# Spatial Optimization in Ecological Applications

John Hof
Michael Bevers
Pages: 520
https://www.jstor.org/stable/10.7312/hof-12544

1. Front Matter
(pp. i-vi)
(pp. vii-x)
3. PREFACE
(pp. xi-xiv)
4. 1 INTRODUCTION
(pp. 1-10)

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...

5. PART I Simple Proximity Relationships
• [PART I Introduction]
(pp. 11-12)

This first part of the book is devoted to capturing simple, static ecological relationships that are based on how close different land areas or landscape features are to each other. We refer to these as “proximity” relationships, which should be distinguished from the “adjacency” relationships discussed throughout the forestry literature. Concerns about adjacent timber harvests emerged from legal and regulatory restrictions on the effective size of harvests; for example, if two seemingly legal-size clearcuts occur within a short time of each other in adjacent areas, the effective size of the clearcut may not be legal. Because the original motivations for...

• 2 SEDIMENTATION
(pp. 13-23)

One environmental impact often associated with timber harvesting is an increased level of sedimentation in nearby water courses just after the harvesting operation and during the subsequent “green-up” period of forest regeneration and growth. Linear programming (and other mathematical programming) models have become standard tools in optimizing the schedule of timber harvests, on the basis of economic returns as well as other objective functions. The typical approach to accounting for sediment effects in these types of models (as early as Bottoms and Bartlett 1975 or Dane et al. 1977 and as recently as Bettinger et al. 1998) is to estimate...

• 3 STORMFLOW MANAGEMENT
(pp. 24-41)

Our focus in this chapter is the effect of management activities such as timber harvesting and prescribed burning on storm event runoff (see Robichaud and Waldrop 1994). Physical soil and water effects such as channelization and sediment transport (and deposition) often can be attributed to the peak flow and total short-term water runoff from storm events (Janda et al. 1975). High peak flows are most commonly associated with storm damage, making confluences of stormflows at the same time and place particularly important. Because forest cover affects the magnitude and speed of storm runoff, the spatial arrangement of forest management activities...

• 4 NATURAL REGENERATION IN ANY-AGED FOREST MANAGEMENT
(pp. 42-58)

The future of forest management in North America probably will involve less and less even-aged management (which leaves a residual forest with significant areas of homogeneous tree ages). Harvest systems other than clearcutting have been applied to a significant portion of the nation’s forests for many years, and this portion can only be expected to increase given recent forest policy trends. Clearcutting is coming under increasing criticism on public lands; legislation has been proposed that would ban clearcutting on all public lands (Anonymous 1998b). On private land, policy trends also indicate reduced even-aged management in the future. For example, MacMillan...

• 5 COMBINING SIMULATION WITH OPTIMIZATION: HABITAT PLACEMENT FOR THE NORTHERN SPOTTED OWL
(pp. 59-72)

Simulation methods have been widely applied in ecology, whereas optimization methods have been more common in economics and management science. Perhaps because of these disciplinary loyalties, simulation and optimization methods sometimes are regarded competitively, and each discipline may not have given enough attention to the methods typically used by other disciplines. Most would agree that in modeling ecological systems, simulation models can include much more detailed (and realistic) biological behavior than optimization models can and still be solvable (Dunning et al. 1995; Turner et al. 1995). At the same time, few could deny that optimization models are intrinsically more powerful...

6. PART II Reaction–Diffusion Models
• [PART II Introduction]
(pp. 73-77)

One of the most fundamental ecological processes that occurs at the landscape scale is the spatial dispersal of organisms over time. In this part, we focus on the strategic placement (usually through protection) of habitat, taking the dispersal behavior of plants and animals into account. This problem is conducive to optimization because areas that are capable of providing habitat typically are a scarce resource, and the habitat placement decision typically determines not only the amount of habitat protected but also its fragmentation.

Fragmented landscapes often occur as habitable areas of varying size and shape, embedded with varying degrees of compaction...

