Stochastic Birth-death Process Research Articles

Fission as a source of variation for group selection.

Without heritable variation natural selection cannot effect evolutionary change. In the case of group selection, there must be variation in the population of groups. Where does this variation come from? One source of variation is from the stochastic birth-death processes that occur within groups. This is where variation between groups comes from in most mathematical models of group selection. Here, we argue that another important source of variation between groups is fission, the (generally random) group-level reproduction where parent groups split into two or more offspring groups. We construct a simple model of the fissioning process with a parameter that controls how much variation is produced among the offspring groups. We then illustrate the effect of that parameter with some examples. In most models of group selection in the literature, no variation is produced during group reproduction events; that is, groups "clone" themselves when they reproduce. Fission is often a more biologically realistic method of group reproduction, and it can significantly increase the efficacy of group selection.

Modelling species invasion using a metapopulation model with variable mortality and stochastic birth-death processes

A species entering a novel landscape must overcome several obstacles that inhibit invasion, which often leads invaders to go extinct while populations are still small and made up of relatively young individuals. Here, we investigate a theoretical model of invader population dynamics to explore the effects of demographic structure on invader success. To do so, we extended the classic Levins metapopulation model by allowing mortality to vary across individual life stages, such that smaller individuals have a higher chance of dying. We also allow size to vary across individuals, using an application of the Gillespie algorithm. Thus, unlike in the classic Levins model, for which successful invasions occur deterministically whenever colonisation exceeds the background mortality rate, invasion success in our model depends on the distribution of sizes found across individuals in the population. Nevertheless, the resulting likelihood of successful invasion can be estimated as a function of just two parameters – the colonisation rate, and a derived index representing the time-averaged mortality rate. This time-averaged mortality can be expressed analytically as a function of individual-level growth rates, the initial size of individuals, and of the impact of disturbances. While demographic stochasticity also contributes to some invasion failures in our model, we find that these effects are rare except when landscapes and initial fraction of occupied sites are both small. Our results demonstrate that invasion success in a complex stochastic model with explicit demographic structure can be predicted using a relatively simple, analytically tractable function. Applications of these results may therefore be particularly useful for studying general patterns of invasion success among sessile organisms with size dependant mortality, such as terrestrial plants.

Inferring density-dependent population dynamics mechanisms through rate disambiguation for logistic birth-death processes.

Density dependence is important in the ecology and evolution of microbial and cancer cells. Typically, we can only measure net growth rates, but the underlying density-dependent mechanisms that give rise to the observed dynamics can manifest in birth processes, death processes, or both. Therefore, we utilize the mean and variance of cell number fluctuations to separately identify birth and death rates from time series that follow stochastic birth-death processes with logistic growth. Our nonparametric method provides a novel perspective on stochastic parameter identifiability, which we validate by analyzing the accuracy in terms of the discretization bin size. We apply our method to the scenario where a homogeneous cell population goes through three stages: (1) grows naturally to its carrying capacity, (2) is treated with a drug that reduces its carrying capacity, and (3) overcomes the drug effect to restore its original carrying capacity. In each stage, we disambiguate whether the dynamics occur through the birth process, death process, or some combination of the two, which contributes to understanding drug resistance mechanisms. In the case of limited sample sizes, we provide an alternative method based on maximum likelihood and solve a constrained nonlinear optimization problem to identify the most likely density dependence parameter for a given cell number time series. Our methods can be applied to other biological systems at different scales to disambiguate density-dependent mechanisms underlying the same net growth rate.

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.

Maximum Likelihood Estimation of Time-Varying Sparsity Level for Dynamic Sparse Signals

In the field of Compressed Sensing, estimation of sparsity level is very essential as the sparsity level determines the minimum number of (i) measurements to be obtained of a sparse signal during acquisition and (ii) iterations to be performed for many of the greedy techniques for the perfect recovery of the sparse signal from the obtained measurements. In this paper, we propose a Maximum Likelihood (ML) estimator to estimate the instantaneous sparsity level during acquisition and an ML sequence (MLS) estimator of sparsity levels during recovery. As the sparsity level varies in time due to the continuous birth of newer supporting components and death of existing supporting components, this paper models the sparsity level variation as a stochastic birth-death process. The real-world applications of the proposed estimators are presented on the compression of aircraft vibration signals and the estimation of wireless channels. The simulation results on real-world and model-generated data demonstrate the performance merits of the proposed estimators compared to other existing methods.

