This book, authored by Benoît Perthame, is a comprehensive treatise on the mathematical study of partial differential equations (PDEs) that arise in biological contexts. The content is divided into several chapters, each focusing on different aspects of PDEs in biology: 1. **From Differential Equations to Structured Population Dynamics**: This section covers various biological models, including invasions, Lotka-Volterra systems, Hodgkin-Huxley and Fitzhugh-Nagumo equations, virus dynamics, epidemiology, and ecological competition. 2. **Adaptive Dynamics: An Asymptotic Point of View**: Here, the author discusses structured populations, selection principles, cannibalism, mutations, and the Hamilton-Jacobi equation, providing a rigorous derivation and numerical examples. 3. **Population Balance Equations: The Renewal Equation**: This chapter delves into the renewal equation, its properties, existence theory, regularity of solutions, and long-time asymptotics. It also explores more realistic models and extensions. 4. **Population Balance Equations: Size Structure**: This part focuses on size-structured models, including equal mitosis, asymmetric cell division, and other nonlinear examples. It covers existence, regularity, and comparison principles. 5. **Cell Motion and Chemotaxis**: This section examines how cells move, the Keller-Segel system, critical mass in dimension 2, radially symmetric solutions, and related chemotactic and angiogenetic systems. It also discusses kinetic equations and their diffusion limits. 6. **General Mathematical Tools**: This chapter provides foundational mathematical tools, including transport equations, generalized relative entropy, BV functions, Sobolev embeddings, and the Krein-Rutman theorem. The book aims to bridge the gap between biological models and mathematical analysis, emphasizing the importance of understanding the natural structure of the models rather than treating mathematics as a black box. The author acknowledges the contributions of various colleagues and students who have influenced the development of the content.This book, authored by Benoît Perthame, is a comprehensive treatise on the mathematical study of partial differential equations (PDEs) that arise in biological contexts. The content is divided into several chapters, each focusing on different aspects of PDEs in biology: 1. **From Differential Equations to Structured Population Dynamics**: This section covers various biological models, including invasions, Lotka-Volterra systems, Hodgkin-Huxley and Fitzhugh-Nagumo equations, virus dynamics, epidemiology, and ecological competition. 2. **Adaptive Dynamics: An Asymptotic Point of View**: Here, the author discusses structured populations, selection principles, cannibalism, mutations, and the Hamilton-Jacobi equation, providing a rigorous derivation and numerical examples. 3. **Population Balance Equations: The Renewal Equation**: This chapter delves into the renewal equation, its properties, existence theory, regularity of solutions, and long-time asymptotics. It also explores more realistic models and extensions. 4. **Population Balance Equations: Size Structure**: This part focuses on size-structured models, including equal mitosis, asymmetric cell division, and other nonlinear examples. It covers existence, regularity, and comparison principles. 5. **Cell Motion and Chemotaxis**: This section examines how cells move, the Keller-Segel system, critical mass in dimension 2, radially symmetric solutions, and related chemotactic and angiogenetic systems. It also discusses kinetic equations and their diffusion limits. 6. **General Mathematical Tools**: This chapter provides foundational mathematical tools, including transport equations, generalized relative entropy, BV functions, Sobolev embeddings, and the Krein-Rutman theorem. The book aims to bridge the gap between biological models and mathematical analysis, emphasizing the importance of understanding the natural structure of the models rather than treating mathematics as a black box. The author acknowledges the contributions of various colleagues and students who have influenced the development of the content.