• 6 CHARACTERISTICS OF THE DISCRETE REACTION–DIFFUSION MODEL
(pp. 78-97)

Before we discuss optimization models based on the discrete reaction–diffusion model developed in the introduction to part II, we explore the characteristics and behavior of this model (using simulation) and compare them with the reaction–diffusion theory literature. We use the cellular landscape definition and explore the effects of spatial habitat structure on population persistence, abundance, and distribution using numerical analysis of the implied “coupled map lattices” (Kaneko 1993). Using this approach to model spatial structure allows us to examine both within-patch and between-patch population dynamics with a single discrete time, discrete space reaction–diffusion model. The two-dimensional habitat...

• 7 THE BASIC MODEL: HABITAT PLACEMENT FOR THE BLACK-FOOTED FERRET
(pp. 98-113)

Early in 1987 the black-footed ferret (Mustela nigripes) became one of the world’s most endangered mammals when the last known free-ranging member of the species was taken into captivity (Thorne and Belitsky 1989). The Wyoming Game and Fish Department was successful in breeding six of the surviving ferrets in captivity (Clark 1989), setting the stage for a national recovery program of releasing captive-bred ferrets back into the wild.

Historically, the black-footed ferret ranged sympatrically with prairie dogs (Cynomys spp.) across much of North America (Anderson et al. 1986). Available evidence strongly supports the conclusion by Henderson et al. (1969) that...

• 8 POPULATION-DEPENDENT DISPERSAL: HABITAT PLACEMENT FOR THE BLACK-TAILED PRAIRIE DOG
(pp. 114-124)

In the late 1800s, an estimated 283 million ha were occupied by a combined population of North American prairie dogs (Cynomys spp.) exceeding some 5 billion individuals (Seton 1929). By 1971, that area had declined to 600,000 ha (Fagerstone and Biggins 1986). Loss of habitat, control programs, and plague (Yersinia pestis) continued to reduce populations to the point that it is currently estimated that prairie dogs have been reduced by 98–99% of their former numbers across the western United States (Miller et al. 1994; Mulhern and Knowles 1997). Ever since Merriam (1902) reported that prairie dogs compete with livestock...

• [Illustrations]
(pp. None)
• 9 TOPOGRAPHY-BASED DISPERSAL: HABITAT LOCATION FOR THE WESTERN PRAIRIE FRINGED ORCHID
(pp. 125-141)

Spatial patchiness is a common feature of the distributions of most plants, particularly those occurring in ephemeral habitats (e.g., Schemske et al. 1994). This factor combined with their immobility through certain stages of life makes plants seem particularly appropriate for spatial landscape analyses. A surprisingly small number of studies have examined plant populations with such an approach, however, particularly in terms of testing theoretical models (Husband and Barrett 1996; Wu and Levin 1997). Plant population dynamics differ from those of animals in a number of ways, including vulnerability levels independent of the size and age of populations (e.g., Husband and...

• 10 HABITAT EDGE EFFECTS
(pp. 142-162)

For decades after Aldo Leopold’s publication of Game Management (1933), wildlife habitat management efforts tended to emphasize creating induced edge (Thomas et al. 1979) and coverts (where three or more habitat types come together) because of the positive effects observed for many wildlife species (Hunter 1990). Although Leopold (1933) recognized that not all species respond positively to habitat edges, the emphasis on creating edge persisted at least into the 1970s (Giles 1978). By the 1970s and 1980s, however, negative edge effects on habitat interior species were becoming recognized (e.g., Soulé 1986) and were of increasing concern in forest management (e.g.,...

