1st Edition
Mathematical Biology Discrete and Differential Equations
This text serves as an exploration of the beautiful topic of mathematical biology through the lens of discrete and differential equations. Intended for students who have completed differential and integral calculus, this book allows students to explore topics such as bifurcation diagrams, nullclines, discrete dynamics, and SIR models for disease spread, which are often reserved for more advanced undergraduate or graduate courses. These exciting topics are sprinkled throughout the text alongside the more typical first and second order linear differential equations and systems of linear differential equations.
This class-tested text is written in a conversational, welcoming voice which should help invite students along as they discover the magic of mathematical biology and both discrete and differential equations. A focus is placed on examples with solutions written out step by step, including computational steps, with the goal of being as easy as possible for students to independently follow along.
Rich in applications, this text can be used for a semester-long course in either differential equations or mathematical biology. Alternatively, it can serve as a companion text for a two-semester sequence beginning with discrete time systems, extending through a wide array of topics in differential equations, and culminating in systems, SIR models, and other applications.
1 Modeling with Discrete Equations
1.1 Introduction to Difference Equations
1.1.1 Exponential, Linear Difference, and Logistic Models
1.1.2 Fishery Models
1.2 First Order Difference Equations and Fixed Points
1.2.1 Cobweb Analysis
1.2.2 Fixed Points and Stability
1.3 Solutions of Linear First Order Difference Equations
1.4 Solutions of Linear Homogeneous Second Order Difference Equations
1.4.1 Distinct Roots
1.4.2 Repeated Roots
1.4.3 Complex Roots
1.5 Systems of Difference Equations: Fixed Points and Stability
1.5.1 The Eigenvalue Approach
1.5.2 The Jury Condition
1.5.3 Discrete Interacting Species Models
1.6 Age-Structured Leslie Matrix Models
2 Introduction to Ordinary Differential Equations
2.1 Classification of ODEs and the Verification of their Solutions
2.2 Existence and Uniqueness of Solutions of Linear First Order ODEs
2.3 Vector Fields
3 Modeling with First Order ODEs
3.1 First Order ODEs and their Applications
3.1.1 Exponential, Migration, and Logistic Models
3.1.2 Newton's Law of Cooling
3.1.3 Mixing Models
3.1.4 Interacting Species
3.2 Autonomous Equations
3.3 Bifurcation Diagrams
3.4 Separable Equations
3.5 Integrating Factors
3.6 Exact Equations
4 Modeling with Second Order ODEs
4.1 The Wronskian and the Fundamental Set
4.2 The Characteristic Equation and Solutions of Linear Homogeneous Second Order ODEs
4.2.1 Distinct Real Roots
4.2.2 Repeated Roots
4.2.3 Complex Roots
4.3 Mechanical and Electrical Vibrations
4.3.1 Mechanical Vibrations: Unforced Springs
4.3.2 Electrical Vibrations: RLC Circuits
4.4 Reduction of Order
4.5 Linear Nonhomogeneous Second Order ODEs: Undetermined Coefficients
4.6 Linear Nonhomogeneous Second Order ODEs: Variation of Parameters
4.7 Forced Vibrations
5 Modeling with Systems of ODEs
5.1 Systems of ODEs and their Applications
5.1.1 Interacting Species
5.1.2 Parallel RLC Circuits
5.1.3 Multiple Tank Mixing Problems
5.2 Stability of Equilibria of Linear Systems using Eigenvalues
5.3 Solutions of Systems of Linear ODEs
5.3.1 Distinct Eigenvalues
5.3.2 Repeated Eigenvalues
5.3.3 Complex Eigenvalues
5.3.4 Solutions of Linear Modeling Problems
5.4 Solutions of Linear Nonhomogeneous Systems: Undetermined Coefficients
5.5 The Stability Criteria for Linear and Nonlinear ODEs
5.6 Phase Planes and Nullclines
6 SIR-Type Models
6.1 The Basic SIR Model with Birth and Death
6.1.1 Equilibria and Stability
6.1.2 The Basic Reproduction Number, R0
6.2 The SEIR Model
6.3 A Model with Vaccination
6.4 Sensitivity Analysis
Biography
Christina Alvey is an Associate Professor of Mathematics at Mount Saint Mary College in Newburgh, New York. She earned a PhD in Mathematics from Purdue University. Her current research investigates mathematical models in biology and epidemiology, as well as current trends and developments in the field of mathematics education.
Daniel Alvey is a data scientist at Accenture Federal Services. Prior to this, he was an Assistant Professor of Mathematics at Manhattan College in the Bronx, New York, and a Visiting Assistant Professor of Mathematics at Trinity College in Hartford, Connecticut. He earned his PhD in Mathematics from Wesleyan University where his research focused on homogeneous dynamics and metric number theory.
