Spatial Optimization in Ecological Applications

By John Hof and Michael Bevers

-- Basic & Applied Ecology

• Language: English
• Publisher: Columbia University Press
• Release date: Jan 22, 2005
• ISBN: 9780231500739

Book preview

Spatial Optimization in Ecological Applications

Complexity in Ecological Systems

Complexity in Ecological Systems Series

Timothy F. H. Allen and David W. Roberts, Editors Robert V. O’Neill, Adviser

Robert Rosen

Life Itself: A Comprehensive Inquiry Into the Nature, Origin, and Fabrication of Life

Timothy F. H. Allen and Thomas W. Hoekstra

Toward a Unified Ecology

Robert E. Ulanowicz

Ecology, the Ascendent Perspective

John Hof and Michael Bevers

Spatial Optimization for Managed Ecosystems

David L. Peterson and V. Thomas Parker, Editors

Ecological Scale: Theory and Applications

Robert Rosen

Essays on Life Itself

Robert H. Gardner, W. Michael Kemp, Victor S. Kennedy, and John E. Petersen, Editors

Scaling Relations in Experimental Ecology

S. R. Kerr and L. M. Dickie

The Biomass Spectrum: A Predator-Prey Theory of Aquatic Production

Spatial Optimization in Ecological Applications

John Hof and Michael Bevers

Columbia University Press

New York

Columbia University Press

Publishers Since 1893

New York     Chichester, West Sussex

cup.columbia.edu

© 2002 Columbia University Press

All rights reserved

E-ISBN 978-0-231-50073-9

Library of Congress Cataloging-in-Publication Data

Hof, John G.

Spatial optimization in ecological applications / John Hof and Michael Bevers.

        p. cm.—(Complexity in ecological systems)

Includes bibliographical references (p.).

ISBN 0-231-12544-5 (alk. paper)—ISBN 0-231-12545-3 (alk. paper)

    1. Spatial ecology—Mathematical models. I. Bevers, Michael. II. Title. III. Complexity in ecological systems series.

QH541.15.S62 H65 2002

577’.01’5118—dc21                               2001047717

A Columbia University Press E-book.

CUP would be pleased to hear about your reading experience with this e-book at [e-mail].

