The students I work with frequently ask what math they need to know. So here’s a list of topics that are particularly useful for a mathematical biologist / quantitative ecologist to know, followed by a list of my favorite math and modeling books.
Foundations
Linear algebra: This is at the very top of the list. Linear algebra is useful in many, many areas including population dynamics, quantitative genetics, statistics, and dynamical systems theory. It is probably more useful than calculus. But for some reason, biology students rarely have any background.
Calculus: Most biology students I talk to had calculus at some point – so you probably just need to refresh a bit. Derivatives and optimization will certainly come in handy.
Dynamics
Ordinary differential equations: Mathematical descriptions of how things change with time or space. Most dynamical models are framed in terms of differential or difference equations.
Advanced: Nonlinear dynamics and chaos, partial differential equations, optimal control theory, stochastic processes
Statistics
Generalized linear models: If you have not had a course in statistics, this is the place to start. Most of the models biologists use for analyzing experimental and ecological data can be written as generalized linear models.
Advanced: Multivariate analysis, nonlinear models, nonparametric regression, time series analysis, state-space models.
Recommended reading
These are some books that I found useful as I was trying to learn mathematical modeling and theoretical ecology.
Constructing and analyzing models
A Biologist’s Guide to Mathematical Modeling, S. Otto and T. Day
This book starts at the beginning and covers a lot of theoretical ground.
Matrix Population Models, H. Caswell
A comprehensive guide to constructing and analyzing models with age or stage-structure. Thought slightly dated, it is still a solid place to start.
Evolutionary Theory: Mathematical and Conceptual Foundations, S.H. Rice
This is a terrific read for those interested in evolution. It does a fabulous job of connecting branches of evolutionary theory that are typically treated independently.
Mathematical Biology, J.D. Murray
This two volume set is a bit more difficult than the others, but well worth the challenge.
Fitting models to data
The Ecological Detective, R. Hilborn and M. Mangel
This is a terrific entry point into quantitative ecology written by one of my favorite people.
Ecological Models and Data in R, B. Bolker
This is another good place to start, replete with example R code to get you going.
Models for Ecological Data, J. Clark
This book is slightly more challenging but covers a lot of good material including an entree into state-space modeling. Well worth the read.
Stochastic Processes
Probability: The Logic of Science, E.T. Jaynes
This is an incredibly well-argued introduction to Bayesian reasoning and maximum entropy. I get a huge kick out of Jaynes’ writing style. This is probably the only math book that I read cover to cover just for fun.
Stochastic Processes, Sheldon Ross
Ross has written books on several topics, all of which are clear and relatively student-friendly.
Stochastic Calculus for Finance I & II, Steven Shreve
Although the examples are from finance, Shreve has a lucid writing style that makes this material accessible.
Nonlinear dynamics and chaos
Nonlinear Dynamics and Chaos, S. Strogatz
A very nice intro to nonlinear dynamics written in an engaging style by one of the most prolific writers of popular math books. Mostly continuous time dynamics.
Chaos: An Introduction to Dynamical Systems, K. Alligood, T. Sauer, and J. Yorke.
An excellent introduction, primarily covering discrete time dynamics.