Start with A biologist's guide to mathematical modeling in ecology and evolution by Sarah Otto and Troy Day. It was written for biologists who need the mathematics rather than mathematicians who need the biology, it builds recursion equations, differential equations and stability analysis from scratch with biological examples throughout, and its appendices double as a refresher on the calculus and linear algebra everything later assumes.
The prerequisite that actually binds in this field is comfort with differential equations and linear algebra — phase-plane analysis, eigenvalues, local stability — and not biology, which these books teach as they go. Get that wrong and the standard courses become impenetrable at chapter three; get it right and the biology looks after itself. So the order below supplies the mathematics first, works through the two standard first courses, specialises into ecology, epidemiology and evolution, and ends at the reference shelf.
Getting the mathematics you will need
Otto and Day is 748 pages and worth all of them. Alongside it, read the first two-thirds of Nonlinear dynamics and Chaos. Steven Strogatz's book is not biology at all but it is the standard source for the dynamical-systems machinery the rest of this path runs on, and it is unusually readable for a text at that level. A separate solutions manual circulates; you want the textbook.
The first course
Two different books here are catalogued under the identical display title Mathematical models in biology, which is the single most confusing thing about this subject's bibliography. The first is Elizabeth Allman and John Rhodes, from 2003: the most accessible of the standard courses, strongest on discrete-time models, matrix population models and molecular-evolution applications, and assuming only calculus. The second is Leah Edelstein-Keshet's classic 1988 course text, now a SIAM reprint, which is still the best single treatment of continuous-time models — Lotka-Volterra systems, chemostats, molecular events, spatially distributed systems. Allman and Rhodes first, then the step up to Edelstein-Keshet. Check the author when you buy either.
Essential Mathematical Biology is Nicholas Britton's compact Springer undergraduate course covering single-species and interacting populations, infectious disease, biological motion and pattern formation. It sits third as the consolidating read: the shortest route to seeing the shape of the whole field before you specialise.
Population and community ecology
Elements of Mathematical Ecology is Mark Kot's graduate text for the ecological half of the field — unstructured and structured population models, delay equations, spatial spread, harvesting theory. It assumes the stability analysis of the previous stage and pushes it hard. An Illustrated Guide to Theoretical Ecology by Ted Case is its conceptual complement: heavily figured, more biological in emphasis, and much better at explaining what the models are claiming about real communities. Read the two alongside each other rather than in sequence.
Epidemics
Infectious Diseases of Humans by Roy Anderson and Robert May founded modern mathematical epidemiology and remains the reference for the biology-plus-model reasoning behind the basic reproduction number, age structure and vaccination thresholds. Read it first in this stage for the conceptual grounding. Modeling Infectious Diseases in Humans and Animals by Matt Keeling and Pejman Rohani is the modern course text and the one to work through computationally — stochastic models, metapopulations, networks, with model code discussed throughout. It updates Anderson and May rather than replacing them.
Evolution, pattern, and the reference shelf
Evolutionary Dynamics is Martin Nowak's clear entry to evolutionary game theory, replicator dynamics, finite-population effects and the evolution of cooperation; read it before the harder book because Nowak explains what the mathematics is for. Evolutionary games and population dynamics by Josef Hofbauer and Karl Sigmund is that harder book — replicator and Lotka-Volterra equations proved properly, with the permanence results, at the rigorous end of the subject.
Mathematical Biology is the first volume of James Murray's two-volume treatise, subtitled An Introduction: single-species and interacting populations, enzyme kinetics, biological oscillators. Mathematical Biology II is the second, Spatial Models and Biomedical Applications, and it is the reason people cite Murray at all — reaction-diffusion equations, Turing pattern formation, animal coat markings, wound healing, tumour growth. It needs partial differential equations, which nothing earlier on this path covers properly. Both are reference works rather than books to read straight through.
Mathematical physiology closes the path. Keener and Sneyd is the other standard multi-volume reference — cellular physiology in one volume, systems physiology in the other, covering Hodgkin-Huxley, calcium dynamics, and cardiac and respiratory models. Open Library carries the set under a single record, so expect one catalogue entry to stand for both volumes. It is the deepest applied treatment here.
Follow the full ordered path here: How to Learn Mathematical Biology from Books, in Order.
