flexible and problem-driven approach to predator–prey modeling. Models to understand population cycles and models to predict the effect of basal enrichment on food chains need not be the same. While ratio-dependent functional responses in ODE models might sometimes be useful as limit cases for food chains, they are not intrinsically more useful than prey-dependent models to understand the effect of a given predator on prey population dynamics—and sometimes less useful, given the small temporal scales considered in many models. “Instantism” is showed to be an invalid criticism when ODEs are interpreted as describing average trajectories of stochastic birth–death processes. Moreover, other modeling frameworks with strong ties to time series statistics, such as stochastic difference equations, should be promoted to improve the feedback loop between field and theoretical research. The main problems of current trophic ecology do not lie in a wrong null model, as ecologists have already several at their disposal. The loose connection of ODE models with empirical data and spatial/temporal scaling up of empirical measurements constitute more serious challenges to our understanding of trophic interactions and their consequences on ecosystem functioning.

Dynamics of a stochastic spatially extended system predicted by comparing deterministic and stochastic attractors of the corresponding birth–death process

Living cells may be considered as biochemical reactors of multiple steady states. Transitions between these states are enabled by noise, or, in spatially extended systems, may occur due to the traveling wave propagation. We analyze a one-dimensional bistable stochastic birth–death process by means of potential and temperature fields. The potential is defined by the deterministic limit of the process, while the temperature field is governed by noise. The stable steady state in which the potential has its global minimum defines the global deterministic attractor. For the stochastic system, in the low noise limit, the stationary probability distribution becomes unimodal, concentrated in one of two stable steady states, defined in this study as the global stochastic attractor. Interestingly, these two attractors may be located in different steady states. This observation suggests that the asymptotic behavior of spatially extended stochastic systems depends on the substrate diffusivity and size of the reactor. We confirmed this hypothesis within kinetic Monte Carlo simulations of a bistable reaction– diffusion model on the hexagonal lattice. In particular, we found that although the kinase–phosphatase system remains inactive in a small domain, the activatory traveling wave may propagate when a larger domain is considered.

Building Development Monitoring in Multitemporal Remotely Sensed Image Pairs with Stochastic Birth-Death Dynamics

In this paper, we introduce a new probabilistic method which integrates building extraction with change detection in remotely sensed image pairs. A global optimization process attempts to find the optimal configuration of buildings, considering the observed data, prior knowledge, and interactions between the neighboring building parts. We present methodological contributions in three key issues: 1) We implement a novel object-change modeling approach based on Multitemporal Marked Point Processes, which simultaneously exploits low-level change information between the time layers and object-level building description to recognize and separate changed and unaltered buildings. 2) To answer the challenges of data heterogeneity in aerial and satellite image repositories, we construct a flexible hierarchical framework which can create various building appearance models from different elementary feature-based modules. 3) To simultaneously ensure the convergence, optimality, and computation complexity constraints raised by the increased data quantity, we adopt the quick Multiple Birth and Death optimization technique for change detection purposes, and propose a novel nonuniform stochastic object birth process which generates relevant objects with higher probability based on low-level image features.

Strategy abundance in [formula omitted] games for arbitrary mutation rates

We study evolutionary game dynamics in a well-mixed populations of finite size, N . A well-mixed population means that any two individuals are equally likely to interact. In particular we consider the average abundances of two strategies, A and B , under mutation and selection. The game dynamical interaction between the two strategies is given by the 2 × 2 payoff matrix ( a c b d ) . It has previously been shown that A is more abundant than B , if a ( N - 2 ) + bN > cN + d ( N - 2 ) . This result has been derived for particular stochastic processes that operate either in the limit of asymptotically small mutation rates or in the limit of weak selection. Here we show that this result holds in fact for a wide class of stochastic birth–death processes for arbitrary mutation rate and for any intensity of selection.

ESTIMATING SPECIATION AND EXTINCTION RATES FROM DIVERSITY DATA AND THE FOSSIL RECORD

Understanding the processes that underlie biodiversity requires insight into the evolutionary history of the taxa involved. Accurate estimation of speciation, extinction, and diversification rates is a prerequisite for gaining this insight. Here, we develop a stochastic birth-death model of speciation and extinction that predicts the probability distribution of both extinct and extant numbers of species in a clade. We present two estimation methods based on this model given data on the number of extinct species (from the fossil record) and extant species (from diversity assessments): a multivariate method of moments approach and a maximum-likelihood approach. We show that, except for some special cases, the two estimation methods produce very similar estimates. This is convenient, because the usually preferred maximum-likelihood approach is much more computationally demanding, so the method of moments can serve as a proxy. Furthermore, we introduce a correction for possible bias that can arise by the mere fact that we will normally only consider extant clades. We find that in some cases the bias correction affects the estimates profoundly. Finally, we show how our model can be extended to incorporate incomplete preservation. Preservation rates can, however, not be reliably estimated on the basis of numbers of extant and extinct species alone.

