mathematical-biology - Mathematical Biology Lecture notes...

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Mathematical Biology Lecture notes for MATH 365 Jeffrey R. Chasnov The Hong Kong University of Science and Technology
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The Hong Kong University of Science and Technology Department of Mathematics Clear Water Bay, Kowloon Hong Kong Copyright c 2009, 2010 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 http://creativecommons.org/licenses/by/3.0/hk/ or send a letter to Creative Commons, 171 Second Street, Suite 300, San Francisco, California, 94105, USA.
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Preface What follows are my lecture notes for Math 365: Mathematical Biology , taught at the Hong Kong University of Science and Technology. This applied mathe- matics course is primarily for final year mathematics major and minor students. Other students are also welcome to enroll, but must have the necessary mathe- matical skills. My main emphasis is on mathematical modeling, with biology the sole ap- plication 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 re- quired to know differential equations and linear algebra, and this usually means having taken two courses in these subjects. I also touch on topics in stochas- tic 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 http://www.math.ust.hk/˜machas/mathematical-biology.pdf and to use them freely for teaching and learning. I welcome any comments, sug- gestions, or corrections sent to me by email ([email protected]). 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 iii
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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 . . . . . . . . . . . . . . 8 2 Age-structured Populations 15 2.1 Fibonacci’s rabbits . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.1.1 The golden ratio Φ . . . . . . . . . . . . . . . . . . . . . . 17 2.2 Rabbits are an age-structured population . . . . . . . . . . . . . 18 2.3 Discrete age-structured populations . . . . . . . . . . . . . . . . . 20 2.4 Continuous age-structured populations . . . . . . . . . . . . . . . 22 2.5 The brood size of a hermaphroditic worm . . . . . . . . . . . . . 26 3 Stochastic Population Growth 33 3.1
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