Mathematical Biology

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