On the linear birth and death processes of biology as Markoff chains

Stochastic Markoff models for the linear birth and death population growth processes of biology are constructed using the Q-matrix method of Doob. The relationship of the stochastic theory to the classical deterministic foundations of these processes is stressed by showing in detail how the classical postulates are mathematically transformed via the Q-matrix elements into the basis for a stationary Markoff process with continuous time parameter and denumerably many “populations states.” It is shown that the resulting stochastic models predict that the population size will fluctuate about the deterministic time curve, the extent of fluctuation being measured by the variance functions. General formulas covering all possible transitions from one population size to another are derived.     

Think globally,act locally: the role of local demographics and vaccination coverage in the dynamic response of measles infection to control

The global reduction of the burden of morbidity and mortality owing to measles has been a major triumph of public health. However, the continued persistence of measles infection probably not only reflects local variation in progress towards vaccination target goals, but may also reflect local variation in dynamic processes of transmission, susceptible replenishment through births and stochastic local extinction. Dynamic models predict that vaccination should increase the mean age of infection and increase inter-annual variability in incidence. Through a comparative approach, we assess national-level patterns in the mean age of infection and measles persistence. We find that while the classic predictions do hold in general, the impact of vaccination on the age distribution of cases and stochastic fadeout are mediated by local birth rate. Thus, broad-scale vaccine coverage goals are unlikely to have the same impact on the interruption of measles transmission in all demographic settings. Indeed, these results suggest that the achievement of further measles reduction or elimination goals is likely to require programmatic and vaccine coverage goals that are tailored to local demographic conditions.     

Stochastic instability and Liapunov stability

Summary The relationship between the deterministic stability of nonlinear ecological models and the properties of the stochastic model obtained by adding weak random perturbations is studied. It is shown that the expected escape time for the stochastic model from a bounded region with nonsingular boundary is determined by a Liapunov function for the nonlinear deterministic model. This connection between stochastic and deterministic models brings together various notions of persistence and vulnerability of ecosystems as defined for deterministically perturbed or randomly perturbed models.     

Understanding the persistence of measles: reconciling theory, simulation and observation

Ever since the pattern of localized extinction associated with measles was discovered by Bartlett in 1957, many models have been developed in an attempt to reproduce this phenomenon. Recently, the use of constant infectious and incubation periods, rather than the more convenient exponential forms, has been presented as a simple means of obtaining realistic persistence levels. However, this result appears at odds with rigorous mathematical theory; here we reconcile these differences. Using a deterministic approach, we parameterize a variety of models to fit the observed biennial attractor, thus determining the level of seasonality by the choice of model. We can then compare fairly the persistence of the stochastic versions of these models, using the 'best-fit' parameters. Finally, we consider the differences between the observed fade-out pattern and the more theoretically appealing 'first passage time'.     

Stochastic amplification in an epidemic model with seasonal forcing

We study the stochastic susceptible-infected-recovered (SIR) model with time-dependent forcing using analytic techniques which allow us to disentangle the interaction of stochasticity and external forcing. The model is formulated as a continuous time Markov process, which is decomposed into a deterministic dynamics together with stochastic corrections, by using an expansion in inverse system size. The forcing induces a limit cycle in the deterministic dynamics, and a complete analysis of the fluctuations about this time-dependent solution is given. This analysis is applied when the limit cycle is annual, and after a period doubling when it is biennial. The comprehensive nature of our approach allows us to give a coherent picture of the dynamics which unifies past work, but which also provides a systematic method for predicting the periods of oscillations seen in whooping cough and measles epidemics.     

Distinguishing error from chaos in ecological time series

Over the years, there has been much discussion about the relative importance of environmental and biological factors in regulating natural populations. Often it is thought that environmental factors are associated with stochastic fluctuations in population density, and biological ones with deterministic regulation. We revisit these ideas in the light of recent work on chaos and nonlinear systems. We show that completely deterministic regulatory factors can lead to apparently random fluctuations in population density, and we then develop a new method (that can be applied to limited data sets) to make practical distinctions between apparently noisy dynamics produced by low-dimensional chaos and population variation that in fact derives from random (high-dimensional) noise, such as environmental stochasticity or sampling error. To show its practical use, the method is first applied to models where the dynamics are known. We then apply the method to several sets of real data, including newly analysed data on the incidence of measles in the United Kingdom. Here the additional problems of secular trends and spatial effects are explored. In particular, we find that on a city-by-city scale measles exhibits low-dimensional chaos (as has previously been found for measles in New York City), whereas on a larger, country-wide scale the dynamics appear as a noisy two-year cycle. In addition to shedding light on the basic dynamics of some nonlinear biological systems, this work dramatizes how the scale on which data is collected and analysed can affect the conclusions drawn.     