To our parents: Gerrit, Ardyce, Guy, and Irene

CONTENTS

Preface

1        Introduction

Perspective

Organization

Methods

Traditional Linear Programming Approaches

Our Approach

PART I      Simple Proximity Relationships

2        Sedimentation

Formulation

Case Example

Results

3        Stormflow Management

Formulation

Case Example

Results

4        Natural Regeneration in Any-Aged Forest Management

Formulation

Case Example

Results

Solvability

Solution Patterns

Incomplete Initial Seeding

5        Combining Simulation with Optimization: Habitat Placement for the Northern Spotted Owl

The Northern Spotted Owl

The Model

The Connectivity (f) Function

The Carrying Capacity (g) Function

Results

Comparison of the Simulation Model with the Optimization Model

Evaluation of the Plan

PART II      Reaction–Diffusion Models

6        Characteristics of the Discrete Reaction–Diffusion Model

Experiments

Populations Isolated in a Single Patch

Effects of Patch Size

Effects of Patch Shape

Effects of Intrapatch Heterogeneity

Populations Occupying Multiple Patches

Fragmentation of Contiguous Habitat

Island Systems

Effects of Patch Shape on Colonization

Discussion

7        The Basic Model: Habitat Placement for the Black-Footed Ferret

The Black-Footed Ferret

The Model

Ferret Reintroduction in South Dakota

Spatial Definition

Ferret Dispersal

Net Population Growth Rate

Ferret Releases

Ferret Carrying Capacity

Results

8        Population-Dependent Dispersal: Habitat Placement for the Black-Tailed Prairie Dog

The Black-Tailed Prairie Dog

The Model

Results

9        Topography-Based Dispersal: Habitat Location for the Western Prairie Fringed Orchid

The Western Prairie Fringed Orchid

The Model

General Formulation

Specific Formulation

Landscape

Parameters

Results

Initial Seed Dispersal Assumptions

Conservative Seed Dispersal Assumptions

10      Habitat Edge Effects

Formulation

Case Example

Results

Forage Constraint Effects

Dynamic Scheduling

Management Scale Effects

PART III      Control Models

11      Strategies for Controlling Exotic Pests

Formulation

Case Example

Results

12      Strategies for Controlling Wildfire

Formulation

Case Example

Results

Extensions

PART IV      Using Optimization to Develop Hypotheses About Ecosystems

13      Multiscaled Ecological Limiting Factors

Formulation

Case Example

Results

Reproduction and Dispersal Effects

Heterogeneous Habitat Effects

Discussion

Appendix

14      Carbon Fixation in Trees as an Optimization Process

Formulation

Case Example

Results

Basic Solution

Sensitivity Analysis

Discussion

15      Postscript

References

Index

PREFACE

When we completed Spatial Optimization for Managed Ecosystems (Hof and Bevers 1998), we immediately identified two shortcomings. First, although we suggested the use of optimization models for developing theoretical hypotheses about ecosystems, we presented no examples, so the point may have been lost. In this book, we include one section devoted to this topic and point to theoretical hypotheses throughout. Second, and more important, we feared that some readers of our last book might conclude that capturing ecological spatial relationships in optimization models requires the use of esoteric integer and nonlinear solution methods, implying that these relationships can be captured only heuristically or in small toy models. In this book, we focus on capturing ecological relationships across a landscape with pragmatic optimization models that can be applied to real-world problems. We use linear programming primarily but also include two formulations for integer programming that are integer-friendly. The model in chapter 14 is nonlinear but is still readily solvable.

Using linear programming makes it possible to include many thousands of choice variables and many thousands of constraints and still be confident of being able to solve problems with widely available software. To capture ecological relationships in linear programs, we think of the problem in terms of discrete difference equations, combined with the production system activity analysis applied in, for example, traditional timber harvest scheduling models. Even with this approach, we must often make simplifying assumptions about ecosystem function. The alternative approach would be to start with a more complex (e.g., nonlinear and nonconvex) model and investigate the use of heuristic procedures to approach optimized solutions (Boston 1999; Jager and Gross 2000). Just as it was difficult to choose between the X-15 and the Mercury program as the best approach for early space exploration (Von Braun et al. 1985), it is difficult to say which approach to optimizing ecological systems has more promise. Our emphasis here is on preserving optimality and exploring how much ecosystem function we can capture. We also emphasize solvability of large problems, including real-world case studies.

The methods applied are quantitative, and the book relies heavily on mathematical presentations. However, the reader can skip over the math and still get the general idea. Most of the chapters focus on examples. The book is written primarily for ecologists, resource economists, and management scientists. It is written for advanced undergraduate and graduate students and practicing scientists. As with our previous book, we hope that ecologists will forgive our many simplifying assumptions for the sake of gaining a different perspective on ecological problems.

This book has strong ties to our previous books (Hof 1993; Hof and Bevers 1998) but actually repeats little of the material found in them. It is put together largely from material we have published elsewhere in smaller pieces. To our coauthors of these previous works (identified with a footnote at the beginning of each chapter) we are gratefully indebted. We also thank the U.S. Department of Agriculture Forest Service, especially Fred Kaiser, Denver Burns, Tom Hoekstra, Marcia Patton-Mallory, John Toliver, and Brian Kent, for supporting this work. Many thanks to Dorothy Martinez and Penny Williams, who worked so hard to complete the word processing through many revisions; to Jill Heiner for computer programming support; to Tony Baltic for analytical and GIS support; to Scott Powell and Beth Galleher for additional GIS support; and to Joyce Van De Water and Mike Knowles for technical assistance with the figures.