Stochastic modeling of cellular colonies with quiescence: An application to drug resistance in cancer

Several cancers are thought to be driven by cells with stem cell like properties. An important characteristic of stem cells, which also applies to primitive tumor cells, is the ability to undergo quiescence, where cells can temporarily stop the cell cycle. Cellular quiescence can affect the kinetics of tumor growth, and the susceptibility of the cells to therapy. To study how quiescence affects treatment, we formulate a stochastic birth–death process with quiescence, on a combinatorial cellular mutation network, and consider the pre-treatment (growth) and treatment (decay) regimes. We find that, in the absence of mutations, treatment (if sufficiently strong) will proceed as a biphasic decline with the first (faster) phase driven by the elimination of the cycling cells and the second (slower) phase limited by the process of cell awakening. Other regimes are possible for weaker treatments. We also describe how the process of mutant generation is influenced by quiescence. Interestingly, for single-drug treatments, the probability to have resistance at start of treatment is independent of quiescence. For two or more drugs, the probability to have generated resistant mutants before treatment grows with quiescence. Finally, we study the influence of quiescence on the treatment phase. Starting from a given composition of mutants, the chances of treatment success are not influenced by the presence of quiescence.

Application of a color-shift model with heterogeneous growth to a rat hepatocarcinogenesis experiment

Several hypotheses are established to describe the formation and progression of foci of altered hepatocytes (FAH). A common model of hepatocarcinogenesis is the mutation model (MM), which is based on the assumption that cells have to undergo multiple successive changes on their way from the normal to the malignant stage. This model describes growth and phenotype change of foci on the cellular level and is based on the assumption that single cells change their phenotype through mutation into the next stage and proliferate according to a linear stochastic birth–death process. In contrast, the color-shift model (CSM) was introduced by Kopp-Schneider et al. to describe that whole colonies of altered hepatocytes simultaneously alter their phenotype. In this paper two modifications of the color-shift model are considered which allow the growth rate to vary from focus to focus. All four models are compared with respect to their ability to predict number and radii of foci in a rat hepatocarcinogenesis experiment, in which rats were treated with the carcinogens N-nitrosomorpholine, 2-acetylaminofluoren and Phenobarbital. Maximum likelihood parameter estimates are given, and predicted and empirical FAH size distributions are visualized. The Cramer–von-Mises distance is used as a measure for the discrepancy between empirical and theoretical size distributions.

A scale invariant clustering of genes on human chromosome 7

BackgroundVertebrate genes often appear to cluster within the background of nontranscribed genomic DNA. Here an analysis of the physical distribution of gene structures on human chromosome 7 was performed to confirm the presence of clustering, and to elucidate possible underlying statistical and biological mechanisms.ResultsClustering of genes was confirmed by virtue of a variance of the number of genes per unit physical length that exceeded the respective mean. Further evidence for clustering came from a power function relationship between the variance and mean that possessed an exponent of 1.51. This power function implied that the spatial distribution of genes on chromosome 7 was scale invariant, and that the underlying statistical distribution had a Poisson-gamma (PG) form. A PG distribution for the spatial scattering of genes was validated by stringent comparisons of both the predicted variance to mean power function and its cumulative distribution function to data derived from chromosome 7.ConclusionThe PG distribution was consistent with at least two different biological models: In the microrearrangement model, the number of genes per unit length of chromosome represented the contribution of a random number of smaller chromosomal segments that had originated by random breakage and reconstruction of more primitive chromosomes. Each of these smaller segments would have necessarily contained (on average) a gamma distributed number of genes.In the gene cluster model, genes would be scattered randomly to begin with. Over evolutionary timescales, tandem duplication, mutation, insertion, deletion and rearrangement could act at these gene sites through a stochastic birth death and immigration process to yield a PG distribution.On the basis of the gene position data alone it was not possible to identify the biological model which best explained the observed clustering. However, the underlying PG statistical model implicated neutral evolutionary mechanisms as the basis for this clustering.

