Mathematical Biology - Department of Mathematics, HKUST

Mathematical Biology

Lecture notes for MATH 4333

Jeffrey R. Chasnov

The Hong Kong University of Science and Technology Department of Mathematics Clear Water Bay, Kowloon Hong Kong

Copyright c 2009?2016 by Jeffrey Robert Chasnov This work is licensed under the Creative Commons Attribution 3.0 Hong Kong License. To view a copy of this license, visit or send a letter to Creative Commons, 171 Second Street, Suite 300, San Francisco, California, 94105, USA.

Preface

What follows are my lecture notes for Math 4333: Mathematical Biology, taught at the Hong Kong University of Science and Technology. This applied mathematics course is primarily for final year mathematics major and minor students. Other students are also welcome to enroll, but must have the necessary mathematical skills.

My main emphasis is on mathematical modeling, with biology the sole application area. I assume that students have no knowledge of biology, but I hope that they will learn a substantial amount during the course. Students are required to know differential equations and linear algebra, and this usually means having taken two courses in these subjects. I also touch on topics in stochastic modeling, which requires some knowledge of probability. A full course on probability, however, is not a prerequisite though it might be helpful.

Biology, as is usually taught, requires memorizing a wide selection of facts and remembering them for exams, sometimes forgetting them soon after. For students exposed to biology in secondary school, my course may seem like a different subject. The ability to model problems using mathematics requires almost no rote memorization, but it does require a deep understanding of basic principles and a wide range of mathematical techniques. Biology offers a rich variety of topics that are amenable to mathematical modeling, and I have chosen specific topics that I have found to be the most interesting.

If, as a UST student, you have not yet decided if you will take my course, please browse these lecture notes to see if you are interested in these topics. Other web surfers are welcome to download these notes from

and to use them freely for teaching and learning. I welcome any comments, suggestions, or corrections sent to me by email (jeffrey.chasnov@ust.hk). Although most of the material in my notes can be found elsewhere, I hope that some of it will be considered to be original.

Jeffrey R. Chasnov Hong Kong May 2009

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Contents

1 Population Dynamics

1

1.1 The Malthusian growth model . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 The Logistic equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 A model of species competition . . . . . . . . . . . . . . . . . . . . . . . 6

1.4 The Lotka-Volterra predator-prey model . . . . . . . . . . . . . . . . . . 7

2 Age-structured Populations

15

2.1 Fibonacci's rabbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.2 The golden ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.3 The Fibonacci numbers in a sunflower . . . . . . . . . . . . . . . . . . . 18

2.4 Rabbits are an age-structured population . . . . . . . . . . . . . . . . . 21

2.5 Discrete age-structured populations . . . . . . . . . . . . . . . . . . . . 22

2.6 Continuous age-structured populations . . . . . . . . . . . . . . . . . . 25

2.7 The brood size of a hermaphroditic worm . . . . . . . . . . . . . . . . 28

3 Stochastic Population Growth

35

3.1 A stochastic model of population growth . . . . . . . . . . . . . . . . . 35

3.2 Asymptotics of large initial populations . . . . . . . . . . . . . . . . . . 38

3.2.1 Derivation of the deterministic model . . . . . . . . . . . . . . . 40

3.2.2 Derivation of the normal probability distribution . . . . . . . . 42

3.3 Simulation of population growth . . . . . . . . . . . . . . . . . . . . . . 45

4 Infectious Disease Modeling

49

4.1 The SI model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4.2 The SIS model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4.3 The SIR epidemic disease model . . . . . . . . . . . . . . . . . . . . . . 51

4.4 Vaccination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

4.5 The SIR endemic disease model . . . . . . . . . . . . . . . . . . . . . . . 54

4.6 Evolution of virulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

5 Population Genetics

59

5.1 Haploid genetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

5.1.1 Spread of a favored allele . . . . . . . . . . . . . . . . . . . . . . 61

5.1.2 Mutation-selection balance . . . . . . . . . . . . . . . . . . . . . 62

5.2 Diploid genetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

5.2.1 Sexual reproduction . . . . . . . . . . . . . . . . . . . . . . . . . 64

5.2.2 Spread of a favored allele . . . . . . . . . . . . . . . . . . . . . . 67

5.2.3 Mutation-selection balance . . . . . . . . . . . . . . . . . . . . . 69

5.2.4 Heterosis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

5.3 Frequency-dependent selection . . . . . . . . . . . . . . . . . . . . . . . 72

5.4 Linkage equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

5.5 Random genetic drift . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

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