Boundaries of Stability: A Potpourri of Dynamic Properties

There is more to a system than its equilibrium points. Associated with every stable equilibrium point (or stable limit cycle) is a region of state-space such that any unperturbed trajectory initiated in the region will stay within that region. This is called the region of stability. The boundaries of stability separate contiguous stability regions. An important property of system behavior near these boundaries is that a very small perturbation can move the state of a system across a boundary and transfer the system entirely from one region to another. The system's state cannot move back across the boundary without a subsequent outside perturbation. The performance of systems near their equilibrium points has been the focus of a considerable amount of investigation. Considerations of optimization, maximization, stable states are examples. The properties of systems far from equilibrium, and particularly near regions of instability (i.e., the boundaries) are not well known. The significant strategic problem that this paper hopes to address is to locate these boundaries and to determine system dynamics near them. On a tactical level, some approaches are suggested and their usefulness discussed

The Fourth Power in the Universe

This paper reviews a collection of non-ecological size distributions that have been observed in nature. The range of sizes covers 37 orders of magnitude. Ecologically significant size distributions are reported elsewhere. What can we hope to find in this collection? First we can ask if there are any generalities that exist, and if so, why? In the examples that follow one particular form of distribution is ubiquitous. What might be the ecological significance of this? First we can ask if these distributions are simply a result of some random statistical process. If not, then what are the specific mechanisms which countervene and lead to these distributions? Then we can compare these distributions with those of ecological significance. If the same types appear, then we may have a clue to an explanation. If some other types are found in animal communities, we must look for the special mechanisms that make ecological size distributions different

The Spruce Budworm Model: Some Questions, Corrections and Comments

The following items have been assembled from some notes I made while working my way through the Budworm Model. This is not intended to be an analysis of the model but merely some questions, corrections and comments. Many of these arise from apparent conflicts between the FORTRAN Coding, the description written by Jeff Stander and my perception of the real system

Explorations in Parameter Space

To date most of the mental and graphic thinking that we have been doing has been with a PHASE-SPACE outlook. (Relative to what follows we would more accurately call it a STATE-SPACE.) In principle, if we knew precisely what was going on everywhere in state-space we would know all that we need to know. But since we don't we must draw our characteristic spirals and express the behavior in qualitative terms. What I am proposing here is to supplement (not replace) the state-space approach with another -- PARAMETER-SPACE

Fail-Safe vs. Safe-Fail Catastrophes

This paper is meant to serve two purposes. First, to extend the usefulness of catastrophe theory as a tool to aid our perception of a partially known world. This theory is a newly emerged branch of topology and, as such, begins to fill a large void in our arsenal of qualitative analytical tools. It is not appropriate for all important and interesting situations, particularly those requiring precise numerical results. But it is hoped that it can provide an important missing element for our environmental management tool kit. The second purpose is to report upon some deliberations precipitated by a recent paper of Beer and Casti (1975). We shall follow, to some degree, their development. We shall also borrow some of their examples and terminology in order to emphasize some fundamentally different strategies for managing unexpected events

Towards a Structural View of Resilience

The result of resilience is persistence: the maintenance of certain characteristic behavioral properties in the face of stress, strain and surprise. But the origins of this resilient behavior lie in the structure of the systems which concern us. Our need as policy analysts may only be one of comparative measures: Which system is more resilient? But as active designers -- as engineers, managers, or responsible policy advisors -- we need to be able to say what mechanisms or relationships make a system resilient, and what actions we can take to make it more or less so. This need for a causal view of resilience led us to a search for persistence-promoting (or "resilient") mechanisms and relationships in a variety of natural and man-made systems

Analysis of a Compact Predator-Prey Model. The Basic Equations and Behavior

This paper is the first of a series dealing with the analysis of a compact, relatively uncomplicated predator-prey model. Here, only the basic equations are given and a selected subset of system behavior illustrated. Written documentation concerning this model and its analytic investigation are being documented as completed to speed communication among interested parties. As this model is becoming a focus for several methodological and conceptual discussions, the need has arisen for a concise description of the equations

Sodium storage facility software configuration control plan

This document describes the plan for ensuring that the SSF Trace Heat Software will be available for use whenever that facility is opened for the use of draining sodium from FFTF

The Application of Catastrophe Theory to Ecological Systems

Catastrophe theory is a new field in mathematical topology that allows the formulation of comprehensive qualitative systems models which have previously eluded rigorous mathematical formulation. Because the models have a topological foundation, many seemingly dissimilar phenomena can be related to a common underlying topological structure. The properties of that structure can then be studied in a convenient form and the conclusions related back to the original problem. This paper provides an introduction to catastrophe theory and defines the principal conditions required for its application. The basic properties of bimodality, discontinuity (catastrophe), hysteresis, and divergence are defined and illustrated using the simplest structures of the theory. The application of catastrophe theory to ecology is illustrated with the spruce budworm system of eastern Canada. With a minimum of descriptive information about the budworm system, a qualitative catastrophe theory model is hypothesized. This model is rich in its ability to provide predictions on the global behavior of the system. To further check and refine the assumptions of this qualitative model, an existing detailed simulation model is analyzed from the perspective of catastrophe theory. The simulation indeed exhibits a basic underlying structure in agreement with the previously hypothesized model. In this instance catastrophe theory provides a consistent framework with which to analyze and interpret the results of the simulation. These interpretations are not at variance with the first rough qualitative model based only on a small set of descriptive information

Biology of the Budworm Model

This paper describes the natural history of a simulation model. The model was constructed to illuminate the determinants of the dynamic behavior for a pest/forest system with particular reference to the New Brunswick Budworm experience. It cannot reproduce the real system in all its richness. Rather, it is meant to be an analog of the important links between the budworm and its principal host, balsam fir