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A brief review of recent metapopulation literature indicates that dispersal functions are generally highly simplified. We suggest that this oversight can often make the results of such models misleading. A number of factors are considered that can affect the patterns and functions of this potentially complex process, including conditional and “intel-ligent” dispersal, and dispersal distance. If a system is to be considered in spatially explicit terms then dispersal is an essential component. It follows therefore that a more thorough consideration of this fundamental population process is to be encouraged.

International audience; We are interested in a stochastic model of trait and age-structured population undergoing mutation and selection. We start with a continuous time, discrete individual-centered population process. Taking the large population and rare mutations limits under a well-chosen time-scale separation condition, we obtain a jump process that generalizes the Trait Substitution Sequence process describing Adaptive Dynamics for populations without age structure. Under the additional assumption of small mutations, we derive an age-dependent ordinary differential equation that extends the Canonical Equation. These evolutionary approximations have never been introduced to our knowledge. They are based on ecological phenomena represented by PDEs that generalize the Gurtin-McCamy equation in Demography. Another particularity is that they involve a fitness function, describing the probability of invasion of the resident population by the mutant one, that can not always be computed explicitly. Examples illustrate how adding an age-structure enrich the modelling of structured population by including life history features such as senescence. In the cases considered, we establish the evolutionary approximations and study their long time behavior and the nature of their evolutionary singularities when computation is tractable. Numerical procedures and simulations are carried.

This paper provides a countable representation for a class of infinite-dimensional diffusions which extends the infinitely-many-neutral-alleles model and is related to the two-parameter Poisson-Dirichlet process. By means of Gibbs sampling procedures, we define a reversible Moran-type population process. The associated process of ranked relative frequencies of types is shown to converge in distribution to the two-parameter family of diffusions, which is stationary and ergodic with respect to the two-parameter Poisson-Dirichlet distribution. The construction provides interpretation for the limiting process in terms of individual dynamics.

Mark-recapture studies are one of the most common methods used to obtain demographic parameters for wildlife populations. Time specific estimates of parameters representing population processes contain both temporal variability in the process (process error) and error in estimating the parameters (observation error). Therefore, to estimate the temporal variation in the population process, it is important to separate these two errors. Traditional random effect models can be used to separate the two errors. However, it is difficult to implement the required simultaneous maximization and integration for dynamic nonlinear non-Gaussian models. An alternative hierarchical Bayesian approach using MCMC integration is easier to apply, but requires priors for all model parameters.

It is widely believed that population outbreaks of the southern pine beetle ( Dendroctonus frontalis Zimm.) are caused by vagaries of climate, such as periods of severe drought. According to this view, D. frontalis population dynamics are dominated by density-independent processes. We have statistically analyzed a 30-yr record of D. frontalis activity in east Texas and have assessed the relative roles of density-independent and density-dependent factors in beetle population fluctuations. Regressions of the rate of population change on three climatic variables were not significant. By contrast, both time-series and regression analyses provided strong and consistent evidence for delayed density regulation of D. frontalis populations. Thus, in contrast to previous analyses, we conclude that D. frontalis outbreaks are driven not by stochastic fluctuations of weather, but by some unknown population process acting in a delayed density-dependent manner. This result provides a starting point for a current study that will experimentally test various hypotheses concerning the role of natural enemies in D. frontalis cycles.