1

INTRODUCTION

Perspective

Turner (1989) broadly defines landscape ecology as the study of the effect of landscape pattern on ecological processes. In this book, we present ideas and methods for taking ecological processes into account in optimizing landscape pattern through the strategic placement of management actions over time and space. Webster’s Third International Dictionary defines optimize as to make as perfect, effective, or functional as possible. Chiang (1974:244) refers to optimization as simply the quest for the best. He notes that "the first order of business is to delineate an objective function in which the dependent variable represents the object of maximization or minimization and in which the set of independent variables indicates the objects whose magnitudes … [we] … can pick and choose.… We shall therefore refer to the independent variables as choice variables. The essence of the optimization process is simply to find the set of values of the choice variables that will yield the desired extremum of the objective function" (1974:244). In unconstrained optimization problems, the choice variables are independent in the sense that the decision made regarding one variable does not impinge on the choices of the remaining variables. In constrained optimization, a set of constraints is included, each of which limits the value of some function of the choice variables to be less than or equal to, equal to, or greater than or equal to a specified constant. All the models in the chapters that follow represent constrained optimization problems.

Why would we want to optimize a landscape pattern? It is important to note that most of the work in this volume applies to managed ecosystems, where human impact is taken as a given and the problem centers around managing that impact. If a given level of human activity is adverse to the ecosystem, it makes sense to minimize its impact. Likewise, if we are in a position to help create positive impacts but with limited resources, then it seems reasonable to maximize that impact subject to the constraints implied by the limited resources. When a model is desired to predict impacts or consequences, simulation approaches are a logical choice. However, if it is desired to prescribe management activities, optimization approaches can implicitly evaluate huge numbers of options and allow tradeoff analyses that might otherwise be impossible. For more reading on the use of optimization in the general problems of multiple-resource management, see Hof (1993).

The most common technique for solving quantitatively defined constrained optimization problems is a set of methods called mathematical programming. This set includes but is not limited to linear programming, integer programming, and nonlinear programming. As applications of mathematical programming in natural resource management have evolved past commercial forestry problems, capturing ecological functions and relationships has been a central challenge. In meeting this challenge, many researchers have resorted to nonlinear and integer programming methods. In fact, in our previous book (Hof and Bevers 1998), Spatial Optimization for Managed Ecosystems, we use nonlinear and integer formulations in all but two chapters. However, these models are difficult to solve, thus limiting the size of the application and limiting the confidence that the analyst has in obtaining the best solution.

In this book, we explore formulations that capture highly nonlinear ecological effects with spatial linear programs that can be solved with simplex algorithms (and two integer-friendly linear mixed-integer programs that can be readily solved with branch-and-bound or heuristic methods). This makes it possible to include many thousands of choice variables and many thousands of constraints and still be confident of obtaining an optimal solution. The feat of capturing nonlinearities in linear programs is accomplished here with a variety of formulation methods, but they all boil down to discretizing the problem so that the difference equations relating one discrete time period to another or one discrete land area to another are linear (at least as first-order approximations).

With the heuristic methods available today (see Reeves 1993), it is possible to approximately solve large nonlinear and integer programs with a degree of suboptimality that, for any particular case, can be difficult to determine. Nonlinear programs can capture ecological relationships more precisely and more directly than the linear programs we develop in this book but often must be solved with an unknown level of suboptimality. This presents the analyst with a difficult choice, to paraphrase Reeves (1993), between obtaining a more exact solution of a more approximate model (as with linear programming) and obtaining a less exact solution of a more precise model (as with nonlinear programming). In this book, we pursue the former course, recognizing the legitimacy of both (see Haight and Monserud 1990a or Bettinger et al. 1997 for examples of the latter course). A practical factor that might tip the scale in our favor is that the heuristic methods for solving nonlinear programs tend to require sophisticated analysts capable of writing their own solution software, whereas linear programming solvers are widely available, are highly automated, and are simpler to operate.

All our models involve ecological processes that are not completely understood and are significantly affected by random events. This may make some of our simplifying assumptions a bit more palatable, but it also points out the importance of using our models (and others) in an adaptive management process (Walters 1986). In such a process, ecological behavior (including the response to management actions) is monitored, and the results are fed back into model revisions and additional analysis to generate adjustments in management strategy. Because our models are process oriented, they are conducive to use in this analytical role.

