Population Projection Matrices Research Articles

Elementary Mathematical Ecology.

The Exponential and Logistic Equations. Population Projection Matrices. Discrete Models of Population Changes. Life Tables I. Life Tables II. The Analysis of Spatial Pattern. Interspecific Competition. Predator-Prey Theory. Species Diversity. Dynamics of Multiple Species Assemblages. Index.

On Cohen's stochastic generalization of the strong ergodic theorem of demography

Cohen has generalized the classical strong ergodic theorem of demography to a stochastic setting. In this setting population projection matrices are chosen according to some homogeneous Markov chain. If this Markov chain converges to the same long-run distribution regardless of its starting point, then one can define an induced Markov chain on the product space of projection matrices and age structure vectors that also has a long-run distribution independent of its starting point. The present paper gives more natural conditions under which Cohen's result holds.

On Cohen's stochastic generalization of the strong ergodic theorem of demography

Cohen has generalized the classical strong ergodic theorem of demography to a stochastic setting. In this setting population projection matrices are chosen according to some homogeneous Markov chain. If this Markov chain converges to the same long-run distribution regardless of its starting point, then one can define an induced Markov chain on the product space of projection matrices and age structure vectors that also has a long-run distribution independent of its starting point. The present paper gives more natural conditions under which Cohen's result holds.

The use of population projection matrices in cell kinetics

Population projection matrices are applied to the study of cycling populations of cells. Simple generalizations of the method are used to study the transit time distributions through each phase: G 1, S, G 2 and M. It is thereby possible to derive many of the common results of cell kinetics without recourse to LaPlace transforms. It is further shown that the method may be used with great ease in numerical generation of FLM curves. Two simulated curves are compared with different distributions of transit times through each phase and it is shown that the differences are experimentally undetectable.

An experimental study of weak ergodicity in human populations

In demography, weak ergodicity is usually taken to mean the tendency of an age distribution to “forget” its past, or, equivalently, the essential dependence of current age distribution on only a recent history of age-specific vital rates. We have been interested in providing a more complete mathematical description of the processes underlying weak ergodicity, and decided to begin our inquiry with an experimental approach. Our initial work has been with the discrete formulation of population dynamics, using 10 × 10 matrices for closed populations of human females as experimental data. The findings of our initial experiments are that a sequence of products of human population projection matrices converges towards the sequence defined by its positive, or Perron-Frobenius, eigenstructure, and that, when appropriately defined, sequences both of left and of right eigenvectors converge to constant vectors: left eigenvector sequences converge “forward” from a fixed point in time, while right eigenvector sequences converge “backward” from a given time point. In practice, this means that proportional age structure can be determined from a knowledge only of recent or “late” vital rates, while the relative contribution of initial age structure to eventual population size can be determined with a knowledge of only “early” vital rates in the sequence. For 10 × 10 human population projection matrices, products of 30 or more matrices appear always to contain closely convergent sequences of right and left eigenvectors, and adequate approximations to limiting values usually can be made using about 20 matrices for each limiting eigenvector. Because convergence is not monotone, and because patterns of convergence are specific to the structure of each product, it appears that stronger results will require strong conditions on the sequence of matrices in the product.