For those who love semantics the title of this chapter really is a godsend. When is an issue an 'emerging' issue? When is it an issue in demographic research? For whom? What is demographic research? What is (contemporary) Europe? The list of questions that can be asked about the meaning of the phrase appears endless. Moreover, not all of those questions are largely irrelevant. Three points in particular deserve some attention. Emerging issues can be identified from two quite different perspectives and the results are not necessarily the same. The first perspective may be called the scientific perspective. The demographer or social scientist involved then acts solely as an observer or an investigator who seeks to understand what is observed and to increase knowledge for its own sake. The approach of such demographers may be completely theoretical. They are not interested in influencing the processes they see unfolding, nor in the utility of their findings for those people who concern themselves with the formulation and design of policies, or for the population at large. It is a discipline oriented perspective. The second perspective may be referred to as the applied or policy perspective. These researchers are concerned with obtaining results that have some practical significance. They focus on issues identified as 'societal problems'. They may want to change them, and to seek a specific solution to them. Influencing the processes observed and highlighting the possible ways in which this can be done effectively may be the main interest. Emerging issues are not identified on the basis of their relevance for the development of the discipline, but on the basis of their (potential) relevance for developments in society. The perspective is problem oriented. In the European setting 'demography' is an ambiguous term. In German, for example, Demographie is usually considered to be quite distinct from Bevo6lkerungswissenschaft. While the first has the connotation of being basically confined to pure or formal demography and to deal essentially with the internal relations between demographic variables sensu stricto the second concept is much broader. It can be translated literally as 'population science' and is then akin to population studies. It deals with variables external to the population process as such and thus with the relations between that process and cultural, social and other conditions in society (Mackensen, 1981a, pp. 19, 20). Epistemologically this has important consequences. For, while demography as the 'skeleton of a discipline' will seek its foundation in the logic of mathematics and statistics, the 'science of population' will build on the theories of sociology, psychology, economics and the like. Other European language areas may have their own concepts and terminology. The heterogeneity of Europe is frequently underestimated. Geographically it can be considered to range from Iceland and Ireland in the West to Turkey and the USSR in the East. Countries like Albania, Liechtenstein and San Marino are hardly visible on a map of Europe, while Norway, Italy and Poland are stretched out over hundreds of

International audience; We are interested in the study of models describing the evolution of a polymorphic population with mutation and selection in the specific scales of the biological framework of adaptive dynamics. The population size is assumed to be large and the mutation rate small. We prove that under a good combination of these two scales, the population process is approximated in the long time scale of mutations by a Markov pure jump process describing the successive trait equilibria of the population. This process, which generalizes the so-called trait substitution sequence, is called polymorphic evolution sequence. Then we introduce a scaling of the size of mutations and we study the polymorphic evolution sequence in the limit of small mutations. From this study in the neighborhood of evolutionary singularities, we obtain a full mathematical justification of a heuristic criterion for the phenomenon of evolutionary branching. To this end we finely analyze the asymptotic behavior of 3-dimensional competitive Lotka-Volterra systems.

We formulate some simple conditions under which a Markov chain may be approximated by the solution to a differential equation, with quantifiable error probabilities. The role of a choice of coordinate functions for the Markov chain is emphasised. The general theory is illustrated in three examples: the classical stochastic epidemic, a population process model with fast and slow variables, and core-finding algorithms for large random hypergraphs., Comment: Published in at http://dx.doi.org/10.1214/07-PS121 the Probability Surveys (http://www.i-journals.org/ps/) by the Institute of Mathematical Statistics (http://www.imstat.org)

Mudulation-doped AlxGa1−x AsGaAs quantum wells (QW) have been prepared where the total carrier density Ns and thus the occupation of the upper subband can be tuned via front gate voltage Vg in a controlled and reproducible way. We performed magneto-capacitance versus voltage (MCV) measurements at very low magnetic fields (B

We provide a general framework for estimating persistence in populations which may be affected by catastrophic events, and which are either unbounded or have very large ceilings. We model the population using a birth-death process modified to allow for downward jumps of arbitrary size. For such processes, it is typically necessary to truncate the process in order to make the evaluation of expected extinction times (and higher-order moments) computationally feasible. Hence, we give particular attention to the selection of a cut-off point at which to truncate the process, and we present a simple method for obtaining quantitative indicators of the suitability of a chosen cut-off.

The demographic variance of an age-structured population is defined. This parameter is further split into components generated by demographic stochasticity in each vital rate. The applicability of these parameters are investigated by checking how an age-structured population process can be approximated by a diffusion with only three parameters. These are the deterministic growth rate computed from the expected projection matrix and the environmental and demographic variances. We also consider age-structured populations where the fecundity at any stage is either zero or one, and there is neither environmental stochasticity nor dependence between individual fecundity and survival. In this case the demographic variance is uniquely determined by the vital rates defining the projection matrix. The demographic variance for a long-lived bird species, the wandering albatross in the southwestern part of the Indian Ocean, is estimated. We also compute estimates of the age-specific contributions to the total demographic variance from survival, fecundity and the covariance between survival and fecundity.

