Compartmental models with multiple sources of stochastic variability: The one-compartment,time invariant hazard rate case

Previous stochastic compartmental models have introduced the primary source of stochasticity through either a probabilistic transfer mechanism or a random rate coefficient. This paper combines these primary sources into a unified stochastic compartmental model. Twelve different stochastic models are produced by combining various sources of stochasticity and the mean value and the covariance for each of the twelve models is derived. The covariance of each model has a different form whereby the individual sources of stochasticity are identificable from data. The various stochastic models are illustrated for certain specified distributions of the rate coefficient and of the initial count. Several properties of the models are derived and discussed. Among these is the fact that the expected count of a model with a random rate coefficient will always exceed the expected count of a model with a fixed coefficient evaluated at the mean rate. A general modeling strategy for the onecompartment, time invariant hazard rate is also proposed.     

Distance decay of similarity,effects of environmental noise and ecological heterogeneity among species in the spatio‐temporal dynamics of a dispersal‐limited community

The joint spatial and temporal fluctuations in community structure may be due to dispersal, variation in environmental conditions, ecological heterogeneity among species and demographic stochasticity. These factors are not mutually exclusive, and their relative contribution towards shaping species abundance distributions and in causing species fluctuations have been hard to disentangle. To better understand community dynamics when the exchange of individuals between localities is very low, we studied the dynamics of the freshwater zooplankton communities in 17 lakes located in independent catchment areas, sampled at end of summer from 2002 to 2008 in Norway. We analysed the joint spatial and temporal fluctuations in the community structure by fitting the two‐dimensional Poisson lognormal model under a two‐stage sampling scheme. We partitioned the variance of the distribution of log abundance for a random species at a random time and location into components of demographic stochasticity, ecological heterogeneity among species, and independent environmental noise components for the different species. Non‐neutral mechanisms such as ecological heterogeneity among species (20%) and spatiotemporal variation in the environment (75%) explained the majority of the variance in log abundances. Overdispersion relative to Poisson sampling and demographic stochasticity had a small contribution to the variance (5%). Among a set of environmental variables, lake acidity was the environmental variable that was most strongly related to decay of community similarity in space and time.     

Stochasticity or the fatal ‘imperfection’ of cloning

The concept of clone is analysed with the aim of exploring the limits to which a phenotype can be said to be determined geneticaly. First of all, mutations that result from the replication, topological manipulation or lesion of DNA introduce a source of heritable variation in an otherwise identical genetic background. But more important, stochastic effects in many biological processes may superimpose a phenotypic variation which is not encoded in the genome. The source of stochasticity ranges from the random selection of alleles or whole chromosomes to be expressed in small cell populations, to fluctuations in processes such as gene expression, due to limiting amounts of the players involved. The picture emerging is that the term clone is a statistical over-simplification representing a series of individuals having essentially the same genome but capable of exhibiting wide phenotypic variation. Finally, to what extent fluctuations in biological processes, usually thought of as noise, are in fact signal is also discussed.     

On environmental variability and limits to similarity

Different model approaches to the problem of environmental variability and limits to similarity are reviewed. It is shown that the assumption of a strictly flat resource spectrum leads to singular results. Hence, the result of May & MacArthur that a qualitative difference exists between a constant and a weakly varying environment is not generally valid. For small variations in the environment, several of the models reduce to a common form and the limits to similarity depend only weakly upon the magnitude of the environmental fluctuations.     

