An elementary approach to the population projection-matrix, to the population reproductive value, and to related topics in the mathematical theory of population growth

Pop-Inference: An educational application to evaluate statistical differences among populations

Pop-Inference is an educational tool designed to help teaching of hypothesis testing using populations. The application allows for the statistical comparison of demographic parameters among populations. Input demographic data are projection matrices or raw demographic data. Randomization tests are used to compare populations. The tests evaluate the hypothesis that demographic parameters differ among groups of individuals more that should be expected from random allocation of individuals to populations. Confidence intervals for demographic parameters are obtained using the bootstrap. Tests may be global or pairwise. In addition to tests on differences, one-way life table response experiments (LTRE) are available for random and fixed factors. Planned (a priori) comparisons are possible. Power of comparison tests is evaluated by constructing the distribution of the test statistic when the null hypothesis is true and when it is false. The relationship between power and sample size is explored by evaluating differences among populations at increasing population sizes, while keeping vital rates constant.

Class Structure, Demography, and Selection: Reproductive-Value Weighting in Nonequilibrium, Polymorphic Populations

In natural populations, individuals of a given genotype may belong to different classes. Such classes can represent different age groups, developmental stages, or habitats. Class structure has important evolutionary consequences because the fitness of individuals with the same genetic background may vary depending on their class. As a result, demographic transitions between classes can cause fluctuations in the trait mean that need to be removed when estimating selection on a trait. Intrinsic differences between classes are classically taken into account by weighting individuals by class-specific reproductive values, defined as the relative contribution of individuals in a given class to the future of the population. These reproductive values are generally constant weights calculated from a constant projection matrix. Here, I show for large populations and clonal reproduction that reproductive values can be defined as time-dependent weights satisfying dynamical demographic equations that depend only on the average between-class transition rates over all genotypes. Using these time-dependent demographic reproductive values yields a simple Price equation where the nonselective effects of between-class transitions are removed from the dynamics of the trait. This generalizes previous theory to a large class of ecological scenarios, taking into account density dependence, ecological feedbacks, arbitrary strength of selection, and arbitrary trait distributions. I discuss the role of reproductive values for prospective and retrospective analyses of the dynamics of phenotypic traits.

Leslie matrix models

A review of the Leslie matrix model theory and its literature 1941-1987 is presented. The point of view is that of a mathematician who focuses on the parts of the theory which are relevant to demography. Works of a decidedly applied nature are not dealt with.

Choosing category size in a stage projection matrix

A basic problem associated with choosing category size in a stage projection matrix is described. "Errors of estimation" are large if the category size is chosen too small and "errors of distribution" are large if the category size is chosen too large. An approximate technique of balancing these two error types is suggested as a solution to this dilemma.