Stationary Probabilities and Asymptotic Behaviour of a non-Markovian Model with Linear Transition Rule

Explicit and recursive formulae are obtained for the n-step stochastic kernel, and its stationarity and asymptotic behaviour is examined for a special class of non-Markovian models with a linear transition rule occuring in learning theory, adaption theory, control theory, and biological research.     

On the Asymptotic Behaviour of the Population Number

We consider the stochastic model of an asexual population in which the number of couples formed in some generation is random variable depending on the number of individuals in that generation only. The conditions of convergence were obtained almost everywhere and in mean square of the normalized number of individuals in the n-th generation. These results may be considered as the generalization of some known statements about the models constructed on the basis of the branching processses theory.     

Generalized stable population theory

In generalizing stable population theory we give sufficient, then necessary conditions under which a population subject to time dependent vital rates reaches an asymptotic stable exponential equilibrium (as if mortality and fertility were constant). If x ₀(t) is the positive solution of the characteristic equation associated with the linear birth process at time t, then rapid convergence of x ₀(t) to x ₀ and convergence of mortality rates produce a stable exponential equilibrium with asymptotic growth rate x ₀–1. Convergence of x ₀(t) to x ₀ and convergence of mortality rates are necessary. Therefore the two sets of conditions are very close. Various implications of these results are discussed and a conjecture is made in the continuous case.     

Mutation accumulation in selfing populations under fluctuating selection

Selfing species are prone to extinction, possibly because highly selfing populations can suffer from a continuous accumulation of deleterious mutations, a process analogous to Muller's ratchet in asexual populations. However, current theory provides little insight into which types of genes are most likely to accumulate deleterious alleles and what environmental circumstances may accelerate genomic degradation. Here, we investigate temporal changes in the environment that cause fluctuations in the strength of purifying selection. We simulate selfing populations with genomes containing a mixture of loci experiencing constant selection and loci experiencing selection that fluctuates in strength (but not direction). Even when both types of loci experience the same average strength of selection, loci under fluctuating selection contribute disproportionately more to deleterious mutation accumulation. Moreover, the presence of loci experiencing fluctuating selection in the genome increases the deleterious fixation rate at loci under constant selection; under most realistic scenarios, this effect of linked selection can be attributed to a reduction in N_e. Fluctuating selection is particularly injurious when selective environments are strongly autocorrelated over time and when selection is concentrated into rare bouts of strong selection. These results imply that loci under fluctuating selection are likely important drivers of extinction in selfing species.     

Extinction times for closed epidemics: the effects of host spatial structure

Although of practical importance, the relationship between the duration of an epidemic and host spatial structure is poorly understood. Here we use a stochastic metapopulation model for the transmission of infection in a spatially structured host population. There are three qualitatively different regimes for the extinction time, which depend on patch population size, the within‐patch basic reproductive number and the strength of coupling between patches. In the first regime, the extinction time for the metapopulation (i.e. from all patches) is approximately equal to the extinction time for a single patch. In the second regime, the metapopulation extinction time is maximal but also highly variable. In the third regime, the extinction time for the metapopulation (T_E) is given by T_E = a + bn^1/2 where a is the local extinction time (i.e. from last patch), b is the transit time (i.e. the time taken for infection to spread from one patch to another) and n is the total number of patches.     

Inference for an age-dependent,multitype branching-process model of mast cells

We consider an age-dependent, multitype model for the growth of mast cells in culture. After a colony of cells is established by an initiator type, the two possible types of cells are resting and proliferative. Using novel inferential procedures, we estimate the generation-time distribution and the offspring distribution of proliferative cells, and the waiting-time distribution of resting cells.List of Notations B _i cumulative distribution function for the time until branching of a cell of type i - b _i probability density function for the time until branching of a cell of type i - b _i b _i (1–D _i ) - D _i cumulative distribution function for the time until death of a cell of type i - d _i probability density function for the time until death of a cell of type i - probability density function of a gamma distribution - G _i cumulative distribution function for the lifetime of a cell of type i - G _1*2 Convolution of G ₁ and G ₂ - ¯G _i 1–G _i - g _i probability density function for the lifetime of a cell of type i - L _i likelihood of a history of type i - m average number of proliferative daughters produced by dividing cells - M _ij (t) the expected number of type-j cells in a colony at time t if that colony began at time 0 with one type-i cell - M _i+ (t) M _i0 (t) + M _i 1(t) + M _i 2(t) - p _rs probability that a dividing cell produces r proliferative and s resting daughters - t _i times defining colony histories. See IV.2.1 - T ₀ time to division of an initiator cell - T ₁, T ₂ times from birth to division of the two daughters of an initiator cell - T ₍₁₎, T ₍₂₎ order statistics of T ₁ and T ₂ - minimum value of a gamma distribution - scale parameter of a gamma distribution or of an exponential distribution - probability per unit time of death for proliferative and resting cells - _rs expected value of p _rs when there is heterogeneity - shape parameter of a gamma distribution     