Data-based stochastic modeling reveals sources of activity bursts in single-cell TGF-β signaling

Cells sense their surrounding by employing intracellular signaling pathways that transmit hormonal signals from the cell membrane to the nucleus. TGF-β/SMAD signaling encodes various cell fates, controls tissue homeostasis and is deregulated in diseases such as cancer. The pathway shows strong heterogeneity at the single-cell level, but quantitative insights into mechanisms underlying fluctuations at various time scales are still missing, partly due to inefficiency in the calibration of stochastic models that mechanistically describe signaling processes. In this work we analyze single-cell TGF-β/SMAD signaling and show that it exhibits temporal stochastic bursts which are dose-dependent and whose number and magnitude correlate with cell migration. We propose a stochastic modeling approach to mechanistically describe these pathway fluctuations with high computational efficiency. Employing high-order numerical integration and fitting to burst statistics we enable efficient quantitative parameter estimation and discriminate models that assume noise in different reactions at the receptor level. This modeling approach suggests that stochasticity in the internalization of TGF-β receptors into endosomes plays a key role in the observed temporal bursting. Further, the model predicts the single-cell dynamics of TGF-β/SMAD signaling in untested conditions, e.g., successfully reflects memory effects of signaling noise and cellular sensitivity towards repeated stimulation. Taken together, our computational framework based on burst analysis, noise modeling and path computation scheme is a suitable tool for the data-based modeling of complex signaling pathways, capable of identifying the source of temporal noise.     

Host–parasitoid spatial models: the interplay of demographic stochasticity and dynamics

Host–parasitoid metapopulation models have typically been deterministic models formulated with population numbers as a continuous variable. Spatial heterogeneity in local population abundance is a typical (and often essential) feature of these models and means that, even when average population density is high, some patches have small population sizes. In addition, large temporal population fluctuations are characteristic of many of these models, and this also results in periodically small local population sizes. Whenever population abundances are small, demographic stochasticity can become important in several ways. To investigate this problem, we have reformulated a deterministic, host–parasitoid metapopulation as an integer-based model in which encounters between hosts and parasitoids, and the fecundity of individuals are modelled as stochastic processes. This has a number of important consequences: (1) stochastic fluctuations at small population sizes tend to be amplified by the dynamics to cause massive population variability, i.e. the demographic stochasticity has a destabilizing effect; (2) the spatial patterns of local abundance observed in the deterministic counterpart are largely maintained (although the area of ''spatial chaos'' is extended); (3) at small population sizes, dispersal by discrete individuals leads to a smaller fraction of new patches being colonized, so that parasitoids with small dispersal rates have a greater tendency for extinction and higher dispersal rates have a larger competitive advantage; and (4) competing parasitoids that could coexist in the deterministic model due to spatial segregation cannot now coexist for any combination of parameters.     

Stochastic modelling of environmental variation for biological populations

We examine stochastic effects, in particular environmental variability, in population models of biological systems. Some simple models of environmental stochasticity are suggested, and we demonstrate a number of analytic approximations and simulation-based approaches that can usefully be applied to them. Initially, these techniques, including moment-closure approximations and local linearization, are explored in the context of a simple and relatively tractable process. Our presentation seeks to introduce these techniques to a broad-based audience of applied modellers. Therefore, as a test case, we study a natural stochastic formulation of a non-linear deterministic model for nematode infections in ruminants, proposed by Roberts and Grenfell (1991). This system is particularly suitable for our purposes, since it captures the essence of more complicated formulations of parasite demography and herd immunity found in the literature. We explore two modes of behaviour. In the endemic regime the stochastic dynamic fluctuates widely around the non-zero fixed points of the deterministic model. Enhancement of these fluctuations in the presence of environmental stochasticity can lead to extinction events. Using a simple model of environmental fluctuations we show that the magnitude of this system response reflects not only the variance of environmental noise, but also its autocorrelation structure. In the managed regime host-replacement is modelled via periodic perturbation of the population variables. In the absence of environmental variation stochastic effects are negligible, and we examine the system response to a realistic environmental perturbation based on the effect of micro-climatic fluctuations on the contact rate. The resultant stochastic effects and the relevance of analytic approximations based on simple models of environmental stochasticity are discussed.     

