Ergodicity of age structure in populations with Markovian vital rates. II. General states

1977 ◽  
Vol 9 (01) ◽  
pp. 18-37 ◽  
Author(s):  
Joel E. Cohen
Keyword(s):  
Markov Chain ◽  
Age Structure ◽  
Joint Distribution ◽  
Sample Path ◽  
Initial Distribution ◽  
Transition Function ◽  
Leslie Matrix ◽  
Vital Rates ◽  
Closed Population ◽  
Leslie Matrices

The age structure of a large, unisexual, closed population is described here by a vector of the proportions in each age class. Non-negative matrices of age-specific birth and death rates, called Leslie matrices, map the age structure at one point in discrete time into the age structure at the next. If the sequence of Leslie matrices applied to a population is a sample path of an ergodic Markov chain, then: (i) the joint process consisting of the age structure vector and the Leslie matrix which produced that age structure is a Markov chain with explicit transition function; (ii) the joint distribution of age structure and Leslie matrix becomes independent of initial age structure and of the initial distribution of the Leslie matrix after a long time; (iii) when the Markov chain governing the Leslie matrix is homogeneous, the joint distribution in (ii) approaches a limit which may be easily calculated as the solution of a renewal equation. A numerical example will be given in Cohen (1977).

Ergodicity of age structure in populations with Markovian vital rates. II. General states

1977 ◽  
Vol 9 (1) ◽  
pp. 18-37 ◽  
Author(s):  
Joel E. Cohen
Keyword(s):  
Markov Chain ◽  
Age Structure ◽  
Joint Distribution ◽  
Sample Path ◽  
Initial Distribution ◽  
Transition Function ◽  
Leslie Matrix ◽  
Vital Rates ◽  
Closed Population ◽  
Leslie Matrices

The age structure of a large, unisexual, closed population is described here by a vector of the proportions in each age class. Non-negative matrices of age-specific birth and death rates, called Leslie matrices, map the age structure at one point in discrete time into the age structure at the next. If the sequence of Leslie matrices applied to a population is a sample path of an ergodic Markov chain, then: (i) the joint process consisting of the age structure vector and the Leslie matrix which produced that age structure is a Markov chain with explicit transition function; (ii) the joint distribution of age structure and Leslie matrix becomes independent of initial age structure and of the initial distribution of the Leslie matrix after a long time; (iii) when the Markov chain governing the Leslie matrix is homogeneous, the joint distribution in (ii) approaches a limit which may be easily calculated as the solution of a renewal equation. A numerical example will be given in Cohen (1977).

Age-structured models in a random environment

2019 ◽  
pp. 194-213
Author(s):  
Louis W. Botsford ◽  
J. Wilson White ◽  
Alan Hastings
Keyword(s):  
Time Series ◽  
Environmental Variability ◽  
Extinction Risk ◽  
Population Viability Analysis ◽  
Leslie Matrix ◽  
Vital Rates ◽  
Viability Analysis ◽  
Practical Concern ◽  
Probability Of Extinction ◽  
Age Structured

Most ecological populations exist in a randomly fluctuating environment, and these fluctuations influence vital rates, thus changing population dynamics. These changes are the focus of this chapter. The primary practical concern about environmental variability is the possibility that it could cause a population to go extinct, so the chapter describes several approaches to estimating the probability of extinction. The first is the small fluctuations approximation (SFA) to describe the growth of a population with a randomly varying Leslie matrix. The results reveal that randomly varying populations grow more slowly on average than the equivalent deterministic population. Further applications of the SFA examine how correlated variation in different vital rates affects the probability of extinction, when variability is too large to use the SFA, and how it has been applied to population time series. Finally, several other approaches to estimating extinction risk—also known as population viability analysis—are compared.

