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Über dieses Buch

This book develops the mathematical tools essential for students in the life sciences to describe interacting systems and predict their behavior. From predator-prey populations in an ecosystem, to hormone regulation within the body, the natural world abounds in dynamical systems that affect us profoundly. Complex feedback relations and counter-intuitive responses are common in nature; this book develops the quantitative skills needed to explore these interactions.

Differential equations are the natural mathematical tool for quantifying change, and are the driving force throughout this book. The use of Euler’s method makes nonlinear examples tractable and accessible to a broad spectrum of early-stage undergraduates, thus providing a practical alternative to the procedural approach of a traditional Calculus curriculum. Tools are developed within numerous, relevant examples, with an emphasis on the construction, evaluation, and interpretation of mathematical models throughout. Encountering these concepts in context, students learn not only quantitative techniques, but how to bridge between biological and mathematical ways of thinking.

Examples range broadly, exploring the dynamics of neurons and the immune system, through to population dynamics and the Google PageRank algorithm. Each scenario relies only on an interest in the natural world; no biological expertise is assumed of student or instructor. Building on a single prerequisite of Precalculus, the book suits a two-quarter sequence for first or second year undergraduates, and meets the mathematical requirements of medical school entry. The later material provides opportunities for more advanced students in both mathematics and life sciences to revisit theoretical knowledge in a rich, real-world framework. In all cases, the focus is clear: how does the math help us understand the science?

Inhaltsverzeichnis

Frontmatter

Chapter 1. Modeling, Change, and Simulation

Abstract
In the 1920s, ecologists began to study the populations of two Arctic species, lynx (a predator) and snowshoe hares (their prey) (Figure 1.1).
Alan Garfinkel, Jane Shevtsov, Yina Guo

Chapter 2. Derivatives and Integrals

Abstract
Let’s focus on the change vector \(X'\).
Alan Garfinkel, Jane Shevtsov, Yina Guo

Chapter 3. Equilibrium Behavior

Abstract
A major clue to the behavior of dynamical systems is given by the existence and location of equilibrium points.
Alan Garfinkel, Jane Shevtsov, Yina Guo

Chapter 4. Nonequilibrium Dynamics: Oscillation

Abstract
We now have to make a detour out of mathematics into science. We have to ask, what are the fundamental kinds of behaviors that can be seen in a scientific system, and what do they look like mathematically?
Alan Garfinkel, Jane Shevtsov, Yina Guo

Chapter 5. Chaos

Abstract
We began our study of dynamics by looking at equilibrium behavior, modeled by stable equilibrium points, or as we learned to call them, point attractors.
Alan Garfinkel, Jane Shevtsov, Yina Guo

Chapter 6. Linear Algebra

Abstract
In this chapter, we will be studying linear functions in n dimensions. \(f: \mathbb {R}^n \longrightarrow \mathbb {R}^n\).
Alan Garfinkel, Jane Shevtsov, Yina Guo

Chapter 7. Multivariable Systems

Abstract
In the previous chapter, we used our knowledge of linear algebra to give us insights into linear differential equations. The key to this approach is that the differential equation is viewed as a vector field, that is, as a function \( V: \mathbb {R}^n \longrightarrow \mathbb {R}^n \).
Alan Garfinkel, Jane Shevtsov, Yina Guo

Backmatter

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