
Product No.
99775 
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The Resilience of Grassland Ecosystems (UMAP)
Kevin Cooper

Mathematics Topic: Differential Equations 
Application Areas: Biology, ecology 
Prerequisites: Introduction to ordinary differential equations covering phaseplane solutions. 

 ©1999 by COMAP, Inc.  The UMAP Journal 20.1  17 pages 

This module introduces students to the stateandtransition theory explaining the succession of plant species on grassland and to the concept of successional thresholds partitioning plant states into those gravitating toward socially desirable or socially undesirable plant compositions over time. Students are shown how the stateandtransition theory is formulated in the mathematical ecology literature as a system of two autonomous differential equations, and how a successional threshold is defined by the stable manifold to an interior saddlepoint equilibrium. A series of exercises direct students toward a qualitative phaseplane solution of the system and an analytical approximation of the stable manifold. Students also gain experience working with the numerical phaseplane plotter Dynasys, which ban be downloaded from the World Wide Web. A discussion section applies the approximated stable manifold to the realworld problem of controlling livestock numbers on public grazing land to reestablish more socially desirable plant varieites.
Table of Contents:
INTRODUCTION
MATHEMATICAL FORMULATION OF THE STATEANDTRANSITION THEORY
SOLUTION ANALYSIS
SUCCESSIONAL THRESHOLDS
ANALYTICAL APPROXIMATION OF THE STABLE MANIFOLD
DISCUSSION
SOLUTIONS TO THE EXERCISES
REFERENCES
ABOUT THE AUTHORS



