Stochastic Population Theories

ISBN-13: 9783540070108 / Angielski / Miękka / 1974 / 114 str.

D. Ludwig; M. Levandowsky

Opis

These notes serve as an introduction to stochastic theories which are useful in population biology; they are based on a course given at the Courant Institute, New York, in the Spring of 1974. In order to make the material. accessible to a wide audience, it is assumed that the reader has only a slight acquaintance with probability theory and differential equations. The more sophisticated topics, such as the qualitative behavior of nonlinear models, are approached through a succession of simpler problems. Emphasis is placed upon intuitive interpretations, rather than upon formal proofs. In most cases, the reader is referred elsewhere for a rigorous development. On the other hand, an attempt has been made to treat simple, useful models in some detail. Thus these notes complement the existing mathematical literature, and there appears to be little duplication of existing works. The authors are indebted to Miss Jeanette Figueroa for her beautiful and speedy typing of this work. The research was supported by the National Science Foundation under Grant No. GP-32996X3. CONTENTS I. LINEAR MODELS . . . 1 1. The Poisson Process . . . 1 2. Birth and Death Processes 5 2. 1 Linear Birth Process 5 2. 2 Linear Birth and Death Process . . . 7 2. 3 Birth and Death with Carrying Capacity . . . . 16 3. Branching Processes . . . 20 3. 1 Continuous Time . . . . . ."

Spis treści

I. Linear Models.- 1. The Poisson Process.- 2. Birth and Death Processes.- 2.1 Linear Birth Process.- 2.2 Linear Birth and Death Process.- 2.3 Birth and Death with Carrying Capacity.- 3. Branching Processes.- 3.1 Continuous Time.- 3.2 Galton-Watson Process.- II. Epidemics.- 1. Reed-Frost Model.- 1.1 Deterministic Version.- 1.2 Two Methods for the Study of the Reed-Frost Model.- 1.3 Backward Equation.- 2. Qualitative Theory for the General Stochastic Epidemic.- 2.1 Approximation by Birth and Death Process.- 2.2 Deterministic Theory (Kermack and McKendrick).- 2.3 Diffusion Approximation.- 2.4 Practice Problem.- 2.5 Gaussian Approximation for General Diffusion Equations.- III. Diffusion Equations.- 1. Introduction.- 2. Derivation of the Forward and Backward Equation.- 3. Random Genetic Drift.- 4. Solutions which are Valid for Small Time.- 5. Random Drift and Selection.- 6. Wright’ s Formula for Equilibrium Distributions.- IV. Dynamical Systems Perturbed by Noise.- 1. One Species.- 2. Several Species-Gradient Fields.- 3. Ray Method for General Systems.