Dynamical System Models in the Life Sciences and Their Underlying Scientific Issues

Broadly speaking, there are two general approaches to teaching mathematical modeling: 1) the case study approach, and 2) the method based approach (that teaches mathematical techniques with applications to relevant mathematical models). This text emphasizes instead the scientific issues for modeling different phenomena. For the natural or harvested growth of a fish population, we may be interested in the evolution of the population, whether it reaches a steady state (equilibrium or cycle), stable or unstable with respect to a small perturbation from equilibrium, or whether a small change in the environment would cause a catastrophic change, etc. Each scientific issue requires an appropriate model and a different set of mathematical tools to extract information from the model. Models examined are chosen to help explain or justify empirical observations such as cocktail drug treatments are more effective and regenerations after injuries or illness are fast-tracked (compared to original developments).

Volume I of this three-volume set limits its scope to phenomena and scientific issues that are modeled by ordinary differential equations (ODE). Scientific issues such as signal and wave propagation, diffusion, and shock formation involving spatial dynamics to be modeled by partial differential equations (PDE) will be treated in Vol. II. Scientific issues involving randomness and uncertainty are examined in Vol. III.

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Contents:

• Mathematical Models and the Modeling Cycle

• Growth of a Population:

  • Evolution and Equilibrium
  • Stability and Bifurcation

• Interacting Populations:

  • Linear Interactions
  • Nonlinear Autonomous Interactions
  • HIV Dynamics and Drug Treatments
  • Index Theory, Bistability and Feedback

• Optimization:

  • The Economics of Growth
  • Optimization over a Planning Period
  • Modifications of the Basic Problem
  • Boundary Value Problems are More Complex

• Constraints and Control:

  • "Do Your Best" and the Maximum Principle
  • Chlamydia Trachomatis
  • Genetic Instability and Carcinogenesis
  • Mathematical Modeling Revisited

• Appendices:

  • First Order ODE
  • Basic Numerical Methods
  • Assignments

Readership: Undergraduates in mathematical biology, mathematical modeling of dynamical systems, optimization and control, viral dynamics (infectious diseases), oncology.
Key Features:

• When mathematical techniques new to the readers are introduced and taught in this volume, they are initiated and motivated by specific questions on biological phenomena so that readers can see the need for the relevant mathematics in the life sciences
• Furthermore, nearly all mathematical methods introduced to analyze specific models in the volume can be traced back to, and explained in terms of basic calculus and/or matrix operations familiar to the readers
• By asking different questions about the same biological phenomenon, a familiar model is routinely extended or modified to a different level of complexity to motivate the need for more mathematics
• 15 pages in color