Integrability of stochastic birth-death processes via differential Galois theory

Stochastic birth-death processes are described as continuous-time Markov processes in models of population dynamics. A system of infinite, coupled ordinary differential equations (the so-called master equation) describes the time-dependence of the probability of each system state. Using a generating function, the master equation can be transformed into a partial differential equation. In this contribution we analyze the integrability of two types of stochastic birth-death processes (with polynomial birth and death rates) using standard differential Galois theory. We discuss the integrability of the PDE via a Laplace transform acting over the temporal variable. We show that the PDE is not integrable except for the case in which rates are linear functions of the number of individuals.

Phase transitions in a two-species model for cell segregation and logistic growth

We study a model of cell segregation in a population composed of two cell types. Starting from a model initially proposed in [3], we aim to understand the impact of a cell division process on the system’s segregation abilities. The original model describes a population of spherical cells interacting with their close neighbors by means of a repulsion potential and which centers are subject to Brownian motion. Here, we add a stochastic birth-death process in the agent-based model, that approaches a logistic growth term in the continuum limit. We address the linear stability of the spatially homogeneous steady states of the macroscopic model and obtain a precise criterion for the phase transition, which links the system segregation ability to the model parameters. By comparing the criterion with the one obtained without logistic growth, we show that the system’s segregation ability is the result of a complex interplay between logistic growth, diffusion and mechanical repulsive interactions. Numerical simulations are presented to illustrate the results obtained at the microscopic scale.

Phase Diagram for Logistic Systems under Bounded Stochasticity.

Extinction is the ultimate absorbing state of any stochastic birth-death process; hence, the time to extinction is an important characteristic of any natural population. Here we consider logistic and logisticlike systems under the combined effect of demographic and bounded environmental stochasticity. Three phases are identified: an inactive phase where the mean time to extinction T increases logarithmically with the initial population size, an active phase where T grows exponentially with the carrying capacity N, and a temporal Griffiths phase, with a power-law relationship between T and N. The system supports an exponential phase only when the noise is bounded, in which case the continuum (diffusion) approximation breaks down within the Griffiths phase. This breakdown is associated with a crossover between qualitatively different survival statistics and decline modes. To study the power-law phase we present a new WKB scheme, which is applicable both in the diffusive and in the nondiffusive regime.

Renormalization of radiobiological response functions by energy loss fluctuations and complexities in chromosome aberration induction: deactivation theory for proton therapy from cells to tumor control.

We employ a multi-scale mechanistic approach built upon our recent phenomenological/computational methodologies [R. Abolfath et al., Sci. Rep. 7, 8340 (2017)] to investigate radiation induced cell toxicities and deactivation mechanisms as a function of linear energy transfer in hadron therapy. Our theoretical model consists of a system of Markov chains in microscopic and macroscopic spatio-temporal landscapes, i.e., stochastic birth-death processes of cells in millimeter-scale colonies that incorporates a coarse-grained driving force to account for microscopic radiation induced damage. The coupling, hence the driving force in this process, stems from a nano-meter scale radiation induced DNA damage that incorporates the enzymatic end-joining repair and mis-repair mechanisms. We use this model for global fitting of the high-throughput and high accuracy clonogenic cell-survival data acquired under exposure of the therapeutic scanned proton beams, the experimental design that considers γ-H2AX as the biological endpoint and exhibits maximum observed achievable dose and LET, beyond which the majority of the cells undergo collective biological deactivation processes. An estimate to optimal dose and LET calculated from tumor control probability by extension to ~106 cells per mm-size voxels is presented. We attribute the increase in degree of complexity in chromosome aberration to variabilities in the observed biological responses as the beam linear energy transfer (LET) increases, and verify consistency of the predicted cell death probability with the in vitro cell survival assay of approximately 100 non-small cell lung cancer (NSCLC) cells. The present model provides an interesting interpretation to variabilities in α and β indices via perturbative expansion of the cell survival fraction (SF) in terms of specific and lineal energies, z and y, corresponding to continuous transitions in pair-wise to ternary, quaternary and more complex recombination of broken chromosomes from the entrance to the end of the range of proton beam.

Finite-time and finite-size scalings in the evaluation of large-deviation functions: Analytical study using a birth-death process.

The Giardinà-Kurchan-Peliti algorithm is a numerical procedure that uses population dynamics in order to calculate large deviation functions associated to the distribution of time-averaged observables. To study the numerical errors of this algorithm, we explicitly devise a stochastic birth-death process that describes the time evolution of the population probability. From this formulation, we derive that systematic errors of the algorithm decrease proportionally to the inverse of the population size. Based on this observation, we propose a simple interpolation technique for the better estimation of large deviation functions. The approach we present is detailed explicitly in a two-state model.

