Sensitivity and Elasticity in Population Models, in Markov Chains, and in Graphs

University Of Connecticut, Storrs CT

Investigators

Abstract

Problems are posed in three areas of application of linear algebra in which the theories and methods of nonnegative matrices and of generalized inverses of matrices play very significant roles. The areas are (i) population models in mathematical biology, (ii) the algebraic measurement of the connectivity of graphs, and (iii) finite homogeneous Markov chains in probability and statistics. In all these areas nonnegative matrices and/or their associated M--matrices are used to represent the basic model according to weights or probabilities connecting between states or vertices. The sensitivities and elasticities of the eigenvalues and eigenvectors, as well as other properties, of such matrices are used to analyze and understand the behavior of the models and their response to changes in the input data. Generalized inverses are needed because they enter into expressions for the derivatives, when they exist, of the eigenvalues and eigenvectors of matrices as functions of the entries and thus generalized inverses help in measuring the effects of various types of perturbation on the models. Many physical phenomena can be modeled by mathematics, and mathematics is a primary tool for understanding and predicting the behavior of the phenomena. Take the simple-looking problem of a population, be it of humans or of animals or of germs, that is divided according to age. At each age the females may have a different birth rate and at each age members of the population may have a different probability of survival to the next age group. If one knows the initial distribution of the population into the different age groups, then can one predict the short and long term evolution trends of the population?! Could it be that the population can reach in time a situation that is called an equilibrium? This is a situation in which the sizes of each age group may continue to change, but the relative size of each age group to the other age groups does not change. There are many problems of this sort which the fields of mathematics called Linear Algebra and Matrix Theory can help model and solve. In this project matrix methods are applied to investigate the behavior of three models, one from population growth, one from a networking problem, and one from physical systems that is governed by probability rather than by surety.