Organization

The book is organized into four parts: Simple Proximity Relationships, Reaction–Diffusion Models, Control Problems, and Using Optimization to Develop Hypotheses About Ecosystems. An introduction develops the basic concepts for each part. In part I, models that account for simple proximity relationships are discussed. In chapters 2 and 3, two related models are presented: a model that accounts for the spatial relationship between timbering activity and the sedimentation effects in nearby stream channels and a model that accounts for the spatial effect of vegetative manipulation on storm-flow during severe precipitation events. In these chapters, the landscape is characterized as a watershed, with land areas defined by their runoff properties relative to stream channels. Chapter 4 treats individual trees as harvest choice variables and addresses mixed-age conditions, taking the spatial aspects of natural regeneration into account. The landscape is thus characterized by the areas occupied by mature trees. Chapter 5 uses a uniform grid of hexagonal cells to represent spatial structure and shows how simulation and optimization can be combined to model spatial (proximity) relationships for animals whose life history is too complex to capture directly in a linear programming model.

Part II presents linear programs based on the reaction–diffusion models in ecology that simultaneously capture population growth and dispersal over time and space. Chapter 6 explores the characteristics of the discrete reaction–diffusion model used in chapters 7–10 for optimization purposes. Chapter 7 discusses the basic model with an example that locates habitat for the black-footed ferret. This chapter is the only overlap with our previous book (Hof and Bevers 1998) and is used here as a point of departure. Chapter 8 presents a case study of black-tailed prairie dogs with a formulation that features population-dependent dispersal behavior. This formulation and the model results are compared with those in chapter 7. Whereas the ferret model in chapter 7 uses uniform square cells to define the landscape, chapter 8 uses irregular shapes to identify patches of potential habitat. Chapter 9 models an ephemeral plant, where multiple life stages and sensitivity to climate are featured in addition to the topography-based dispersal of seeds (one of the life stages). The landscape is structured according to topographic features (hummocks and swales) that define habitat and dispersal under different climate scenarios. Chapter 9 adds habitat edge effects to the reaction–diffusion model, contributing a definition of edge based on multiple habitat needs that is usable in a dynamic allocation model.

In part III the focus is control, contrasted with the preceding models, which try to maximize populations. In a mathematical programming sense (Luenberger 1984), our preceding models are also control (i.e., spatial control) models, but we use the term control here to emphasize that we are now trying to minimize rather than maximize a result. Chapter 11 shows how a linear programming model can be used to capture reaction–diffusion relationships when it is desired to minimize a population instead of maximize it, as one might want to do in trying to control an invading exotic species. Chapters 11 and 12 both use uniform square cells to define the landscape, but chapter 12 features a model formulation that tracks timing of fire spread through the landscape cells as opposed to using discrete time periods with associated diffusion distances.

In part IV we demonstrate how ecological theory and empirical investigations might be enhanced through the development of refutable hypotheses with optimization models. In chapter 13, hypotheses regarding the impacts on populations of multiple limiting factors operating at different scales are developed with a linear program, treating the population as a long-term optimizer and using linear programming to find equilibria. In chapter 14, optimization analysis is applied to models of carbon fixation in trees, treating the organism as an optimizer with several different behavioral assertions considered. The hypotheses that result are not empirically tested but are demonstrated in numerical examples.

The central purpose of the book is to describe case studies and pragmatic examples. Chapters 5 and 7–9 are case studies and have sections that describe the specific case, the model, and the results. Chapters 2–4 and 10–14 describe pragmatic examples and contain formulation, case example, and results sections. Chapters 2–4 focus on forest management practices (and mitigating their ecological effects), and all use a similar forest management component in their models. Chapters 2 and 3 are closely tied together, as are chapters 7 and 8, with similar study areas and closely related problem definitions. Reaction–diffusion formulations are used not only throughout part II, but also in chapters 11 and 13 (in their respective contexts). Throughout the book, six different approaches for modeling fauna are demonstrated, six different approaches for modeling flora are demonstrated, six different methods of characterizing the landscape are demonstrated, and two different methods of handling dynamics are demonstrated.

