Spatial Optimization in Ecological Applications

John Hof, Michael Bevers

• Publisher: Columbia University Press
• Year: 2002
• eBook ISBN: 9780231500739
• Topic: Biological Sciences
• Subtopic: Ecology
• Index: Biological Sciences

Whether discussing habitat placement for the northern spotted owl or black-tailed prairie dog or strategies for controlling exotic pests, this book explains how capturing ecological relationships across a landscape with pragmatic optimization models can be applied to real world problems. Using linear programming, Hof and Bevers show how it is possible for the researcher to include many thousands of choice variables and many thousands of constraints and still be quite confident of being able to solve the problem in hand with widely available software. The authors' emphasis is to preserve optimality and explore how much ecosystem function can be captured, stressing the solvability of large problems such as those in real world case studies.

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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...

Table of contents

• Cover 
• Half title
• Series Page
• Title
• Copyright
• Dedication
• Contents 
• Preface
• 1. Introduction
• Part I: Simple Proximity Relationships
• Part II: Reaction–Diffusion Models
• Part III: Control Models
• Part IV: Using Optimization to Develop Hypotheses About Ecosystems
• 15. Postscript
• References
• Index