Multi-patch deterministic and stochastic models for wildlife diseases

Spatial heterogeneity and host demography have a direct impact on the persistence or extinction of a disease. Natural or human-made landscape features such as forests, rivers, roads, and crops are important to the persistence of wildlife diseases. Rabies, hantaviruses, and plague are just a few examples of wildlife diseases where spatial patterns of infection have been observed. We formulate multi-patch deterministic and stochastic epidemic models and use these models to investigate problems related to disease persistence and extinction. We show in some special cases that a unique disease-free equilibrium exists. In these cases, a basic reproduction number ?(0) can be computed and shown to be bounded below and above by the minimum and maximum patch reproduction numbers ?(j), j=1, …, n. The basic reproduction number has a simple form when there is no movement or when all patches are identical or when the movement rate approaches infinity. Numerical examples of the deterministic and stochastic models illustrate the disease dynamics for different movement rates between three patches.     

Population extinction and quasi-stationary behavior in stochastic density-dependent structured models

Density-independent and density-dependent, stochastic and deterministic, discrete-time, structured models are formulated, analysed and numerically simulated. A special case of the deterministic, density-independent, structured model is the well-known Leslie age-structured model. The stochastic, density-independent model is a multitype branching process. A review of linear, density-independent models is given first, then nonlinear, density-dependent models are discussed. In the linear, density-independent structured models, transitions between states are independent of time and state. Population extinction is determined by the dominant eigenvalue λ of the transition matrix. If λ ≤ 1, then extinction occurs with probability one in the stochastic and deterministic models. However, if λ > 1, then the deterministic model has exponential growth, but in the stochastic model there is a positive probability of extinction which depends on the fixed point of the system of probability generating functions. The linear, density-independent, stochastic model is generalized to a nonlinear, density-dependent one. The dependence on state is in terms of a weighted total population size. It is shown for small initial population sizes that the density-dependent, stochastic model can be approximated by the density-independent, stochastic model and thus, the extinction behavior exhibited by the linear model occurs in the nonlinear model. In the deterministic models there is a unique stable equilibrium. Given the population does not go extinct, it is shown that the stochastic model has a quasi-stationary distribution with mean close to the stable equilibrium, provided the population size is sufficiently large. For small values of the population size, complete extinction can be observed in the simulations. However, the persistence time increases rapidly with the population size. This author received partial support by the National Science Foundation grant # DMS-9626417.     

Species selection and random drift in macroevolution

Species selection resulting from trait‐dependent speciation and extinction is increasingly recognized as an important mechanism of phenotypic macroevolution. However, the recent bloom in statistical methods quantifying this process faces a scarcity of dynamical theory for their interpretation, notably regarding the relative contributions of deterministic versus stochastic evolutionary forces. I use simple diffusion approximations of birth‐death processes to investigate how the expected and random components of macroevolutionary change depend on phenotype‐dependent speciation and extinction rates, as can be estimated empirically. I show that the species selection coefficient for a binary trait, and selection differential for a quantitative trait, depend not only on differences in net diversification rates (speciation minus extinction), but also on differences in species turnover rates (speciation plus extinction), especially in small clades. The randomness in speciation and extinction events also produces a species‐level equivalent to random genetic drift, which is stronger for higher turnover rates. I then show how microevolutionary processes including mutation, organismic selection, and random genetic drift cause state transitions at the species level, allowing comparison of evolutionary forces across levels. A key parameter that would be needed to apply this theory is the distribution and rate of origination of new optimum phenotypes along a phylogeny.     

Hybrid stochastic and deterministic simulations of calcium blips

Intracellular calcium release is a prime example for the role of stochastic effects in cellular systems. Recent models consist of deterministic reaction-diffusion equations coupled to stochastic transitions of calcium channels. The resulting dynamics is of multiple time and spatial scales, which complicates far-reaching computer simulations. In this article, we introduce a novel hybrid scheme that is especially tailored to accurately trace events with essential stochastic variations, while deterministic concentration variables are efficiently and accurately traced at the same time. We use finite elements to efficiently resolve the extreme spatial gradients of concentration variables close to a channel. We describe the algorithmic approach and we demonstrate its efficiency compared to conventional methods. Our single-channel model matches experimental data and results in intriguing dynamics if calcium is used as charge carrier. Random openings of the channel accumulate in bursts of calcium blips that may be central for the understanding of cellular calcium dynamics.     

Allee effects and resilience in stochastic populations

Allee effects, or positive functional relationships between a population’s density (or size) and its per unit abundance growth rate, are now considered to be a widespread if not common influence on the growth of ecological populations. Here we analyze how stochasticity and Allee effects combine to impact population persistence. We compare the deterministic and stochastic properties of four models: a logistic model (without Allee effects), and three versions of the original model of Allee effects proposed by Vito Volterra representing a weak Allee effect, a strong Allee effect, and a strong Allee effect with immigration. We employ the diffusion process approach for modeling single-species populations, and we focus on the properties of stationary distributions and of the mean first passage times. We show that stochasticity amplifies the risks arising from Allee effects, mainly by prolonging the amount of time a population spends at low abundance levels. Even weak Allee effects become consequential when the ubiquitous stochastic forces affecting natural populations are accounted for in population models. Although current concepts of ecological resilience are bound up in the properties of deterministic basins of attraction, a complete understanding of alternative stable states in ecological systems must include stochasticity.     

