Sources Of Stochasticity Research Articles

Life-history models of extinction: a test with island spiders.

This study analyzes extinction patterns for two species of orb spiders monitored annually on 77 islands over a continuous 20-yr period. One species, Argiope argentata, has large populations sometimes crashing quickly to extinction and a much weaker relation of extinction likelihood to population size than does the other species, Metepeira datona. Demographic models were built for both species and matched against observations. Differences between the species in life-history traits-estimated with measurements from the field-together with incorporation of demographic stochasticity, a population ceiling, and environmental stochasticity, were necessary to fit the observed extinction curves. As predicted from life-history patterns, long-term population growth rates (and hence predicted extinction probabilities) are relatively very sensitive to values of juvenile survivorship. Models are also sensitive to variation in the population ceiling and environmental noise, which tend to act in a complementary manner. A simple model with no age structure was able to fit the data on large initial population sizes but not on small initial population sizes, showing that life cycle characteristics interact with the various sources of stochasticity and hence have to be taken into account to produce a precise model of the extinction process.

INCORPORATING MULTIPLE SOURCES OF STOCHASTICITY INTO DYNAMIC POPULATION MODELS

Many standard statistical models used to examine population dynamics ignore significant sources of stochasticity. Usually only process error is included, and uncertainty due to errors in data collection is omitted or not directly specified in the model. We show how standard time-series models for population dynamics can be extended to include both observational and process error and how to perform inference on parameters in these models in the Bayesian setting. Using simulated data, we show how ignoring observation error can be misleading. We argue that the standard Bayesian techniques used to perform inference, including freely available software, are generally applicable to a variety of time-series models. Corresponding Editor: O. N. Bjornstad.

MUTATIONAL MELTDOWNS IN SEXUAL POPULATIONS.

Although it is widely acknowledged that the gradual accumulation of mildly deleterious mutations is an important source of extinction for asexual populations, it is generally assumed that this process is of little relevance to sexual species. Here we present results, based on computer simulations and supported by analytical approximations, that indicate that mutation accumulation in small, random-mating monoecious populations can lead to mean extinction times less than a few hundred to a few thousand generations. Unlike the situation in obligate asexuals in which the mean time to extinction (t̄e ) increases more slowly than linearly with the population carrying capacity (K), t̄e increases approximately exponentially with K in outcrossing sexual populations. The mean time to extinction for obligately selfing populations is shown to be equivalent to that for asexual populations of the same size, but with half the mutation rate and twice the mutational effect; this suggests that obligate selfing, like obligate asexuality, is inviable as a long-term reproductive strategy. Under all mating systems, the mean time to extinction increases relatively slowly with the logarithm of fecundity, and mutations with intermediate effects (similar to those observed empirically) cause the greatest risk of extinction. Because our analyses ignore sources of demographic and environmental stochasticity, which have synergistic effects that exacerbate the accumulation of deleterious mutations, our results should yield liberal upper bounds to the mean time to extinction caused by mutational degradation. Thus, deleterious mutation accumulation cannot be ruled out generally as a significant source of extinction vulnerability in small sexual populations or as a selective force influencing mating-system evolution.

Fokker–Planck equations in the simulation of complex systems

The present state of affairs in the description of real systems with chaotic dynamics calls for a new approach aimed at a more complete description of experiments. Such a description takes into account explicitly the inevitable uncertainties by means of an appropriate mathematical formalism and unites thereby deterministic kinetics and the various sources of stochasticity. A mathematical tool for such an approach is available in principle, since almost all real dynamical systems in continuous time can be described properly by a master equation. The master equation of many systems perturbed by external/internal noise and fluctuations can be appropriately approached by a Fokker-Planck form. A derivation of these equations is presented which emphasizes the main assumptions and the applicability conditions involved. A method of solution of the Fokker-Planck equation is proposed which is based on the Green's function of the equation. The algorithm has been incorporated into a library of FORTRAN-77 programs. The performance of the library is tested by solving several physical problems with solution known a priori. A set of applications in the simulation of synergetic, self-organizing and dynamical systems are commented.

The coercivity of particle arrays

Thermal effects in an array of nonidentical, noninteracting single domain ferromagnetic particles are studied. Two sources of stochasticity are recognized within the array: the intrinsic stochastic nature of thermal relaxation and variations within the particles’ volume and orientation with respect to applied field, both are discussed in turn. We use the log-normal volume distribution {∼(σV)−1exp[−ln2(V/V0)/(2σ2)]} and a cosν ψ angular distribution to calculate the mean coercivity B̄c of a particle array. We find that B̄c increases rapidly with increasing average volume <V≳=V0 exp(σ2/2) while for misaligned particles (ν≤∞) we observe a sudden decrease in B̄c accompanied by a gradual loss of frequency dependence.

On the relationship between the reliability of parameter estimates and hydrologic time series data used in calibration

Typically, some of the parameters of conceptual hydrologic models are calibrated using limited hydrologic information, namely, input‐output time series data such as precipitation and streamflow. The first part of this paper examines the sources of stochasticity in these models and then explores the conditions under which parameter estimates are consistent when only input‐output hydrologic time series data are used in calibration. This complements other work done on the stability of parameter estimates. Because the conditions for consistency are stringent, two ways of redressing this situation and also improving the stability of parameter estimates are considered in the second part. Two levels of additional information are considered. The first considers the use of the first two moments of measurement errors to make large sample bias corrections. The second employs time series data corresponding to storage volumes such as groundwater and soil moisture to remove the source of inconsistency due to inferring erroneously unobserved storage volumes and to improve the stability of parameter estimates. Proper use of such information must exploit the interdependence in model equations arising from coupled model structure and correlated disturbances. It is suggested that generalized least squares offers a promising approach for efficiently exploiting all available time series information in model calibration. Finally, a simple hydrologic example is given to illustrate the relationship between estimator reliability and time series data used in calibration.

Stochastic Models of Compartmental Systems

This paper reviews a stochastic approach to compartmental modeling. The needfor stochas ticity in the model is motivated by two examples one concerned with vanadium depuration in marine organisms and the other with the thermal resistance of the green sunfish Lepomis cyanellus. A stochastic formulation is suggested which includes several sources of stochasticity due to some variability among particles in a given experiment and also to some variability between replicates of an experiment. Many digerent stochastic models of compartmental systems may be obtained frosn various combinations of these sources of stochasticity and the mean and autocovariance functions of these models are derived for some one-conspartment systems. The mean value functions are shown to be generally tractable forms which may be readily fitted to data however even in the case of time-invariant coefficients the mean functions are often diffierent frossl the sums of exponentialfunctions given by the deterministic solution. The causal model cannot be identified on the basis of the mean value function alone however the covariance structure is unique for each of the proposed causal models. The paper also illustrates some basic statistical analysis with these stochastic models. The estimation of parass1eters by weighted nonlinear least squares is illustrated by fitting some mean functions of these models to data from the previous examples. The procedure nzay generate RBAN estimators for both of the examples and an asymptotic goodness-of-fit statistic is forsnulated. The paper concludes with a short summary of prosnising areas for future research.

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.