Mathematical Models in Biology
Serving as a basic introduction to concepts in deterministic biological modeling, this text shows how relatively simple mathematics can be applied to a variety of models to draw interesting conclusions.
TABLE OF CONTENTS Mathematical Models in Biology
List of Examples
Chapter 1: The Theory of Linear Difference Equations Applied to Population Growth
- Review of Complex Numbers
- Example
Chapter 2: Nonlinear Difference Equations
- Condition for Stability
- Comparison with a Continuous Equation
- Bifurcation Diagrams
- Summary and Applications of the Logistic Difference Equation (11)
- Derivation of Condition (32)
- Jury Test
Chapter 3: Applications of Nonlinear Difference Equations to Population Biology
- The Poisson Distribution and Escape from Parasitism
- Nicholson-Bailey Model: Equilibrium and Stability
- Example
Chapter 4: An Introduction to Continuous Models
- Terminology
- Remarks
- Stability Condition
Chapter 5: Phase-Plane Methods and Qualitative Solutions
- A Summary of Facts about Vector Functions (from Calculus)
- Example 5.10: Example 6
- An Interpretation of Eigenvectors
- The Shape of Trajectories Close to (N1, C2)
- Summary of the Chemostat Model
Chapter 6: Applications of Continuous Models to Population Dynamics
- Example 6.5: For Further Study
- Example 6.9: Gause: Empirical Tests of the Species-Competition Model
- The Routh-Hurwitz Criteria
- Routh-Hurwitz Criteria for k=2, 3, 4
- Example 3: (Levins, 1977)
- Example 4: (Jeffries, 1974)
- Primitive Model for a Viral Infection
- Assumption
- Qualitative Analysis of a SIRS Model: Epidemic with Temporary Immunity and No Vital Dynamics
- How Vaccinations Raise the Age of First Acquiring a Disease
Chapter 7: Models for Molecular Events
- Summary of Steps Leading to the Michaelis-Menten Rate Equation
Chapter 8: Limit Cycles, Oscillations, and Excitable Systems
- A Simple Electric Circuit
- Theorem 1: The Poincar -Bendixson Theorem
- Existence of Periodic Solutions
- Corollary of Bendixson s Criterion:
- Conclusions
- The Hopf Bifurcation Theorem for the Case n=2
- The Hopf Bifurcation Theorem for the Case n>2
- Calculations of Stability of...
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