Management of a Buffalo Herd (UMAP)

A unit that uses linear algebra in the topic of buffalo herd management. This module shows students how to: 1) calculate linear difference equations on a computer; 2) identify overdetermined linear equations and decide when they have a solution; 3) set up and solve matrix equations from word problems; 4) sum finite geometric series for matrix case; 5) describe an application that uses linear equations to model birth, aging and death in a population, specifically, detail use of a matrix to transform that population through time; 6) differentiate between matrix level and entry level calculations and give examples where both are helpful; 7) describe contexts where polynomial functions of a matrix inevitably arise; 8) discuss major strengths and weaknesses of a linear model in a nonlinear reality; and 9) simulate several policies of harvesting by making varied use of a computer simulation.

Table of Contents:

1. INTRODUCTION
1.1 What Harvest Should You Take?
1.2 What Herd Should You Start With?
1.3 Wildlife Management
1.4 The Task Ahead

2. THE MODEL
2.1 Herd Components and Their Survival Rates
2.2 Basic Model Equations
2.3 The After-Harvest Model in Vector and Matrix Notation
2.4 The Before-Harvest Model
2.5 Survival Rates Would be Larger on a Ranch
2.6 Exercises and Computer Projects

3. ASSUMPTIONS, STRENGTHS, AND WEAKNESSES OF THE MODEL
3.1 The Model's Basic Strengths
3.2 Limitations of the Model

4. APPLICATIONS: ESTABLISHING A HERD
4.1 A Herd and Harvest That Continue Year After Year
4.2 Calculation of (M - I) Inverse
4.3 A Steady Harvest Plus Controlled Growth of the Herd
4.4 Exercises
4.5 Mathematical Insights
4.6 An Efficiently Small Herd
4.7 Exercises

5. APPLICATIONS: CALCULATING THE HARVEST
5.1 Steady Annual Harvests and Herd
5.2 Constant Harvests From a Growing Herd
5.3 Exercises

6. REFERENCES

7. ACKNOWLEDGEMENTS

8. ANSWERS TO EXERCISES