# Mathematical Modeling in Biological Systems

## 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 in order to gain insight to 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 the biological topics is necessary, but a strong interest in both mathematics and biology is essential. 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. Students also get hands on experience with a lab used in the introductory biology core at Davis.

## Core Courses

### Dynamics of Biological Systems: Patterns in Time and Space

Most biological systems are dynamic, producing fascinating patterns in both 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 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 computer simulation and mathematical analysis 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), epidemiological theory uses dynamical models to provide insights into the spread and control of diseases as well as 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. Computer simulations and mathematical analysis will be used to explore evolutionary games, the structure and function of biological networks, and disease dynamics.