种群统计随机性和环境随机性对种群绝灭的影响

影响种群绝灭的随机干扰可分为种群统计随机性、环境随机性和随机灾害三大类。在相对稳定的环境条件下和相对较短的时间内,以前两类随机干扰对种群绝灭的影响为生态学家关注的焦点。但是,由于自然种群动态及其影响因子的复杂特征,进一步深入研究随机干扰对种群绝灭的作用在理论上和实践上都必须发展新的技术手段。本文回顾了种群统计随机性与环境随机性的概念起源与发展,系统阐述了其分析方法。归纳了两类随机性在种群绝灭研究中的应用范围、作用方式和特点的异同和区别方法。各类随机作用与种群动态之间关系的理论研究与对种群绝灭机理的实践研究紧密相关。根据理论模型模拟和自然种群实际分析两方面的研究现状,作者提出了进一步深入研究随机作用与种群非线性动态方法的策略。指出了随机干扰影响种群绝灭过程的研究的方向：更多的研究将从单纯的定性分析随机干扰对种群动力学简单性质的作用,转向结合特定的种群非线性动态特征和各类随机力作用特点具体分析绝灭极端动态的成因,以期做出精确的预测。     

种群生存力分析研究进展和趋势

种群生存力分析（PVA）是正在迅速发展的新方法,已成为保护生物学研究的热点。它主要研究随机干扰对小种群绝灭的影响,其目的是制定最小可存活种群（MVP）,把绝灭减少到可接受的水平。随机干扰可分四类;统计随机性,环境随机性,自然灾害和遗传随机性。确定MVP的方法有三种：理论模型,模拟模型,模拟模型和岛屿生物地理学方法。理论模型主要研究理想或特定条件下随机因素对种群的影响;模拟模型是利用计算机模拟种群绝灭过程;岛屿生物地理学方法主要分析岛屿物种的分布和存活,证实分析模型和模拟模型。已有大量的文献研究统计随机性,环境随机性和自然灾害的行为特征,但遗传因素与种群生存力之间的关系还不清楚。建立包括四种随机性的综合性模型,广泛地检验PVA模型,系统地研制目标种的遗传和生态特性以及MVP的实际应用是PVA的发展趋势。     

From the stochasticity of molecular processes to the variability of synaptic transmission

The variability of the postsynaptic response following a single action potential arises from two sources: the neurotransmitter release is probabilistic, and the postsynaptic response to neurotransmitter release has variable timing and amplitude. At individual synapses, the number of molecules of a given type that are involved in these processes is small enough that the stochastic (random) properties of molecular events cannot be neglected. How the stochasticity of molecular processes contributes to the variability of synaptic transmission, its sensitivity and its robustness to molecular fluctuations has important implications for our understanding of the mechanistic basis of synaptic transmission and of synaptic plasticity.     

Does increased stochasticity speed up extinction?

We consider the impact of increased stochastic fluctuations on the extinction date of an unstructured population subject to either environmental or demographical stochasticity (or both). By modelling the population density as a general linear diffusion, we state a set of typically satisfied conditions under which the decreasing minimal r-excessive mapping (and, therefore, the moment generating function) of the considered diffusion process is convex and, consequently, under which the impact of increased stochastic fluctuations on the expected date at which the density becomes arbitrarily small is unambiguously negative. In other words, we establish a set of sufficient conditions under which increased stochasticity speeds up the extinction process independently of whether stochasticity is environmental or demographic. In this way, we are able to confirm that increased stochasticity is detrimental for population growth. Received: 25 April 2000 / Revised version: 18 April 2001 / Published online: 12 October 2001     

Adaptive and nonadaptive plasticity in changing environments: Implications for sexual species with different life history strategies

