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org/terms-privacy

Competition Models
in Population Biology
 CBMS-NSF REGIONAL CONFERENCE SERIES
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IN APPLIED MATHEMATICS
A series of lectures on topics of current research interest in applied mathematics under the direction of
the Conference Board of the Mathematical Sciences, supported by the National Science Foundation and
published by SIAM.

GARREIT BIRKHOFF, The Numerical Solution of Elliptic Equations

D. V. LINDLEY, Bayesian Statistics, A Review
R. S. VARGA, Functional Analysis and Approximation Theory in Numerical Analysis
R. R. BAHADUR, Some Limit Theorems in Statistics
PATRICK BILLINGSLEY, Weak Convergence of Measures: Applications in Probability
J. L. LIONS, Some Aspects of the Optimal Control of Distributed Parameter Systems
ROGER PENROSE, Techniques of Differential Topology in Relativity
HERMAN CHERNOFF, Sequential Analysis and Optimal Design
1. DURBIN, Distribution Theory for Tests Based on the Sample Distribution Function
SOL I. RUBINOW, Mathematical Problems in the Biological Sciences
P. D. LAX, Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock
Waves
I. J. SCHOENBERG, Cardinal Spline Interpolation
IVAN SINGER, The Theory of Best Approximation and Functional Analysis
WERNER C. RHEINBOLDT, Methods of Solving Systems of Nonlinear Equations
HANS F. WEINBERGER, Variational Methodsfor Eigenvalue Approximation
R. TYRRELL ROCKAFELLAR, Conjugate Duality and Optimization
SIR JAMES LIGHTHILL, Mathematical Biofluiddynamics
GERARD SALTON, Theory of Indexing
CATHLEEN S. MORAWETZ, Notes on Time Decay and Scattering for Some Hyperbolic Problems
F. HOPPENSTEADT, Mathematical Theories of Populations: Demographics, Genetics and Epidemics
RICHARD ASKEY, Orthogonal Polynomials and Special Functions
L. E. PAYNE, Improperly Posed Problems in Partial Differential Equations
S. ROSEN, Lectures on the Measurement and Evaluation of the Performance of Computing Systems
HERBERT B. KELLER, Numerical Solution of Two Point Boundary Value Problems
J. P. LASALLE, The Stability of Dynamical Systems - Z. ARTSTEIN, Appendix A: Limiting Equations
and Stability of Nonautonomous Ordinary Differential Equations
D. GOTTLIEB AND S. A. ORSZAG, Numerical Analysis of Spectral Methods: Theory and Applications
PETER J. HUBER, Robust Statistical Procedures
HERBERT SOLOMON, Geometric Probability
FRED S. ROBERTS, Graph Theory and Its Applications to Problems of Society
JURIS HARTMANIS, Feasible Computations and Provable Complexity Properties
ZOHAR MANNA, Lectures on the Logic of Computer Programming
ELLIS L. JOHNSON, Integer Programming: Facets, Subadditivity, and Duality for Group and Semi-
Group Problems
SHMUEL WINOGRAD, Arithmetic Complexity of Computations
J. F. C. KINGMAN, Mathematics of Genetic Diversity
MORTON E. GURTIN, Topics in Finite Elasticity
THOMAS G. KURTZ, Approximation of Population Processes

(continued on inside back cover)

 (continued from inside front cover)
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JERROLD E. MARSDEN, Lectures on Geometric Methods in Mathematical Physics

