# How animals get their stripes

### From NW-DCSD

## 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.