Stochastic model of CO oxidation on platinum surfaces and deterministic limit

We present a general model for the oxidation of CO on low-index platinum single crystal surfaces. In order to take fluctuations into account, it is first formulated in terms of stochastic birth–death processes. A corresponding deterministic limit for large particle numbers can be derived rigorously. The dynamical behavior of the reaction kinetics is investigated by means of a numerical bifurcation analysis of the deterministic limit and stochastic simulations. The computed bifurcation diagrams for Pt(110) and Pt(100) are in qualitative and reasonable quantitative agreement with experimental results. For Pt(110), several types of noise-induced oscillations are obtained.

Spatial Structure and Fluctuations in the Contact Process and Related Models

The contact process is used as a simple spatial model in many disciplines, yet because of the buildup of spatial correlations, its dynamics remain difficult to capture analytically. We introduce an empirically based, approximate method of characterizing the spatial correlations with only a single adjustable parameter. This approximation allows us to recast the contact process in terms of a stochastic birth–death process, converting a spatiotemporal problem into a simpler temporal one. We obtain considerably more accurate predictions of equilibrium population than those given by pair approximations, as well as good predictions of population variance and first passage time distributions to a given (low) threshold. A similar approach is applicable to any model with a combination of global and nearest-neighbor interactions.

Population Dynamics and Spatial Scale: Effects of System Size on Population Persistence

This paper explores the limitations of ecological models based on differential equations. We test three different model implementations of the processes represented by the Lotka–Volterra predator–prey equations. We use these models to investigate the system dynamics suppressed by the assumptions contained in differential-equation-based models; specifically, we examine the effects of space and demographic stochasticity. The differential equation implementation of the Lotka–Volterra processes assumes that there are enough individuals in the system that the effects of demographic stochasticity can be ignored and that the interactions between individuals always occur as though the system were continually well mixed or spatially random. The model predicts a neutrally stable equilibrium and infinite persistence, regardless of the size of the system. We use a stochastic birth–death (SBD) model to explore the possible effects of demographic stochasticity on the basic Lotka–Volterra model. The results show that, over a wide range of system sizes, the time to extinction is finite and increases linearly with the total size of the system. We introduce a new type of spatially explicit individual-based model called the Heuristic Asynchronous Discrete Event Simulation or HADES. HADES adds an explicit component of space to the SBD version of our process. We compare the system dynamics of the SBD and HADES models over the same system sizes, using the same demographic parameters, thereby partitioning the effects of demographic stochasticity and space. As system sizes are increased, the dynamics of the HADES model with respect to the SBD model, as measured by time to extinction, move from equivalent to less stable to much more stable. We analyze the effects of space on the system dynamics using animation, statistical point process analysis, and predicted system dynamics (spatial effects on the predation rate). We then show that both the destabilizing and stabilizing effects of space with respect to the SBD dynamics can be accounted for by specific nonrandom spatial patterns. We contrast our results with three commonly invoked mechanisms that affect the stability of predator–prey systems. We comment on the strengths and limitations of differential-equation-based models.

POPULATION DYNAMICS AND SPATIAL SCALE: EFFECTS OF SYSTEM SIZE ON POPULATION PERSISTENCE

This paper explores the limitations of ecological models based on differential equations. We test three different model implementations of the processes represented by the Lotka–Volterra predator–prey equations. We use these models to investigate the system dynamics suppressed by the assumptions contained in differential-equation-based models; specifically, we examine the effects of space and demographic stochasticity. The differential equation implementation of the Lotka–Volterra processes assumes that there are enough individuals in the system that the effects of demographic stochasticity can be ignored and that the interactions between individuals always occur as though the system were continually well mixed or spatially random. The model predicts a neutrally stable equilibrium and infinite persistence, regardless of the size of the system. We use a stochastic birth–death (SBD) model to explore the possible effects of demographic stochasticity on the basic Lotka–Volterra model. The results show that, over a wide range of system sizes, the time to extinction is finite and increases linearly with the total size of the system. We introduce a new type of spatially explicit individual-based model called the Heuristic Asynchronous Discrete Event Simulation or HADES. HADES adds an explicit component of space to the SBD version of our process. We compare the system dynamics of the SBD and HADES models over the same system sizes, using the same demographic parameters, thereby partitioning the effects of demographic stochasticity and space. As system sizes are increased, the dynamics of the HADES model with respect to the SBD model, as measured by time to extinction, move from equivalent to less stable to much more stable. We analyze the effects of space on the system dynamics using animation, statistical point process analysis, and predicted system dynamics (spatial effects on the predation rate). We then show that both the destabilizing and stabilizing effects of space with respect to the SBD dynamics can be accounted for by specific nonrandom spatial patterns. We contrast our results with three commonly invoked mechanisms that affect the stability of predator–prey systems. We comment on the strengths and limitations of differential-equation-based models.