Derivation of stochastic partial differential equations for size- and age-structured populations

Stochastic partial differential equations (SPDEs) for size-structured and age- and size-structured populations are derived from basic principles, i.e. from the changes that occur in a small time interval. Discrete stochastic models of size-structured and age-structured populations are constructed, carefully taking into account the inherent randomness in births, deaths, and size changes. As the time interval decreases, the discrete stochastic models lead to systems of Itô stochastic differential equations. As the size and age intervals decrease, SPDEs are derived for size-structured and age- and size-structured populations. Comparisons between numerical solutions of the SPDEs and independently formulated Monte Carlo calculations support the accuracy of the derivations.     

Stochastic modelling of wall stresses in abdominal aortic aneurysms treated by a gene therapy

A stochastic mechanical model using the membrane theory was used to simulate the in vivo mechanical behaviour of abdominal aortic aneurysms (AAAs) in order to compute the wall stresses after stabilisation by gene therapy. For that, both length and diameter of AAAs rats were measured during their expansion. Four groups of animals, control and treated by an endovascular gene therapy during 3 or 28 days were included. The mechanical problem was solved analytically using the geometric parameters and assuming the shape of aneurysms by a ‘parabolic–exponential curve’. When compared to controls, stress variations in the wall of AAAs for treated arteries during 28 days decreased, while they were nearly constant at day 3. The measured geometric parameters of AAAs were then investigated using probability density functions (pdf) attributed to every random variable. Different trials were useful to define a reliable confidence region in which the probability to have a realisation is equal to 99%. The results demonstrated that the error in the estimation of the stresses can be greater than 28% when parameters uncertainties are not considered in the modelling. The relevance of the proposed approach for the study of AAA growth may be studied further and extended to other treatments aimed at stabilisation AAAs, using biotherapies and pharmacological approaches.     

Empirical Modelling of the Population Dynamics of a Small Population of the Threespine Stickleback, <Emphasis Type="Italic">Gasterosteus aculeatus</Emphasis>

Synopsis We estimated the abundance of a small population of threespine stickleback, Gasterosteus aculeatus, by mark-recapture over a 21 year period. Length-frequency analysis showed that the population in October consisted almost entirely of young-of-the-year. The per capita annual rate of increase was inversely related to abundance in October. Time series analysis suggested the presence of a cycle of abundance with a period of about 6 years. There was a significant inverse relationship between abundance in year t and in year t + 3. A simple, empirical, deterministic model based on this inverse relationship and run for 100 years predicted that population abundance showed damped oscillations leading to a stable abundance. When a stochastic component was added to the model, seven of 10 runs included a component with a period of about 6 years. These simulations suggest that the dynamics of this population are driven by an interaction between a deterministic (density-dependent) component and a stochastic component. We compare these results with time series of abundance of threespine stickleback obtained from the Thames Estuary in south-east England and Loch Lomond in Scotland.     

Top predators determine how biodiversity is partitioned across time and space

Natural ecosystems are shaped along two fundamental axes, space and time, but how biodiversity is partitioned along both axes is not well understood. Here, we show that the relationship between temporal and spatial biodiversity patterns can vary predictably according to habitat characteristics. By quantifying seasonal and annual changes in larval dragonfly communities across a natural predation gradient we demonstrate that variation in the identity of top predator species is associated with systematic differences in spatio‐temporal β‐diversity patterns, leading to consistent differences in relative partitioning of biodiversity between time and space across habitats. As the size of top predators increased (from invertebrates to fish) habitats showed lower species turnover across sites and years, but relatively larger seasonal turnover within a site, which ultimately shifted the relative partitioning of biodiversity across time and space. These results extend community assembly theory by identifying common mechanisms that link spatial and temporal patterns of β‐diversity.     

