## Introduction

This cluster will introduce students to a wide variety of mathematical models used in biology. Students will learn how to construct mathematical models and use mathematical techniques to analyze these models to gain insight into biological phenomena. Mathematical topics covered include difference equations, differential equations, probability, network theory, and game theory. Biological topics covered range from ecology and epidemiology to physiology and cell biology. No prior knowledge of these mathematical modeling methods or biological topics is necessary, but a strong interest in mathematics and biology is essential. Computer programming is integral to cluster activities. While no prior experience in programming is required, a  willingness to learn is expected. In addition to the core courses described below, this cluster will have weekly guest lectures by UC Davis faculty working at the interface of mathematics and biology.

### Core Courses - Dynamics of Biological Systems: Patterns in Time and Space

Most biological systems are dynamic, producing fascinating patterns in time and space. Examples include outbreaks of epidemics, the development of spots on a leopard, the synchronization of flashing fireflies, and pathological rhythms in the heart and the brain. Identifying the mechanisms that underlie the “Spatio-temporal” dynamics of biological systems can not only lead to a better understanding of natural phenomena but also help us to design more effective interventions when necessary. Mathematical modeling plays a fundamental role in identifying these mechanisms. In this course, students will use mathematical analysis and write computer programs to explore the dynamics in models of a variety of biological processes and to gain insight into the mechanisms that produce complex temporal and spatial patterns.

### Networks and Games in Biology

Biological systems often involve many interacting components that form complex networks. These networks occur at all biological scales ranging from genes to ecosystems. Network theory provides a collection of mathematical and computational methods to understand the structure and function of these networks. When networks consist of interacting individuals, the structure of the network may determine (i) whether a disease spreads rapidly through the population and (ii) what genetically determined strategies are favored by natural selection. For (i), the epidemiological theory uses dynamic models to provide insights into the spread and control of diseases and the evolutionary emergence of new pathogens. For (ii), evolutionary game theory examines the long-term outcomes of interactions of competing strategies and has provided insights into the evolution of cooperation, social learning, animal conflicts, and language. In this course, students will learn some of the fundamentals of network theory, probability, disease dynamics, and evolutionary game theory. Students will develop mathematical models of evolutionary games, biological networks, and disease dynamics. They will develop computer programs and perform mathematical analyses of these models to understand the underlying biology.