Limit theorems for sums of chain-dependent processes

1974 ◽  
Vol 11 (3) ◽  
pp. 582-587 ◽  
Author(s):  
G. L. O'Brien
Keyword(s):  
Markov Chain ◽  
Limit Theorem ◽  
Limit Theorems ◽  
Law Of Large Numbers ◽  
Initial Distribution ◽  
Stationary Case ◽  
Strong Law ◽  
Iterated Logarithm ◽  
Large Numbers ◽  
Dependent Processes

Chain-dependent processes, also called sequences of random variables defined on a Markov chain, are shown to satisfy the strong law of large numbers. A central limit theorem and a law of the iterated logarithm are given for the case when the underlying Markov chain satisfies Doeblin's hypothesis. The proofs are obtained by showing independence of the initial distribution of the chain and by then restricting attention to the stationary case.

Death and Life in a Colonial Immigrant City: A Demographic Analysis of Philadelphia

1977 ◽  
Vol 37 (4) ◽  
pp. 863-889 ◽  
Author(s):  
Billy G. Smith
Keyword(s):  
Age Structure ◽  
Elevated Level ◽  
Colonial America ◽  
Demographic Characteristics ◽  
Demographic Analysis ◽  
Vital Rates ◽  
Urban Center ◽  
Death Rates ◽  
Primary Responsibility

This study analyzes the demographic characteristics of a previously neglected area in colonial America—the urban center. Growth, birth, and death rates in Philadelphia between 1720 and 1775 are estimated using a variety of sources. Immigration, smallpox, economic vacillations, and a skewed age structure are attributed primary responsibility in determining the level of and changes in Philadelphia's vital rates. The elevated level of these rates is evident in a comparison with vital rates in Andover and Boston, Massachusetts, and Nottingham, England.

scholarly journals Convergence of Conditional Metropolis-Hastings Samplers

2014 ◽  
Vol 46 (2) ◽  
pp. 422-445 ◽  
Author(s):  
Galin L. Jones ◽  
Gareth O. Roberts ◽  
Jeffrey S. Rosenthal
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Markov Chain Monte Carlo ◽  
Diffusion Process ◽  
Sample Path ◽  
Discrete Observations ◽  
Entire Sample ◽  
Bayesian Inferences ◽  
Monte Carlo Algorithms

We consider Markov chain Monte Carlo algorithms which combine Gibbs updates with Metropolis-Hastings updates, resulting in a conditional Metropolis-Hastings sampler (CMH sampler). We develop conditions under which the CMH sampler will be geometrically or uniformly ergodic. We illustrate our results by analysing a CMH sampler used for drawing Bayesian inferences about the entire sample path of a diffusion process, based only upon discrete observations.

scholarly journals The importance of life history and population regulation for the evolution of social learning

2020 ◽  
Vol 375 (1803) ◽  
pp. 20190492 ◽  
Author(s):  
Dominik Deffner ◽  
Richard McElreath
Keyword(s):  
Life History ◽  
Social Learning ◽  
Age Structure ◽  
Life Histories ◽  
Population Regulation ◽  
Individual Learning ◽  
Vital Rates ◽  
Strategic Learning ◽  
Cognition And Culture ◽  
Exemplary Model

Social learning and life history interact in human adaptation, but nearly all models of the evolution of social learning omit age structure and population regulation. Further progress is hindered by a poor appreciation of how life history affects selection on learning. We discuss why life history and age structure are important for social learning and present an exemplary model of the evolution of social learning in which demographic properties of the population arise endogenously from assumptions about per capita vital rates and different forms of population regulation. We find that, counterintuitively, a stronger reliance on social learning is favoured in organisms characterized by ‘fast’ life histories with high mortality and fertility rates compared to ‘slower’ life histories typical of primates. Long lifespans make early investment in learning more profitable and increase the probability that the environment switches within generations. Both effects favour more individual learning. Additionally, under fertility regulation (as opposed to mortality regulation), more juveniles are born shortly after switches in the environment when many adults are not adapted, creating selection for more individual learning. To explain the empirical association between social learning and long life spans and to appreciate the implications for human evolution, we need further modelling frameworks allowing strategic learning and cumulative culture. This article is part of the theme issue ‘Life history and learning: how childhood, caregiving and old age shape cognition and culture in humans and other animals’.