We consider the evolution of a simple population process governed by deaths and multiple immigrations that arrive with rates particular to their order. For a particular choice of rates, the equilibrium solution has a discrete power‐law form. The model is a generalization of a process investigated previously where immigrants arrived in pairs [1]. The general properties of this model are discussed in a companion paper. The population is initiated with precisely M individuals present and evolves to an equilibrium distribution with a power‐law tail. However the power‐law tails of the equilibrium distribution are established immediately, so that moments and correlation properties of the population are undefined for any non‐zero time. The technique we develop to characterize this process utilizes external monitoring that counts the emigrants leaving the population in specified time intervals. This counting distribution also possesses a power‐law tail for all sampling times and the resulting time series exhibits two features worthy of note, a large variation in the strength of the signal, reflecting the power‐law PDF; and secondly, intermittency of the emissions. We show that counting with a detector of finite dynamic range regularizes naturally the fluctuations, in effect ‘clipping’ the events. All previously undefined characteristics such as the mean, autocorrelation and probabilities to the first event and time between events are well defined and derived. These properties, although obtained by discarding much data, nevertheless possess embedded power‐law regimes that characterize the population in a way that is analogous to box averaging determination of fractal‐dimension.

Latitudinal gradients in population dynamics can arise through regional variation in the deterministic components of the population dynamics and the stochastic factors. Here, we demonstrate an increase with latitude in the contribution of a large-scale climate pattern, the North Atlantic Oscillation (NAO), to the fluctuations in size of populations of two European hole-nesting passerine species. However, this influence of climate induced different latitudinal gradients in the population dynamics of the two species. In the great tit the proportion of the variability in the population fluctuations explained by the NAO increased with latitude, showing a larger impact of climate on the population fluctuations of this species at higher latitudes. In contrast, no latitudinal gradient was found in the relative contribution of climate to the variability of the pied flycatcher populations because the total environmental stochasticity increased with latitude. This shows that the population ecological consequences of an expected climate change will depend on how climate affects the environmental stochasticity in the population process. In both species, the effects will be larger in those parts of Europe where large changes in climate are expected.

The use of Markovian models is an established way for deriving the complete distribution of the size of a population and the probability of extinction. However, computationally impractical transition matrices frequently result if this mathematical approach is applied to natural populations. Binning, or aggregating population sizes, has been used to permit a reduction in the dimensionality of matrices. Here, we present three deterministic binning methods and study the errors due to binning for a metapopulation model. Our results indicate that estimation errors of the investigated methods are not consistent and one cannot make generalizations about the quality of a method. For some compared output variables of populations studied, binning methods that caused a strong reduction in dimensionality of matrices resulted in better estimations than methods that produced a weaker reduction. The main problem with deterministic binning methods is that they do not properly take into account the stochastic population process itself. Straightforward usage of binning methods may lead to substantial errors in population-dynamical predictions.

This paper investigates population of active users in a cell and proposes a traffic model for mobile multimedia networks. This model describes a population process of active users in a cell in diffusion model, and its characteristics include self-similarity and activity of mobility. We also made an evaluation of network performance by using simulation, in order to show that characteristics of the proposed traffic model have impact upon planning and designing networks.

En este trabajo se analizan los cambios de la población de la ciudad de Mexicali, ciudad que por razones de su evolución y efectos del medio registra un notable dinamismo territorial, Se alude primero a la dimensión histórica, para explicar el proceso de poblamiento acaecido a inicios del presente siglo. A continuación se analizan las transformaciones poblacionales que se han sucedido hasta la actualidad. Asimismo, en el último apartado se relaciona el crecimiento de la población con el del espacio y su importancia político-administrativa como localidad sede de la capital del estado de Baja California, y por su situación fronteriza con Estados Unidos de América. La metodología aplicada fue el modelo general de gradiente de densidad, que permite determinar dimensiones fundamentales del espacio, social y sus variaciones intraurbanas.

