The Resilience of Grassland Ecosystems (UMAP)

This module introduces students to the state-and-transition 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 state-and-transition 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 saddle-point equilibrium. A series of exercises direct students toward a qualitative phase-plane solution of the system and an analytical approximation of the stable manifold. Students also gain experience working with the numerical phase-plane plotter Dynasys, which ban be downloaded from the World Wide Web. A discussion section applies the approximated stable manifold to the real-world problem of controlling livestock numbers on public grazing land to reestablish more socially desirable plant varieites.

Table of Contents:

INTRODUCTION

MATHEMATICAL FORMULATION OF THE STATE-AND-TRANSITION THEORY

SOLUTION ANALYSIS

SUCCESSIONAL THRESHOLDS

ANALYTICAL APPROXIMATION OF THE STABLE MANIFOLD

DISCUSSION

SOLUTIONS TO THE EXERCISES

REFERENCES

ABOUT THE AUTHORS