THE PROBABILITY OF FIXATION OF A NEW KARYOTYPE IN A CONTINUOUS POPULATION

We investigate the probability of fixation of a chromosome rearrangement in a subdivided population, concentrating on the limit where migration is so large relative to selection (m ? s) that the population can be thought of as being continuously distributed. We study two demes, and one- and two-dimensional populations. For two demes, the probability of fixation in the limit of high migration approximates that of a population with twice the size of a single deme: migration therefore greatly reduces the fixation probability. However, this behavior does not extend to a large array of demes. Then, the fixation probability depends primarily on neighborhood size (Nb), and may be appreciable even with strong selection and free gene flow (≈exp(-B ≈ Nb√s) in one dimension, ≈exp(-B ≈ Nb) in two dimensions). Our results are close to those for the more tractable case of a polygenic character under disruptive selection.     

A comparison of a deterministic and stochastic model for Hepatitis C with an isolation stage

We formulate a deterministic epidemic model for the spread of Hepatitis C containing an acute, chronic and isolation class and analyse the effects of the isolation class on the transmission dynamics of the disease. We calculate the basic reproduction number R₀ and show that for R₀≤1, the disease-free equilibrium is globally asymptotically stable. In addition, it is shown that for a special case when R₀>1, the endemic equilibrium is locally asymptotically stable. Furthermore, an analogous stochastic epidemic model for Hepatitis C is formulated using a continuous time Markov chain. Numerical simulations are used to estimate the mean, variance and probability distributions of the discrete random variables and these are compared to the steady-state solutions of the deterministic model. Finally, the expected time to disease extinction is estimated for the stochastic model and the impact of isolation on the time to extinction is explored.     

A model for the study of Wolbachia pipientis Hertig (Rickettsiales: Rickettsiaceae)- induced cytoplasmic incompatibility in arrhenotokous haplodiploid populations: consequences for biological control

Wolbachia is an endocytoplasmic bacterium responsible for various reproductive modifications in arthropods. In several species, Wolbachia induces a phenomenon called cytoplasmic incompatibility (CI), whereby crosses between a Wolbachia-infected male and a healthy female are incompatible. In haplodiploid species reproducing with arrhenotokous parthenogenesis, CI crosses produce only parthenogenetic males, inducing a male-biased sex ratio in the population. Here, we used two modeling approaches to evaluate the respective influences of demographic and biological parameters on Wolbachia fixation probability and on the sex ratio peak occurring during a Wolbachia invasion, and compared these parameters to values reported in the literature. Results suggest that the impact of Wolbachia invasion on population dynamics remains relatively limited, especially for parasitoids with high rates of sib-mating. The consequences for introduction of the parasitoids for biological control are discussed.     

Strategy maps: Generalised giving-up densities for optimal foraging

Finding a common currency for benefits and hazards is a major challenge in optimal foraging theory, often requiring complex computational methods. We present a new analytic approach that builds on the Marginal Value Theorem and giving-up densities while incorporating the nonlinear effect of predation risk. We map the space of all possible environments into strategy regions, each corresponding to a discrete optimal strategy. This provides a generalised quantitative measure of the trade-off between foraging rewards and hazards. This extends a classic optimal diet choice rule-of-thumb to incorporate the hazard of waiting for better resources to appear. We compare the dynamics of optimal decision-making for three foraging life-history strategies: One in which fitness accrues instantly, and two with delays before fitness benefit is accrued. Foragers with delayed-benefit strategies are more sensitive to predation risk than resource quality, as they stand to lose more fitness from a predation event than instant-accrual foragers.     