Models of populations in which a type or location, represented by a point in a metric space $E$, is associated with each individual in the population are considered. A population process is neutral if the chances of an individual replicating or dying do not depend on its type. Measure-valued processes are obtained as infinite population limits for a large class of neutral population models, and it is shown that these measure-valued processes can be represented in terms of the total mass of the population and the de Finetti measures associated with an $E^{\infty}$ -valued particle model $X=(X_1, X_2\ldots)$ such that, for each $t \geq 0,(X_1(t),X_2(t),\ldots)$ is exchangeable. The construction gives an explicit connection between genealogical and diffusion models in population genetics. The class of measure-valued models covered includes both neutral Fleming-Viot and Dawson-Watanabe processes. The particle model gives a simple representation of the Dawson-Perkins historical process and Perkins’s historical stochastic integral can be obtained in terms of classical semimartingale integration. A number of applications to new and known results on conditioning, uniqueness and limiting behavior are described.

▪ Abstract Microbial population dynamics on leaves in time and space are a function of immigration, emigration, growth, and death. Insight into the relative significance of each population process to the generation of specific dynamics for individual microorganisms is necessary to understanding the ecology and life history strategy of the microorganism and to developing effective control strategies. Additionally, information on the significance of within-leaf versus extra-leaf processes to the generation of phyllosphere dynamics is important to determining the range of spatial scales over which a population should be studied. Unfortunately, such information is difficult to obtain due to the lack of effective methodologies for distinguishing these processes within phyllosphere populations. Future research efforts should focus on the quantification of immigration, emigration, growth, and death relative to the population dynamics of phyllosphere microorganisms.

We present a central limit theorem for the population process of a simple genetic algorithm. This theory approximates the discrete population process with finite population size by a continuous Gaussian — process. As a special application we consider a search space with only two elements. For this case we present a complete solution of the differential equation arising in the calculation of the Gaussian — process. This analysis gives first insight into the theory of the general case. Some applications of the theory and extensions to the general case are mentioned, too.

The processes by which disease spreads in a population of individuals are inherently stochastic. The master equation has proven to be a useful tool for modeling such processes. Unfortunately, solving the master equation analytically is possible only in limited cases (e.g., when the model is linear), and thus numerical procedures or approximation methods must be employed. Available approximation methods, such as the system size expansion method of van Kampen, may fail to provide reliable solutions, whereas current numerical approaches can induce appreciable computational cost. In this paper, we propose a new numerical technique for solving the master equation. Our method is based on a more informative stochastic process than the population process commonly used in the literature. By exploiting the structure of the master equation governing this process, we develop a novel technique for calculating the exact solution of the master equation – up to a desired precision – in certain models of stochastic epidemiology. We demonstrate the potential of our method by solving the master equation associated with the stochastic SIR epidemic model. MATLAB software that implements the methods discussed in this paper is freely available as Supporting Information S1.

A two-sex age-dependent demographic model is formulated within the framework of a stochastic population process, including both time-homogeneous and time-inhomogenous laws of evolution. An outline of the parametric components of the system, which expedite computer implementation and experimentation, is also given. New features of the model include a component for couple formation, using the class of Farlie-Morgenstern bivariate distributions to accommodate age preferences in selecting marriage partners, a component for couple dissolution due to separation or divorce, and an outline of techniques for initializing a two-sex projection given scanty information. For the case of time-homogeneous laws of evolution, stability properties of two-sex models that are analogs of those for one-sex models are difficult to prove mathematically due to nonlinearities. But computer experiments in this case suggest that these properties continue to hold for two-sex models for such widely used demographic indicators as period crude birth rates, period rates of natural increase, and period age distributions, which converge to constant forms in long-term projections. The values of the stable crude birth rate, rate of natural increase, and quantiles of the stable age distribution differ markedly among projections that differ only in selected values of parameters governing couple formation and dissolution. Such experimental results demonstrate that two-sex models are not merely intellectual curiosities but exist in their own right and lead to insights not attainable in simpler one-sex formulations.

