About these notes¶
These are notes for MATH 456, Mathematical Modeling, given by Andreas Buttenschoen at the University of Massachusetts Amherst. They introduce nonlinear dynamics through models from cell and population biology. The emphasis is on building a model, reducing it to something simple enough to analyze, and checking what the reduction cost.
Contents¶
What Is a Model. What a mathematical model is, and how estimating the size of each term tells us which terms to keep.
From Newton’s Second Law to a Force Balance. A cell’s equation of motion, its scaling, and the singular limit in which inertia is dropped: two time scales, inner and outer solutions, and the error of the reduction.
Flows on the Line. One-dimensional nonlinear ODEs: balance equations, the phase line, steady states and their stability, and the transcritical and saddle-node bifurcations, with examples from antibiotic resistance.
Background¶
Calculus and an introductory course in ordinary differential equations. No biology is assumed: the biology each model needs is explained where it is used.
Computation¶
The figures are produced by Python code (numpy, scipy, matplotlib) that
is included on each page and can be expanded.
License¶
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.