A Mathematical Model for Invasion Range of Population Dispersion Through a Patchy Environment

We focus on the question of how the dispersion of an invading population is affected by the spatial distribution of patches that have resource available for the population’s settlement and reproduction. We have developed and analyzed a mathematical model with a simple stochastic process. The patches are grouped into three classes – free, occupied and abandoned – depending on the state of the patch used by the population. We especially consider the range expanded by invaded patches, the invaded range R, assuming a certain generalized relation between R and the total number of invaded patches k, making use of an index, a sort of fractal dimension, to characterize the spatial distribution of invaded patches. We show that the expected velocity is significantly affected by the nature of spatial distribution of resource patches, and is temporally variable. When the invading population finally becomes extinct at a certain moment, the terminal size of the invaded range at that the moment is closely related to the nature of the spatial distribution of resource patches, which is explicitly demonstrated by our analysis.     

Molecular mechanic study of nerve agent O-ethyl S-[2-(diisopropylamino)ethyl]methylphosphonothioate (VX) bound to the active site of Torpedo californica acetylcholinesterase

Herein a molecular mechanic study of the interaction of a lethal chemical warfare agent, O-ethyl S-[2-(diisopropylamino)ethyl]methylphosphonothioate (also called VX), with Torpedo californica acetylcholinesterase (TcAChE) is discussed. This compound inhibits the enzyme by phosphonylating the active site serine. The chirality of the phosphorus atom induces an enantiomeric inhibitory effect resulting in an enhanced anticholinesterasic activity of the S_P isomer (VX^S) versus its R_P counterpart (VX^R). As formation of the enzyme-inhibitor Michaelis complex is known to be a crucial step in the inhibitory pathway, this complex was addressed by stochastic boundary molecular dynamics and quantum mechanical calculations. For this purpose two models of interaction were analyzed: in the first, the leaving group of VX was oriented toward the anionic subsite of TcAChE, in a similar way as it has been suggested for the natural substrate acetylcholine; in the second, it was oriented toward the gorge entrance, placing the active site serine in a suitable position for a backside attack on the phosphorus atom. This last model was consistent with experimental data related to the high inhibitory effect of this compound and the difference in activity observed for the two enantiomers. Proteins 28:543–555, 1997. © 1997 Wiley-Liss, Inc.     

A scaling analysis in the SIRI epidemiological model

For the spatial stochastic epidemic reinfection model SIRI, where susceptibles S can become infected I, then recover and remain only partial immune against reinfection R, we determine the phase transition lines using pair approximation for the moments derived from the master equation. We introduce a scaling argument that allows us to determine analytically an explicit formula for these phase transition lines and prove rigorously the heuristic results obtained previously.     

随机进化稳定性研究进展

在过去的三十多年, 演化博弈理论及其进化稳定对策的概念不仅被广泛地应用于解释动物行为的进化, 而且也被成功地应用于分子生物学、经济学、政治学和社会学等诸多学科。然而, 在随机波动环境中演化博弈动态的随机动力学性质始终没有被清晰地认识, 并且这是一个极具挑战性的理论问题。本文简单介绍了我们最近所提出的随机进化稳定性(stochastic evolutionary stability, SES)的概念。随机进化稳定性不仅是经典进化稳定对策(evolutionarily stably strategy, ESS)概念在随机环境下的自然扩展, 而且为揭示在随机环境中动物行为的演化动态提供一个基本的理论框架。     

Life‐history syndromes: Integrating dispersal through space and time

Recent research has highlighted interdependencies between dispersal and other life‐history traits, i.e. dispersal syndromes, thereby revealing constraints on the evolution of dispersal and opportunities for improved ability to predict dispersal by considering suites of dispersal‐related traits. This review adds to the growing list of life‐history traits linked to spatial dispersal by emphasising the interdependence between dispersal through space and time, i.e. life‐history diversity that distributes individuals into separate reproductive events. We reviewed the literature that has simultaneously investigated spatial and temporal dispersal to examine the prediction that traits of these two dispersal strategies are negatively correlated. Our results suggest that negative covariation is widely anticipated from theory. Empirical studies often reported evidence of weak negative covariation, although more complicated patterns were also evident, including across levels of biological organisation. Existing literature has largely focused on plants with dormancy capability, one or two phases of the dispersal process (emigration and/or transfer) and a single level of biological organisation (theory: individual; empirical: species). We highlight patterns of covariation across levels of organisation and conclude with a discussion of the consequences of dispersal through space and time and future research areas that should improve our understanding of dispersal‐related life‐history syndromes.