A theoretical system is usually developed from a few premises. In this book, these are the notions of persistence and regulation of populations. The particular intention here is to deduce what conditions are required for a given population to be regulated so as to persist. In deducing such conditions, we shall gain insight into the basic properties and structure of animal population processes. The practical value of our deduction depends, of course, on how we set up the premises.

In the previous two chapters, I discussed statistical properties of population processes without referring to their ecological mechanisms. In this chapter, I shall discuss several types of mechanisms and examine their interrelations. Through such comparative studies, we gain insight into the way our ideas evolve. Understanding the ecological meaning of a model promotes further improvement in its applicability through systematic generalizations.

In Chapter 2, I have shown various patterns in population dynamics generated by different types of model structures and parameter values. I shall now reverse the procedure and discuss how to guess the generating mechanism of an observed series. This is a much more demanding, challenging task, and we must admit that no method is as powerful or as widely applicable as we wish it to be. We need to utilize all available information, statistical and nonstatistical, such as life-history patterns, food-web structure and environmental influences. In this chapter I shall concentrate on statistical methods and discuss their merits and limitations. Many observed population time series may not be long enough to apply the statistical methods introduced here. Nonetheless, learning the methods will enrich perception of animal population dynamics.

In this chapter, I set up some basic theoritical models and illustrate the patterns that they generate. By such exercises, we gain knowledge essential for guessing the unknown generating mechanism of an observed pattern (to be discussed in Chapter 3).

A non-age-dependent model, describing the evolution of a bisexual population, is developed in this paper and applied to projecting an AIDS epidemic in a heterosexual population. Included in the formulation are frequency- and non-frequency-dependent rules of partnership formation as well as five states of HIV disease, affecting the probability of infection per sexual contact. Results from computer experiments, designed to study the development of an AIDS epidemic in a heterosexual population fed by single males with a 50% prevalence of HIV infection prior to becoming active in heterosexual partnerships, are reported. In these experiments, the only source of HIV infection for females was sexual contacts with infected males within partnerships. Data on the probability of infection per sexual contact with an infected partner and the number of sexual contacts per month were incorporated into the model. However, the numbers used for the initial population of singles, couples, and those becoming sexually active per month were hypothetical. Even though the prevalence of HIV infection among males entering heterosexual partnerships was high, after 30 years the projected prevalence of HIV infection among females ranged from about 10 to 15% depending in part of the expected duration of partnerships and on whether the frequency- or non-frequency-dependent model was used. In these experiments, solutions of the embedded, nonlinear, deterministic equations for the incidence of HIV infection and the cumulative number of deaths due to AIDS proved to be good measures of central tendency for the sample functions of the stochastic population process.

Phylloplane fungal communities are dynamic in time and space. The patterns of spatial and temporal change in the complement of microbes on the plant surface reflect successive balances among the population processes of immigration (arrival of propagules), emigration (physical loss or removal of propagules), growth (increase in biomass or number of propagules), and death for individual species. The individual population processes are in turn driven by physical forces and biotic interactions. Community study may focus on description of patterns, quantifying the population processes that determine community dynamics, or on the manner in which biotic and abiotic factors influence microbial population processes and therefore community dynamics.

The dynamics of vegetation are, above all, dynamics of populations and phytocoenoses. The aim of a study of vegetation dynamics is to gain a knowledge of the fate of a given phytocoenosis from its beginning to the achievement of dynamic equilibrium, and of the fate of the populations of species which built the phytocoenosis. In investigations on succession it is endeavoured to find out the fates of the successive phytocoenoses and to become acquainted with the mechanisms of species turnover (Table 5.1). It is suggested recently, that for gaining a better knowledge of the vegetation dynamics it is necessary to understand the population structure, its variability in time and space (Noble and Slatyer 1980, Houssard et al. 1980, Peet and Christensen 1980, Falinski 1980, 1986, White 1985, Falinska 1989a, b, Peart 1989a, b,c).

