The bulletin of mathematical biophysics

, Volume 28, Issue 3, pp 333–345 | Cite as

A multidimensional birth process and its application to some problems in the dynamics of biological populations

  • Charles J. Mode
Article
  • 31 Downloads

Abstract

This paper is a sequel to a paper by the author entitled “Restricted Transition Probabilities and Their Applications to Some Problems in the Dynamics of Biological Populations” (Bull. Math. Biophysics, 1966,28, 315–331). The paper is divided into two parts. In part one some aspects of the maximum size attained by the population during a finite time interval are studied for the case the stochastic process underlying the evolution of the population is a birth process. Two interesting by-products emerge from the study presented in part one; namely a combinatorial method of finding solutions to the Kolmogorov differential equations in special cases, and secondly, a set of criteria for the optimum allocation of genotypes in the host population of a host-pathogen system. The optimum allocation of genotypes in the host population is a problem of practical importance in controlling plant pathogens. In part two the theory of restricted transition probabilities developed in the companion paper is applied in finding the distribution of the time to the appearance of the first mutation for the case of a two dimensional birth process. The distribution of the time to the appearance of the first mutation is of importance in understanding the role mutation plays in the evolution of a population, particularly in the pathogen population of a host-pathogen system.

Keywords

Optimum Allocation Host Population Companion Paper Pathogen Population Biological Population 

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Literature

  1. Feller, W. 1957.An Introduction to Probability Theory and Its Applications. New York: John Wiley and Sons, Inc.Google Scholar
  2. Loe've, M. 1960.Probability Theory. Princeton, New Jersey: Van Nostrand, Inc.Google Scholar
  3. Mode, C. J. 1966. “Restricted Transition Probabilities and Their Applications to Some Problems in the Dynamics of Biological Populations.”Bull. Math. Biophysics,28, 1966.Google Scholar

Copyright information

© N. Rashevsky 1966

Authors and Affiliations

  • Charles J. Mode
    • 1
  1. 1.Department of MathematicsMontana State UniversityBozeman

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