The distributions of cluster functionals of extreme events in a dth-order Markov chain

2000 ◽  
Vol 37 (01) ◽  
pp. 29-44 ◽  
Author(s):  
Seokhoon Yun
Keyword(s):  
Markov Chain ◽  
Weak Convergence ◽  
Extreme Events ◽  
Joint Distribution ◽  
Asymptotic Distributions ◽  
Absolutely Continuous ◽  
Stationary Markov Chain ◽  
Order Markov Chain ◽  
Distributional Assumptions

The paper concerns the asymptotic distributions of cluster functionals of extreme events in a dth-order stationary Markov chain {X n , n = 1,2,…} for which the joint distribution of (X 1,…,X d+1) is absolutely continuous. Under some distributional assumptions for {X n }, we establish weak convergence for a class of cluster functionals and obtain representations for the asymptotic distributions which are well suited for simulation. A number of examples important in applications are presented to demonstrate the usefulness of the results.

Age-Grouping, Time-Specific Life-Tables, and Predictive Population Models

2021 ◽  
pp. 362-383
Author(s):  
Peter A. Henderson
Keyword(s):  
Developmental Stage ◽  
Food Availability ◽  
Life Tables ◽  
Population Models ◽  
Leslie Matrix ◽  
Population Modelling ◽  
Temperature Dependent ◽  
Animal Groups ◽  
Leslie Matrices ◽  
Time Specific

This chapter describes techniques to create life-tables for animals whose generations overlap widely. Age-grouping is a prerequisite for these methods, which have been most widely applied to vertebrate populations. Age cannot be inferred from the developmental stage without reference to the environment. The speed of development may be temperature-dependent or influenced by factors such as oxygen and food availability. The methods for ageing animal groups, including invertebrates, fish, reptiles, amphibians, birds, and mammals, are reviewed. Time-specific life-tables, population modelling, and Leslie matrices are described. R code to analyze Leslie matrix dynamics is presented.

Sex Preferences and Two-Sex Models

1998 ◽  
Author(s):  
C. Y. Cyrus Chu
Keyword(s):  
Population Dynamics ◽  
Technical Difficulty ◽  
Leslie Matrix ◽  
State Variables ◽  
Vital Rates ◽  
Male And Female ◽  
Demographic Models ◽  
Sex Preferences ◽  
Males And Females ◽  
Almost All

The demographic models I reviewed in previous chapters are all one-sex models, in which the sex referred to is usually the female. This setting can be justified if we assume either that the life-cycle vital rates (as functions of state variables) for both sexes are the same or that the population dynamics are determined by one sex alone, independent of the possibly relative abundance of the other sex. However, at least for human population, neither assumption is valid. The ratio of newborn girls and newborn boys is close to one, but is less than one for almost all countries in the world. The age-specific mortality rates of women are also lower than those of men worldwide. This is called sexual dimorphism in the demography literature. Such a dimorphism makes the study of two-sex models indispensable. If we look at the male and female vital rates, we find that the differences are small. Despite this small difference, population dynamics derived solely from male vital rates and those derived solely from female vital rates will show ever-increasing differences with the passage of time. Furthermore, because the intrinsic growth rates derived from male and female lines, respectively, are distinct, we cannot avoid the undesirable conclusion that, if we do not incorporate males and females in a unified model, eventually the sex ratio will become either zero or infinity, which is never the case in reality. This is the inconsistency we have to overcome while dealing with population models with two sexes. Another technical difficulty with two-sex modeling has to do with the irreducibility of the state-transition matrix. I mentioned in chapters 2 and 3 that in an age-specific one-sex model, because people older than a particular age, say β, are not fertile anymore, the age group older than β is an absorbing set; hence, our focus of population dynamics can be restricted to the age set [0, β]. This is why we can transform the n × n Leslie matrix to a Lolka renewal equation. In a two-sex model, however, there does not exist a common upper bound for the reproduction of both sexes, for a male older than β can marry a female younger than β and become fertile again.

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