The objective of the present chapter is to sketch the elements of a general approach to testing probabilistic theories of micro spatial behaviour in terms of macro statistical observations. This approach, designated as most-probable-state (MPS) analysi., unifies many of the existing concepts and methods employed in maximum-entropy and minimum-information techniques. In addition, MPS-analysis provides a general analytical framework for testing a wide range of spatial models which extend well beyond these traditional techniques, and indeed, which have in some cases not even been recognised as probabilistic models at all (as illustrated in the examples of section 6 below).

The QuaTI System, sponsored by AMIB and DIXTAL, is developing an important database for intensive care in Brazil. The QuaTI System offers significant information for the participating ICUs for their continuous improvement, and gives data, although partial, of demographic characteristics of patients seen at participant ICUs. In our country, there is a steady decline in natality and mortality in the past 30 years, showing an irreversible and accelerated aging population process. Life expectancy at birth is 68 years and expectation at 60 years is more 18 years on average. In 1970, 5.07% of the population was older than 60; in 2000, 8.56%; and in 2002, 9.3%. In the South and Southeast, 10.2% of population are more than 60 years old.

SUMMARY It is known that in the limit of large population sizes certain classes of Markov population processes can be approximated by a deterministic process plus a noise term. In the usual approximation procedures the noise term is proportional to a Wiener process. This allows us to derive an estimation procedure for Markov population processes that are observed periodically at times A, 2A,.... 1. INTRODIUCTION In this paper we are concerned with the general problem of estimating parameters in a time homogeneous Markov population process. If the trajectory of the process can be observed in continuous time, then parameter estimation is straightforward in principle, for it is possible to write down the likelihood explicitly, and even if explicit maximum likelihood estimators for the required parameters are not obtainable they can be obtained numerically. We refer to Billingsley (1975) for a general approach to the problem, and to Moran (1951, 1953), Bartlett, Brennan & Pollock (1971), Bailey & Thomas (1971) and Keiding (1974) for detailed analyses in particular cases. In many situations, it is not possible to observe the process continuously but rather, observations are sampled at regular or irregular epochs. In this case the likelihood involves a product of factors, where a typical factor is the conditional distribution of the process at time t + A, say, given its value at time t. In most applications these distributions are intractable, and some approximation is required. Fortunately, there is a straightforward Gaussian approximation which is valid whenever the number of jumps between observation epochs is large. There have been a number of recent investigations into the asymptotic behaviour of Markov population processes under these conditions (Barbour, 1972, 1974; Kurtz, 1971; McNeil, 1972; McNeil & Schach, 1973; van Kampen, 1961, 1973). In ?? 2 and 3, below, we specify the class of processes to which our results will be usefully applied, and outline the general estimation procedure. In the remaining sections the procedure is illustrated using the example of the linear growth birth and death process, and an example is given in which a problem of identifiability of parameters arises because of the Gaussian approximation used.

Markovian contact processes (mcp's) include some of the commoner models for the spatial spread of population processes, such as the ‘simple’ and ‘general’ epidemics, percolation processes, and birth, death and migration processes. They are here set in a framework which relates them to each other, and to a particular basic model, the contact birth process (cbp); indeed they are defined as modifications of the cbp. The immediate advantage of this is that bounds for the velocity of the cbp, which can be obtained from the linear equation describing its expected numbers (provided the contact distribution has exponentially bounded tail) apply also to general mcp's. Existence of moments (and for some processes their time-derivatives) of St (θ), the distance to the furthest individual in an arbitrary direction θ, can also be deduced whenever the corresponding moment of the contact distribution exists. An important subclass consists of population-monotone mcp's, for which the distributions of St can be shown to be subconvolutive, so that the work of Hammersley (1974) can be applied to obtain convergence theorems for St /t; and these can be extended to convergence of the convex hull of the set of inhabited points. These results are particularly valuable because they apply to some non-linear processes, e.g. the simple epidemic. Some special results on the cbp (Section 6) emphasize the differences between linear and non-linear spatial stochastic processes. Although the paper is written throughout in terms of continuous-time processes, the results on subconvolutive distributions are actually more easily applied in the discrete-time case. (Conversely, the approach used in the accompanying paper by Biggins (pp. 62–84), which is written in terms of discrete-time processes, can be extended to deal also with continuous-time processes.)