BRADLEY EFRON, The Jackknife, the Bootstrap, and Other Resampling Plans
M. WOODROOFE, Nonlinear Renewal Theory in Sequential Analysis
D. H. SATIINGER, Branching in the Presence of Symmetry
R. TEMAM, Navier-Stokes Equations and Nonlinear Functional Analysis
MIKLOS CSORGO, Quantile Processes with Statistical Applications
J. D. BUCKMASTER AND G. S. S. LUOFORD, Lectures on Mathematical Combustion
R. E. TARJAN, Data Structures and Network Algorithms
PAUL WALTMAN, Competition Models in Population Biology
S. R. S. V ARADHAN, Large Deviations and Applications
KlYOSI ITO, Foundations of Stochastic Differential Equations in Infinite Dimensional Spaces
ALAN C. NEWELL, Solitons in Mathematics and Physics
PRANAB KUMAR SEN, Theory and Applications of Sequential Nonparametrics
LASZLO Lov Asz, An Algorithmic Theory of Numbers, Graphs and Convexity
E. W. CHENEY, Multivariate Approximation Theory: Selected Topics
JOEL SPENCER, Ten Lectures on the Probabilistic Method
PAUL C. FIFE, Dynamics of Internal Layers and Diffusive Interfaces
CHARLES K. CHUI, Multivariate Splines
HERBERT S. WlLF, Combinatorial Algorithms: An Update
HENRY C. TUCKWELL, Stochastic Processes in the Neurosciences
FRANK H. CLARKE, Methods of Dynamic and Nonsmooth Optimization
ROBERT B. GARDNER, The Method of Equivalence and Its Applications
GRACE W AHBA, Spline Models for Observational Data
RICHARD S. VARGA, Scientific Computation on Mathematical Problems and Conjectures
INGRID DAUBECHIES, Ten Lectures on Wavelets
STEPHEN F. MCCORMICK, Multilevel Projection Methods for Partial Differential Equations
HARALD NIEDERREITER, Random Number Generation and Quasi-Monte Carlo Methods
JOEL SPENCER, Ten Lectures on the Probabilistic Method, Second Edition
CHARLES A. MICCHELLl, Mathematical Aspects of Geometric Modeling
ROGER TEMAM, Navier-Stokes Equations and Nonlinear Functional Analysis, Second Edition
GLENN SHAFER, Probabilistic Expert Systems
PETER J. HUBER, Robust Statistical Procedures, Second Edition
J. MICHAEL STEELE, Probability Theory and Combinatorial Optimization
Downloaded 10/22/24 to 103.181.227.57 . Redistribution subject to SIAM license or copyright; see https://epubs.siam.org/terms-privacy

Paul Waltman
Emory University
Atlanta, Georgia

Competition Models
in Population Biology

SOCIETY FOR INDUSTRIAL AND APPLIED MATHEMATICS

PHILADELPHIA
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Copyright ©1983 by the Society for Industrial and Applied Mathematics.

1098765432

All rights reserved. Printed in the United States of America. No part of this book may be
reproduced, stored, or transmitted in any manner without the written permission of the
publisher. For information, write to the Society for Industrial and Applied Mathematics,
3600 University City Science Center, Philadelphia, PA 19104-2688.

Library of Congress Card Catalog Number: 83-50665

ISBN 0-89871-188-6

S LBJIL is a registered trademark.

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Contents

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

Chapter 1
SINGLE POPULATION GROWTH MODELS
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2. Logistic growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
3. The simple chemostat . . . . . . . . . . . . . . . . . . . . . . . . . 6

Chapter 2
INTERACTING POPULATIONS
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2. The Lotka-Volterra competition model . . . . . . . . . . . . . . . . 14
3. Competition in the chemostat . .................. 18
4. A three-level fond chain . . . . ....
................... 25
5. Bifurcation from a limit cycle . . . . . . . . . . . . . . . . . . . . . 30
6. Two predators competing for a renewable resource . . . . . . . . . . 34
7. Two predators feeding on a prey growing in a chemostat . . . . . . . 41

Chapter 3
SOME DETERMINISTIC PROBLEMS IN GENETICS
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
2. Predator influence on the growth of a population with three genotypes 51
3. A model of cystic fibrosis . . . . . . . . . . . . . . . . . . . . . . . . 58
4. A parental selection problem . . . . . . . . . . . . . . . . . . . . . . 65

References .................................. 75

v
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Pretace

These lecture notes are from a series of lectures given at a regional conference at
Logan, Utah, in July of 1982. The audience had a wide variety of backgrounds, and
principal groups included those interested in biology or forestry, those interested in
ordinary differential equations but not particularly in biology, and those whose
major interest was in mathematical biology. The lectures began on familiar
territory to ease some mathematical anxiety and proceeded to new material. The
single theme was the use of relatively elementary ideas of dynamical systems to
answer questions of a biological nature, in particular questions as to the nature of
the eventual (asymptotic) behavior of populations given a relatively few hypotheses
as to the nature of their growth and their interactions.
The success of the conference depended very much on the enthusiasm of the
participants, the extremely strong supporting lectures and the work of the organiz-
ing committee, in particular the untiring efforts of Bob Gunderson and Bob
McKelvey. The very lovely surroundings of the campus of Utah State University
and the assistante of the Mathematics Faculty contributed a great deal to the
enjoyment of the conference.
Finally, I would like to acknowledge, with thanks, the efforts of K. Beck, H.
Freedman, J. A. Gatica and S. B. Hsu, who read and commented on portions of the
original manuscript.

vii