Demography in an increasingly variable world

Recent advances in stochastic demography provide unique insights into the probable effects of increasing environmental variability on population dynamics, and these insights can be substantially different compared with those from deterministic models. Stochastic variation in structured population models influences estimates of population growth rate, persistence and resilience, which ultimately can alter community composition, species interactions, distributions and harvesting. Here, we discuss how understanding these demographic consequences of environmental variation will have applications for anticipating changes in populations resulting from anthropogenic activities that affect the variance in vital rates. We also highlight new tools for anticipating the consequences of the magnitude and temporal patterning of environmental variability.     

Modeling the presence probability of invasive plant species with nonlocal dispersal

Mathematical models for the spread of invading plant organisms typically utilize population growth and dispersal dynamics to predict the time-evolution of a population distribution. In this paper, we revisit a particular class of deterministic contact models obtained from a stochastic birth process for invasive organisms. These models were introduced by Mollison (J R Stat Soc 39(3):283, 1977). We derive the deterministic integro-differential equation of a more general contact model and show that the quantity of interest may be interpreted not as population size, but rather as the probability of species occurrence. We proceed to show how landscape heterogeneity can be included in the model by utilizing the concept of statistical habitat suitability models which condense diverse ecological data into a single statistic. As ecologists often deal with species presence data rather than population size, we argue that a model for probability of occurrence allows for a realistic determination of initial conditions from data. Finally, we present numerical results of our deterministic model and compare them to simulations of the underlying stochastic process.     

On the stochastic theory of compartments: III. General time-dependent reversible systems

General formulation of stochastic single- and multi-compartment reversible systems with time-dependent transitions is made. The correspondence between the stochastic mean and the deterministic value is established in case of time-dependence and it is shown how the consequence of this can be utilized to compute the distribution and the moments of each individual compartment of the system. A two-compartment reversible system previously proposed by Cardenas and Matis (1975a) is analyzed on the basis of the theory.     

Environmental changes affect the assembly of soil bacterial community primarily by mediating stochastic processes

Both ‘species fitness difference’‐based deterministic processes, such as competitive exclusion and environmental filtering, and ‘species fitness difference’‐independent stochastic processes, such as birth/death and dispersal/colonization, can influence the assembly of soil microbial communities. However, how both types of processes are mediated by anthropogenic environmental changes has rarely been explored. Here we report a novel and general pattern that almost all anthropogenic environmental changes that took place in a grassland ecosystem affected soil bacterial community assembly primarily through promoting or restraining stochastic processes. We performed four experiments mimicking 16 types of environmental changes and separated the compositional variation of soil bacterial communities caused by each environmental change into deterministic and stochastic components, with a recently developed method. Briefly, because the difference between control and treatment communities is primarily caused by deterministic processes, the deterministic change was quantified as (mean compositional variation between treatment and control) – (mean compositional variation within control). The difference among replicate treatment communities is primarily caused by stochastic processes, so the stochastic change was estimated as (mean compositional variation within treatment) – (mean compositional variation within control). The absolute of the stochastic change was greater than that of the deterministic change across almost all environmental changes, which was robust for both taxonomic and functional‐based criterion. Although the deterministic change may become more important as environmental changes last longer, our findings showed that changes usually occurred through mediating stochastic processes over 5 years, challenging the traditional determinism‐dominated view.     

Predator-prey interactions in a nonequilibrium context: the metapopulation approach to modeling "hide-and-seek" dynamics in a spatially explicit tri-trophic system

Predators and prey are usually heterogeneously distributed in space so that the ability of the predators to respond to the distribution of their prey may have a profound influence on the stability and persistence of a predator‐prey system. A special type of dynamics is “hide‐and‐seek” characterized by a high turnover rate of local populations of prey and predators, because once the predators have found a patch of prey they quickly overexploit it, whereupon the starving predators either should move to better places or die. Continued persistence of prey and predators thus hinges on a long‐term balance between local extinctions and founding of new subpopulations. The colonization rate depends on the rate of emigration from occupied patches and the likelihood of successfully arriving at a suitable new patch, while extinction rate depends on the local population dynamics. Since extinctions and colonizations are both discrete probabilistic events, these phenomena are most adequately modeled by means of a stochastic model. In order to demonstrate the qualitative differences between a deterministic and stochastic approach to population dynamics, a spatially explicit tritrophic predator‐prey model is developed in a deterministic and a stochastic version. The model is parameterized using data for the two‐spotted spider mite (Tetranychus urticae) and the phytoseiid mite predator Phytoseiulus persimilis inhabiting greenhouse cucumbers.
Simulations show that the deterministic and stochastic approaches yield different results. The deterministic version predicts that the populations will exhibit violent fluctuations, implying that the system is fundamentally unstable. In contrast, the stochastic version predicts that the two species will be able to coexist in spite of frequent local extinctions of both species, provided the system consists of a sufficiently large number of local populations. This finding is in agreement with experimental results. It is therefore concluded that demographic stochasticity in combination with dispersal is capable of producing and maintaining sufficient asynchrony between local populations to ensure long‐term regional (metapopulation) persistence.