How animals get their stripes

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Contents

Overview

J. D. Murray (1981) proposed a mathematical model for the biological process
that produces animal coat patterns. The model consists of a system of nonlinear
partial differential equations. Approximating these equations by a finite
difference method requires the solution of large systems of algebraic equations
by the common technique known as Gaussian elimination. However, these systems
are highly susceptible to large round-off error without the technique of
Gaussian elimination with partial pivoting. This module is to give students
in mathematics and computer science an interdisciplinary project from biology
for which the chosen computational algorithm must handle round off error.
The final program executes for ten to fifteen minutes while steady state
is being reached before the user knows whether spots or stripes have formed.
They must make many experimental runs – some which do not yield a coat pattern
and some which do. Thus, students have experience with a sufficiently complex
computational algorithm.

Relationship to NWDCSD Goals

The module involves an interdisciplinary project (biology, mathematics,
computer science) that brings computational thinking to non-computer
science students as well as computer science students.

Developer

Kathie Yerion, Gonzaga University

Subscribed Users

Kathie Yerion, Gonzaga University

Potential Users

Teachers of mathematics courses like numerical analysis, differential
equations, linear algebra, and mathematical biology.
Teachers of computer science courses like algorithms.

Assessment

Survey Links

Instructor and Student Survey Links

Learning Objectives

  • The students will learn the techniques of approximating differential equations
by finite differences and approximating the resulting algebraic systems.
  • They will compare the techniques of “Gaussian elimination” and
“Gaussian elimination with partial pivoting.”
  • They will observe that “Gaussian elimination” causes sufficient round-off error
to ruin approximations, even in this day of powerful computers.
  • They will experience a computational method that takes some time to complete,
even in this day of fast computers.
  • They will do experimental runs and analysis of the results, seeing experimental
mathematics and computer science.

Learning Outcomes

Through the assignments in the module, we want the students to be able

  • to approximate derivatives by finite differences
  • to perform the techniques of Gaussian elimination and
Gaussian elimination with partial pivoting on small systems
  • to use programs that implement these two techniques of Gaussian elimination
  • to understand the systems that result from the finite difference method
for the nonlinear partial differential equations
  • to understand the computational method
  • to use the program that implements the computational method for the
formation of animal stripes and spots
  • to experience that there are inputs for the program that do not yield
“good” results and ones that do.

Dependencies / Prerequisites

Module Versions and Dependencies

Status

Description

Module Length

2 weeks (Shortened version 1 week)

Resources

A finite difference method for modeling the formation of animal coat patterns paper

Assignments For this Module

media:Outline of Assignments for Module on Animal.doc

Community Comments

If you reviewed this module but decided to not try and use it, we would appreciate feedback as to why. Please send an email to Dr. Kathie Yerion (yerion@gonzaga.edu) with any feedback and/or questions.

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