Methods

Many improved solution algorithms have become available in recent years for solving integer and nonlinear programs (see Reeves 1993), but we maintain that the simplex algorithm still is one of the most powerful tools in management science. It can solve huge linear problems (on the order of 50,000 constraints and 100,000 choice variables, depending on model structure) and reliably obtain an optimal solution in reasonable computing time. Simplex solvers are widely available, as are very powerful matrix generators that build linear programs efficiently. To take advantage of this capability, however, it must be possible to formulate the problem within the proportionality and additivity assumptions of linear programming. This means that the research challenge in this book is to formulate the problems in the first place: If linear approximation is possible and useful, solution is routine. For context, we quickly review the traditional approach to natural resource allocation with linear programming. We then begin to develop our ecological approach to landscape-level optimization.

Traditional Linear Programming Approaches

The basic structure of the linear programs historically used to analyze managed natural resource planning problems is depicted in table 1.1. For simplicity, the example in table 1.1 includes only a single discrete time period and ignores many constraints, such as budget limitations and minimum output levels. Also, table 1.1 only includes three outputs: timber, recreation, and forage.

In table 1.1, the major column headings are types of land and resource products. The X1,1 through X2,3 columns represent choice variables for the number of hectares allocated to alternative management prescriptions that could be applied in type I (X1,1 and X1,2) and type II (X2,1, X2,2, X2,3) land. The timber, recreation, and forage rows (equations) in the matrix represent the resource flows that result from implementation of the management prescriptions. For example, A1,1,1 is the output of timber for each hectare of land type I on which management prescription X1,1 is implemented. The type I and type II rows are the land inputs to this production system. L1 hectares of type I land are available, and L2 hectares of type II land are available.

TABLE 1.1

A Simple Depiction of Traditional Linear Programs Used in Multiple-Use Forest Resource Management and Planning

The products (P1, P2, P3) are accounting columns (variables) that collect the outputs described in the first three rows (equations) into aggregate outputs for the area being analyzed. K1, K2, and K3 are set at zero to force all product output levels into P1, P2, and P3. The coefficients in the last row, the net benefits equation, describe the cost if 1 unit of Xhj is applied on land type h and the benefit if 1 unit of Pi is produced. Thus, for example, C1,1 is the cost of prescription X1,1 on 1 hectare, and B1 is the benefit derived from 1 unit of timber output (P1). This row is the objective function to be maximized in this example.

An algebraic representation of the model in table 1.1 would be as follows:

Maximize

subject to

where Jh (the number of prescriptions) is 2 for type I (h = 1) and 3 for type II (h = 2) lands. Lower bounds of zero are implicitly assumed for all variables in most linear programming problems, a convention we adopt throughout this book. If scheduling for, say, four time periods is to be included, then the model would be modified as follows:

Maximize

subject to

where

Xhj = the number of hectares allocated to the jth management schedule (i.e., a prescribed schedule of treatments over time) for land type h,

Aihjt = the amount of the ith output in the tth time period that results from 1 hectare being allocated to the jth management prescription for land type h,

Pit = the total amount of the ith output produced in time period t,

Bit = the discounted benefit per unit of Pit,

Chj = the discounted cost per hectare (over all four time periods) of the jth management schedule for land type h.

When the land units are given a well-defined spatial context rather than just being defined collectively (as all type I lands, for example), such models provide a starting point for spatial optimization. The obvious limitation in this approach is that many ecological relationships are not captured, especially with regard to spatial relationships over time and across the landscape. Capturing these relationships in linear programs is the challenge set forth for this book.

Our Approach

The model just described is typical of activity analysis approaches, where the choice variables are activities that create a vector of outcomes. Our methods certainly include activity analyses but also involve functional (usually mechanistic) relationships between the activity choice variables, between the state variables (described later) in different time periods and land units, or between the choice variables in one time period or land unit and the state variables in another time period or land unit. Because our variables represent segments or pieces of larger variables