Kinetic theory of age-structured stochastic birth-death processes.

Classical age-structured mass-action models such as the McKendrick-von Foerster equation have been extensively studied but are unable to describe stochastic fluctuations or population-size-dependent birth and death rates. Stochastic theories that treat semi-Markov age-dependent processes using, e.g., the Bellman-Harris equation do not resolve a population's age structure and are unable to quantify population-size dependencies. Conversely, current theories that include size-dependent population dynamics (e.g., mathematical models that include carrying capacity such as the logistic equation) cannot be easily extended to take into account age-dependent birth and death rates. In this paper, we present a systematic derivation of a new, fully stochastic kinetic theory for interacting age-structured populations. By defining multiparticle probability density functions, we derive a hierarchy of kinetic equations for the stochastic evolution of an aging population undergoing birth and death. We show that the fully stochastic age-dependent birth-death process precludes factorization of the corresponding probability densities, which then must be solved by using a Bogoliubov--Born--Green--Kirkwood--Yvon-like hierarchy. Explicit solutions are derived in three limits: no birth, no death, and steady state. These are then compared with their corresponding mean-field results. Our results generalize both deterministic models and existing master equation approaches by providing an intuitive and efficient way to simultaneously model age- and population-dependent stochastic dynamics applicable to the study of demography, stem cell dynamics, and disease evolution.

AN EVOLUTIONARY MODEL OF TUMOR CELL KINETICS AND THE EMERGENCE OF MOLECULAR HETEROGENEITY DRIVING GOMPERTZIAN GROWTH.

We describe a cell-molecular based evolutionary mathematical model of tumor development driven by a stochastic Moran birth-death process. The cells in the tumor carry molecular information in the form of a numerical genome which we represent as a four-digit binary string used to differentiate cells into 16 molecular types. The binary string is able to undergo stochastic point mutations that are passed to a daughter cell after each birth event. The value of the binary string determines the cell fitness, with lower fit cells (e.g. 0000) defined as healthy phenotypes, and higher fit cells (e.g. 1111) defined as malignant phenotypes. At each step of the birth-death process, the two phenotypic sub-populations compete in a prisoner's dilemma evolutionary game with the healthy cells playing the role of cooperators, and the cancer cells playing the role of defectors. Fitness, birth-death rates of the cell populations, and overall tumor fitness are defined via the prisoner's dilemma payoff matrix. Mutation parameters include passenger mutations (mutations conferring no fitness advantage) and driver mutations (mutations which increase cell fitness). The model is used to explore key emergent features associated with tumor development, including tumor growth rates as it relates to intratumor molecular heterogeneity. The tumor growth equation states that the growth rate is proportional to the logarithm of cellular diversity/heterogeneity. The Shannon entropy from information theory is used as a quantitative measure of heterogeneity and tumor complexity based on the distribution of the 4-digit binary sequences produced by the cell population. To track the development of heterogeneity from an initial population of healthy cells (0000), we use dynamic phylogenetic trees which show clonal and sub-clonal expansions of cancer cell sub-populations from an initial malignant cell. We show tumor growth rates are not constant throughout tumor development, and are generally much higher in the subclinical range than in later stages of development, which leads to a Gompertzian growth curve. We explain the early exponential growth of the tumor and the later saturation associated with the Gompertzian curve which results from our evolutionary simulations using simple statistical mechanics principles related to the degree of functional coupling of the cell states. We then compare dosing strategies at early stage development, mid-stage (clinical stage), and late stage development of the tumor. If used early during tumor development in the subclinical stage, well before the cancer cell population is selected for growth, therapy is most effective at disrupting key emergent features of tumor development.

Emergence of periodic behaviours from randomness

This paper discusses how periodic behaviours can arise in discrete systems where the underlying dynamics are purely random. We consider non-interacting particles moving randomly on a network of nodes forming a closed loop. The population dynamics describing the number of particles at a node is a stochastic birth-death process, augmented by particles migrating randomly to adjacent nodes. This can result in the emergence of periodic behaviours occurring because of the interaction between the dynamics of the particles and the spatial structure through which they move. The conditions for this requires the network to comprise of three or more nodes and the migration to have a preferred direction. Moreover there are three classes of equilibria for the populations at nodes that depend on the relative values of the migration and birth rates.