Some random environment population processes are considered. Using a coupling argument, a criticality condition is obtained. The criticality parameter governs whether the process dies out or fluctuates about a positive finite constant, which is explicitly computed.

This paper considers a population process where individuals reproduce according to an age-dependent branching process and immigrants enter the population at the event epochs of an ergodic point process. A limit theorem is proven for what corresponds to the supercritical case, and the limit random variable is investigated.

Island biogeographic theory has been applied to the design of nature reserves. However, immigration, which is important in maintaining species equilibrium on true islands, will not contribute significantly to the maintenance of equilibrium on reserves in the future because of the disappearance of recolonisation sources. Consequently, extinction becomes the dominant population process, and the internal disturbance dynamics become the critical design feature of reserves. The design of reserves should be based on ‘minimum dynamic area’, the smallest area with a natural disturbance regime which maintains internal recolonisation sources and hence minimises extinctions. Determination of minimum dynamic area must be based on knowledge of disturbance-generated patch size, frequency, and longevity, and the mobilities of the preserved species. These features have not all been explicitly considered in the previous island biogeographic design recommendations.

We consider either a discrete time or an age-dependent branching process where the population consists of k types of individuals. Each time an individual is born, an action is chosen, for him which affects his lifetime, the number and types of his offspring, and the reward received. The problem of maximizing the expected reward is shown to be equivalent to a generalized Markov decision problem where the (k × k) transition, matrices are non-negative but not necessarily substochastic. It is shown that this branching process decision model can account for immigration, that it can be viewed as a controlled population process or infinite particle system, and that it has a number of applications including marketing and biology.

An Interactive Markov Chain is a population process in which each individuals's transitions depend on the population's distribution over the various states. We investigate a certain aspect of such process’ dynamics for a fixed population size. Conditions for convergence to steady‐state regardless of population size are provided.

A discrete time stochastic growth process for human populations accommodating marriages is introduced. In the introduction the motivations underlying the formulation of the model are discussed and the basic random functions of the process are defined in Sec. 2. The basic random functions of the model change because of deaths, marriages, the dissolutions of couples, and births, and Sects. 3, 4, 5, and 6 are devoted to stochastic processes describing these aspects of population dynamics. Finally, Sec. 7 is devoted to the derivation of a system of nonlinear renewal type equations satisfied by the mean functions of the process. The techniques used in the construction of the population process introduced in this paper are related to those used in the construction of multitype age-dependent branching processes.

This paper is concerned with a class of population growth processes in discrete time; the simple epidemic process is considered as a specific example. A Markov chain model is constructed and standard Markov methods are used to study the main biological concepts. A simple and explicit formula is obtained for the transient distribution of the population size. Then, the cost of the process is defined and the joint probability generating function of its components is derived. Finally, the results are extended to the case where the inter-transition periods are bounded i.i.d. random variables.