Populations adapt to novel environmental conditions by genetic changes or phenotypic plasticity. Plastic responses are generally faster and can buffer fitness losses under variable conditions. Plasticity is typically modeled as random noise and linear reaction norms that assume simple one‐to‐one genotype–phenotype maps and no limits to the phenotypic response. Most studies on plasticity have focused on its effect on population viability. However, it is not clear, whether the advantage of plasticity depends solely on environmental fluctuations or also on the genetic and demographic properties (life histories) of populations. Here we present an individual‐based model and study the relative importance of adaptive and nonadaptive plasticity for populations of sexual species with different life histories experiencing directional stochastic climate change. Environmental fluctuations were simulated using differentially autocorrelated climatic stochasticity or noise color, and scenarios of directional climate change. Nonadaptive plasticity was simulated as a random environmental effect on trait development, while adaptive plasticity as a linear, saturating, or sinusoidal reaction norm. The last two imposed limits to the plastic response and emphasized flexible interactions of the genotype with the environment. Interestingly, this assumption led to (a) smaller phenotypic than genotypic variance in the population (many‐to‐one genotype–phenotype map) and the coexistence of polymorphisms, and (b) the maintenance of higher genetic variation—compared to linear reaction norms and genetic determinism—even when the population was exposed to a constant environment for several generations. Limits to plasticity led to genetic accommodation, when costs were negligible, and to the appearance of cryptic variation when limits were exceeded. We found that adaptive plasticity promoted population persistence under red environmental noise and was particularly important for life histories with low fecundity. Populations producing more offspring could cope with environmental fluctuations solely by genetic changes or random plasticity, unless environmental change was too fast.     

Stochasticity in Protein Levels Drives Colinearity of Gene Order in Metabolic Operons of Escherichia coli

In bacterial genomes, gene order is not random. This is most evident when looking at operons, these often encoding enzymes involved in the same metabolic pathway or proteins from the same complex. Is gene order within operons nonrandom, however, and if so why? We examine this issue using metabolic operons as a case study. Using the metabolic network of Escherichia coli, we define the temporal order of reactions. We find a pronounced trend for genes to appear in operons in the same order as they are needed in metabolism (colinearity). This is paradoxical as, at steady state, enzymes abundance should be independent of order within the operon. We consider three extensions of the steady-state model that could potentially account for colinearity: (1) increased productivity associated with higher expression levels of the most 5′ genes, (2) a faster metabolic processing immediately after up-regulation, and (3) metabolic stalling owing to stochastic protein loss. We establish the validity of these hypotheses by employing deterministic and stochastic models of enzyme kinetics. The stochastic stalling hypothesis correctly and uniquely predicts that colinearity is more pronounced both for lowly expressed operons and for genes that are not physically adjacent. The alternative models fail to find any support. These results support the view that stochasticity is a pervasive problem to a cell and that gene order evolution can be driven by the selective consequences of fluctuations in protein levels.     

The influence of habitat boundaries on evolutionary branching along environmental gradients

It is well known that habitat boundaries affect ecological dynamics, but their influence on evolutionary dynamics is less well understood. Here, we study the effects of different kinds of boundaries on evolutionary branching in clonal populations along environmental gradients by systematically analyzing individual-based stochastic models in small- and large-range systems, as well as their large-population-size limits through deterministic approximations. Specifically, we examine four prototypical kinds of boundaries: impermeable boundaries at which individuals stop (“stopping”), or from which they continue back into the interior as if bouncing back mechanically (“reflecting”), or that let them abort the dispersal attempt, return to their original position and try a different direction (“reprising”), and semipermeable boundaries that can be crossed without hindrance, but do not allow the crossing individual to return (“absorbing”).We find that boundary conditions shape branching patterns only in small-range systems, where stopping boundaries generate disruptive selection for a wide range of parameters, whereas absorbing boundaries always generate stabilizing selection. Reflecting and reprising boundaries generate disruptive selection at low individual mobilities, and stabilizing selection at high mobilities. To further analyze these findings, we introduce a simple approximation of the invasion fitness in a mobile population, which predicts the observed outcome. The effect of stochasticity on evolutionary outcomes is small even in small populations: stochasticity causes random branch extinctions at steeper slopes and higher mobilities. In large-range systems, frequency-dependent interactions alone induce evolutionary branching for almost all parameters and independent of boundary conditions.     