Asymptotic properties of maximum likelihood estimators for a generalized Pareto-type distribution

Astola and Danielian [1], using stochastic birth-death process, have proposed a regular four-parameter discrete probability distribution, called generalized Pareto-type model, which is an appealing distribution for modeling phenomena in Bioinformatics. Farbod and Gasparian [5], fitted this distribution to the two sets of real data, and have derived conditions under which a solution for the system of likelihood equations exists and coincides with the maximum likelihood estimators (MLE) for the model unknown parameters. Also, in [5], an accumulation method for approximate computation of the MLE has been considered with simulation studies. In this paper we show that for sufficiently large sample size the system of likelihood equations has a solution, which according to [5], coincides with the MLE of vector-valued parameter for the underlying model. Besides, we establish asymptotic unbiasedness, weak consistency, asymptotic normality, asymptotic efficiency, and convergence of arbitrary moments of the MLE, by verifying the so-called regularity-conditions.

Modeling delay in genetic networks: from delay birth-death processes to delay stochastic differential equations.

Delay is an important and ubiquitous aspect of many biochemical processes. For example, delay plays a central role in the dynamics of genetic regulatory networks as it stems from the sequential assembly of first mRNA and then protein. Genetic regulatory networks are therefore frequently modeled as stochastic birth-death processes with delay. Here, we examine the relationship between delay birth-death processes and their appropriate approximating delay chemical Langevin equations. We prove a quantitative bound on the error between the pathwise realizations of these two processes. Our results hold for both fixed delay and distributed delay. Simulations demonstrate that the delay chemical Langevin approximation is accurate even at moderate system sizes. It captures dynamical features such as the oscillatory behavior in negative feedback circuits, cross-correlations between nodes in a network, and spatial and temporal information in two commonly studied motifs of metastability in biochemical systems. Overall, these results provide a foundation for using delay stochastic differential equations to approximate the dynamics of birth-death processes with delay.

Stochastic desertification

The process of desertification is usually modeled as a first-order transition, where a change of an external parameter (e.g., precipitation) leads to a catastrophic bifurcation followed by an ecological regime shift. However, vegetation elements like shrubs and trees undergo a stochastic birth-death process with an absorbing state; such a process supports a second-order continuous transition with no hysteresis. Here we study a minimal model of a first-order transition with an absorbing state. When the external parameter varies adiabatically the transition is indeed continuous, and we present some empirical evidence that supports this scenario. The front velocity renormalizes to zero at the extinction transition, leaving a finite “quantum” region where domain walls are stable and the desertification takes place via accumulation of local extinctions. A catastrophic regime shift may occur as a dynamical hysteresis, if the pace of environmental variations is too fast.

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.

Wildlife mortality from infrastructure collisions: statistical modeling of count data from carcass surveys

Anthropogenic infrastructure is a mortality source for many vertebrate species. Mortality is often measured using periodic counts of carcasses or remains at infrastructure segments, and bias from carcass removal is estimated via field experiments with wildlife carcasses. We describe a model for combining removal experiment and carcass count data to estimate underlying process parameters using joint likelihood. In the model, the instantaneous number of carcasses present is a stochastic birth-death process with Poisson arrivals (carcass addition) and proportional deaths (removal of carcasses). The approach accommodates modeling heterogeneity in the addition and removal processes using generalized regression. Results of fitting the model to a Greater Sage-Grouse (Centrocercus urophasianus) fence collision data set show that order of magnitude differences in expected carcass counts can be a function of spatial differences in removal and suggest caution for interpretation of many published studies. While the model assumption of negligible detection error may be tenable for some systems, the modeling framework provides a starting point for future state-space versions incorporating detection error.

The balance between adaptation to catalysts and competition radius shapes the total wealth, time variability and inequality

The globalization of modern markets has led to the emergence of competition between producers in ever growing distances. This opens the interesting question in population dynamics of the effect of long-range competition. We here study a model of non-local competition to test the effect of the competition radius on the wealth distribution, using the framework of a stochastic birth-death process, with non-local interactions. We show that this model leads to non-trivial dynamics that can have implications in other domains of physics. Competition is studied in the context of the catalyst induced growth of autocatalytic agents, representing the growth of capital in the presence of investment opportunities. These agents are competing with all other agents in a given radius on growth possibilities. We show that a large scale competition leads to an extreme localization of the agents, where typically a single aggregate of agents can survive within a given competition radius. The survival of these aggregates is determined by the diffusion rates of the agents and the catalysts. For high and low agent diffusion rates, the agent population is always annihilated, while for intermediate diffusion rates, a finite agent population persists. Increasing the catalyst diffusion rate always leads to a decrease in the average agent population density. The extreme localization of the agents leads to the emergence of intermittent fluctuations, when a large aggregate of agents disappear. As the competition radius increases, so does the average agent density and its spatial variance as well as the volatility.