This paper presents some concepts and methodology essential for the analysis of population dynamics of univoltine species. Simple stochastic difference equations, comprised of endogenous and exogenous components, are introduced to provide a basic structure for density—dependent population processes. The endogenous component of a population process is modelled as a function of density in the past p generations, including the most recent. The exogenous component of the process consists of all density—independent components of the ecological factors involved, including enhance variations. The model is called a pᵗ ʰ order density—dependent process. For a successful analysis of a population process by the above model, it is important that the process be in a state of statistical equilibrium, or stationarity. The simplest notion of stationarity is introduced, and the average behavior of the process, under this assumption, discussed. The order of density dependence in the population process of a given species depends on its interaction with other species involved in the food web. Considering certain attributes of the food web, in particular the limited number of trophic levels, the pyrmaid of numbers, the linear linkages between closely interacting species, and niche separation among competing species, it is argued that the order of density dependence is probably not much higher than three. A second—order model is perhaps adequate in many practical cases. The dynamics of some lower order density—dependent processes are compared by simulations, with a view to showing the effect of density—dependent and density—independent components at different orders. Several types of density dependence are discussed. If a given factor influencing the temporal variation in density is by itself influenced by density, it is called "causally density—dependent," which may reveal itself by some degree of correlation with density. A density—independent factor, however, may also show some sort of correlation with density in the recent past. This is called "statistically density—dependent." Such statistical density dependence may be due to: (1) spurious correlation, (2) bias in an estimator of the correlation coefficient, (3) autocorrelations in the density—independent factor, and (4) an intriguing mathematical property of the stochastic process. Particularly because of the last two reasons, it is often difficult to distinguish, by correlation method, between causal and statistical density dependence. Distinction also exists between temporal and spatial density dependence, the latter not necessarily implying the involvement of the former. The importance of the distinction between these types of density dependence is discussed in relation to the data analysis and model building. A Statistical analysis of the effect of ecological factors on population dynamics is attempted. Since it is often difficult to determine, by correlation, the causally density—dependent structure of a population process under the influence of some unknown density—independent factors, it is suggested that we reverse the procedure to determine the effect of the density—independent factors first. To confirm the involvement of some suspected density—independent factors in the species dynamics, I propose several methods of correlation between annual fluctuations in some population parameters, such as density, rate of change in density, and their transforms, and those in suitable indices of the suspected factors. Merits, demerits, and limitations of these methods are also discussed. To simplify the arguments, the correlation models are set up first without stage division, and then are elaborated to those in which the whole generation span is divided into several life—cycle stages, so that life table information can be used effectively for the identification of the density—independent factors involved in each stage. A set of life tables of the spruce budworm, Choristoneura fumiferana (Clem.), is analyzed to provide an example of the application of the above concepts and methods. Concluding remarks include some notes on designing life table studies.

Algorithms for a stochastic population process, based on assumptions underlying general age-dependent branching processes in discrete time with time inhomogeneous laws of evolution, are developed through the use of a new representation of basic random functions involving birth cohorts and random sums of random variables. New algorithms provide a capability for computing the mean age structure of the process as well as variances and covariances, measuring variation about means. Four exploratory population projections, testing the implications of the algorithms for the case of time-homogeneous laws of evolution, are presented. Formulas extending mean and variance functions for unit population projections to an arbitrary initial population size are also presented. These formulas show that, in population processes with non-random laws of evolution, stochastic fluctuations about the mean function are negligible when initial population size is large. Further extensions of these formulas to the case of randomized laws of evolution suggest that stochastic fluctuations about the mean function can be significant even for large initial populations.

We consider the problem of number and weight distributions for breakage-mechanism branching processes whose break distributions are general. We derive a recursive relation between the expected empirical distribution after (n + 1) breaks and after n breaks, making use of length-biased sampling. Using this relation and the strong law of large numbers, we derive integro-differential equations for the asymptotic expected empirical distribution and its associated weight distribution. The mean of the asymptotic number distribution is derived using the integro-differential equation. We then provide approximate solutions to these equations and the moments of these approximations. Finally we apply these results to the case where the break distribution is a symmetric beta distribution.

We study the problem of maximum-likelihood estimation in some random-environment population models, defined through nonlinear ltǒ stochastic differential equations. It is shown that a criticality parameter can be estimated consistently. The asymptotic behavior of the estimators is analyzed, and a goodness-of-fit test is proposed. Cet article est consacre a l'etude des problemes d'estimation qui surviennent dans le cadre de modeles de population, lorsque ceux-ci sont definis dans un environnement aleatoire au moyen d'equations stochastiques differentielles non-lineaires d'itǒ. On demontre qu'un certain parametre critique peut ětre estime de facon consistante par la methode du maximum de vraisemblance. Le comportement asymptotique des estimateurs est analyse, et un test d'ajustement statistique est propose.

A population process {Zn } is defined, with offspring distribution depending on both population size and generation number. As is well known, in models where the offspring distribution depends only on the size, under most conditions extinction surely results. The model we now introduce would mean that if history has an influence on evolution of the population, it is possible to obtain models for which a probability as high as desired exists that the population size balances between two given values.