随机时滞Lotka-Volterra模型数值解的收敛性

通常情况下,随机时滞Lotka-Volterra模型没有解析解,因而数值逼近方法是研究其性质的有效工具.本文根据Euler数值方法,利用鞅不等式和Ito公式讨论了一类随机时滞Lotka-Volterra模型数值解的收敛性,给出了数值解收敛于解析解的条件.最后通过数值算例对数值计算方法进行了验证.     

Durable Resistance to Crop Pathogens: An Epidemiological Framework to Predict Risk under Uncertainty

Increasing the durability of crop resistance to plant pathogens is one of the key goals of virulence management. Despite the recognition of the importance of demographic and environmental stochasticity on the dynamics of an epidemic, their effects on the evolution of the pathogen and durability of resistance has not received attention. We formulated a stochastic epidemiological model, based on the Kramer-Moyal expansion of the Master Equation, to investigate how random fluctuations affect the dynamics of an epidemic and how these effects feed through to the evolution of the pathogen and durability of resistance. We focused on two hypotheses: firstly, a previous deterministic model has suggested that the effect of cropping ratio (the proportion of land area occupied by the resistant crop) on the durability of crop resistance is negligible. Increasing the cropping ratio increases the area of uninfected host, but the resistance is more rapidly broken; these two effects counteract each other. We tested the hypothesis that similar counteracting effects would occur when we take account of demographic stochasticity, but found that the durability does depend on the cropping ratio. Secondly, we tested whether a superimposed external source of stochasticity (for example due to environmental variation or to intermittent fungicide application) interacts with the intrinsic demographic fluctuations and how such interaction affects the durability of resistance. We show that in the pathosystem considered here, in general large stochastic fluctuations in epidemics enhance extinction of the pathogen. This is more likely to occur at large cropping ratios and for particular frequencies of the periodic external perturbation (stochastic resonance). The results suggest possible disease control practises by exploiting the natural sources of stochasticity.     

Stochastic Models of Gene Expression with Delayed Degradation

In many biochemical reactions occurring in living cells, number of various molecules might be low which results in significant stochastic fluctuations. In addition, most reactions are not instantaneous, there exist natural time delays in the evolution of cell states. It is a challenge to develop a systematic and rigorous treatment of stochastic dynamics with time delays and to investigate combined effects of stochasticity and delays in concrete models.We propose a new methodology to deal with time delays in biological systems and apply it to simple models of gene expression with delayed degradation. We show that time delay of protein degradation does not cause oscillations as it was recently argued. It follows from our rigorous analysis that one should look for different mechanisms responsible for oscillations observed in biological experiments.We develop a systematic analytical treatment of stochastic models of time delays. Specifically we take into account that some reactions, for example degradation, are consuming, that is: once molecules start to degrade they cannot be part in other degradation processes.We introduce an auxiliary stochastic process and calculate analytically the variance and the autocorrelation function of the number of protein molecules in stationary states in basic models of delayed protein degradation.     

The evolution of dispersal under demographic stochasticity

Temporal and spatial variations of the environment are important factors favoring the evolution of dispersal. With few exceptions, these variations have been considered to be exclusively fluctuations of habitat quality. However, since the presence of conspecifics forms part of an individual's environment, demographic stochasticity may be a component of this variability as well, in particular when local populations are small. To study this effect, we analyzed the evolution of juvenile dispersal in a metapopulation model in which habitat quality is constant in space and time but occupancy fluctuates because of demographic stochasticity. Our analysis extends previous studies in that it includes competition of resources and competition for space. Also, juvenile dispersal is not given by a fixed probability but is made conditional on the presence of free territories in a patch, whereas individuals born in full patches will always disperse. Using a combination of analytical and numerical approaches, we show that demographic stochasticity in itself may provide enough variability to favor dispersal even from patches that are not fully occupied. However, there is no simple relationship between the evolution of dispersal and various indicators of demographic stochasticity. Selected dispersal depends on all aspects of the life-history profile, including kin selection.