Extinction in Fragmented Habitats Predicted from Stochastic Birth–death Processes with Density Dependence

Habitat loss, the reduction of the habitat area available, is known to greatly reduce resident species' expected time to extinction. This process is widely recognized, if not adequately understood or quantified except in very simple models. However, it is not well understood how the time to extinction will change if the remaining habitat is distributed across a set of smaller, isolated patches, instead of being left in one single, continuous tract. The effect of habitat fragmentation on population persistence under demographic stochasticity has not been resolved. Specifically, it is not known whether a single large population will persist longer than an aggregate set of several smaller populations (with the same total size). Analytical studies of birth–death processes typically report the mean time to extinction for a single population as a function of the maximum population size, but omit higher moments. To estimate the overall persistence time, or the probability of extinction as a function of time, for a set of small populations, the entire distribution of extinction times must be known for a single population of each size. Knowing all the moments of the distribution of extinction times is not adequate, unless one can reconstruct the distribution from them. Here I analyse stochastic birth–death processes with linear density dependence in per capita birth and death rates, and obtain analytical expressions and numerical solutions for the distribution of extinction times in both subdivided and continuous populations. This is a single-species model that deals with demographic stochasticity only, and assumes independence of extinction events in different patches. These assumptions are relaxed elsewhere. Habitat fragmentation, even without any loss of overall area, has a great and detrimental effect on the persistence time of populations across all temporal and spatial scales. The effect is similar across spatial scales, but shifted in time—larger populations take longer to go extinct but the extinction risk relative to that of a smaller or more fragmented population is the same across spatial scales for the available habitat.

Two stage model for carcinogenesis: number and size distributions of premalignant clones in longitudinal studies

The two stage clonal expansion model of carcinogenesis provides a convenient biologically based framework for the description of toxicologic and epidemiologic data on carcinogenesis. Under this model, a cancer cell is generated following the occurrence of two critical mutations in a single stem cell. Initiated cells that have sustained the first mutation undergo a stochastic birth–death process resulting in clonal expansion of the initiated cell population. In this article, we consider the analysis of longitudinal data on the number and size of premalignant clones, formed by clonal expansion of initiated cells. In particular, the joint distribution of the number of premalignant clones observed at different points in time in the same subject is derived. The application of these results in the statistical analysis of longitudinal data on the number and size of premalignant clones observed in initiation-promotion experiments is indicated.

The relation of fluxes and forces to work in nonequilibrium systems

Prior work has shown that an excess work is necessary to displace a chemical or physical system from a stationary state, and this excess work determines the stationary distribution of a stochastic birth–death master equation. We derive the augmentation of this master equation for a one-variable system in the presence of external noise. When this noise is much larger than internal noise, but still small compared to macroscopic averages, then the stationary distribution reduces to a form suggested by Landau and Schlögl, which is the integral of the flux of the deterministic kinetic equation. A similar result was obtained on the basis of an assumed Fokker–Planck equation. Hence, in the presence of external forces exceeding in intensity the internal fluctuations, fluxes are proportional to forces without linearization in concentrations.

Reproductive potential of Gyrodactylus bullatarudis (Monogenea) on guppies (Poecilia reticulata)

SUMMARYTwo simple experiments were undertaken using the viviparous ectoparasite, Gyrodactylus bullatarudis, and laboratory-reared guppies, Poecilia reticulata, whereby detailed records of the number and temporal sequence of all births and the age at death were obtained for flukes on isolated guppies. Gyrodactylus bullatarudis has an average fecundity of 1·68 offspring during its expected life-span of 4·20 days. The instantaneous birth rate is independent of generation but dependent on age. The first offspring is born approximately 1 day after the birth of the parent and subsequent offspring are born at 2–2·5 day intervals. The average instantancous birth rate (under given experimental conditions) is 0.43/parasite/day. The death rate increases exponentially with the age of the fluke and has an average value of 0.24/parasite/day. A simple deterministic model incorporating age structure and an age-dependent death rate was found to provide a good fit to the observed exponential increase in G. bullatarudis numbers. The variability observed in this host–parasite system was found to be largely a function of chance, as shown by Monte Carlo simulations of stochastic birth–death processes.