7. PART III Control Models
• [PART III Introduction]
(pp. 163-166)

We have identified this part as “Control Models” because it actually makes an enormous difference to optimization modeling whether a population or other entity is to be maximized or minimized. In part II, populations were determined by whichever factor (e.g., carrying capacity, population growth, dispersal) is limiting. We are able to model this by maximizing the population (s) subject to the minimum limiting factor (fi) as a function of management (x):

Maximize

s,

subject to

s ≤ the smallest fi(x),

which we can formulate as follows:

Maximize

$s\quad\caption(\rm{III}.1)$

subject to

$\quad s\leq f_{i}(x)\quad\forall i.\quad\caption(\rm{III}.2)$

Incidentally, we can also solve optimization models that minimize...

• 11 STRATEGIES FOR CONTROLLING EXOTIC PESTS
(pp. 167-182)

Recent literature documents the important and expanding problem of exotic pests invading forest ecosystems. As Haack and Byler (1993:34) summarize, “most native insects and pathogens reach a dynamic state of equilibrium with their hosts and natural enemies. However, this situation may not be true for . . . newly introduced exotic insects and pathogens.” These authors also state, “Exotic insects and pathogens have dramatically altered forest ecosystem diversity, function, and productivity. More than 20 exotic fungal pathogens and 360 exotic insects now attack woody trees and shrubs in North America” (1993:33). These numbers have increased (and probably will continue to...

• 12 STRATEGIES FOR CONTROLLING WILDFIRE
(pp. 183-200)

The use of mathematical models for managing fires has a rich and varied history in North America. Martell (1982) traces the application of operations research methods in forest fire studies back to the early 1960s. The earliest potential applications of operations research techniques to wildland fire management were mentioned by Shephard and Jewell (1961). Follow-up work by Parks and Jewell (1962) generated considerable interest by examining the use of differential equations and calculus to identify the optimal suppression force for a forest fire. Swersey (1963) and McMasters (1966) extended Parks and Jewell’s work by focusing on the optimal mix of...

8. PART IV Using Optimization to Develop Hypotheses About Ecosystems
• [PART IV Introduction]
(pp. 201-202)

Up to this point, we have treated optimization models as prescriptive devices: tools to help choose the most efficient way to allocate certain scarce resources. Our optimization models obviously contain a large component that could be regarded as simulation to predict the ecosystem response to alternative management actions. However, it is possible that ecosystems actually optimize themselves. In this case, optimization could be used to simulate ecosystem behavior per se. The reaction–diffusion simulations of chapter 6, for example, produce identical results over time and converge to the same equilibria as do optimization solutions from models using the same (fixed)...

• 13 MULTISCALED ECOLOGICAL LIMITING FACTORS
(pp. 203-220)

Although we have modeled carrying capacity for single populations as a simple equilibrium or saturation point, the concept is widely recognized as being more complex (Dhondt 1988). For more than a century, biogeographers have recognized that population distributions tend to be limited by a multitude of factors (Haeckel 1866; Shelford 1911; Udvardy 1969). These factors also affect population abundance (as in chapter 10) and can affect populations at different spatial scales (Morris 1987; Levin 1992). For example, many species forage over spatially extensive areas but reproduce in specific locales because of obligate relationships with habitat characteristics that occur sporadically across...

• 14 CARBON FIXATION IN TREES AS AN OPTIMIZATION PROCESS
(pp. 221-232)

In chapter 13 we used optimization as a device for finding model equilibria and then used that model to develop hypotheses about ecosystem function. In this chapter we directly apply optimization as a simulation of ecological optimizing behavior. To this end, we investigate economic analogues to ecological behavior. Applying economic models to ecological problems is not a new concept. For example, Rapport and Turner (1977:367) wrote,

Ecological processes have traditionally been studied from several vantage points. . . . None of these approaches, however, explicitly address what some . . . have regarded as one of the central problems of...

• 15 POSTSCRIPT
(pp. 233-234)

The principal thesis of this book has been that readily solvable mathematical programming models can play an important role in spatial modeling for landscape management and ecological research. Numerous spatial optimization examples were presented in the first three parts, and two theoretical ecology examples were presented in part IV. Linear programming models that can be solved for large-scale spatial problems were used in nearly all these examples.

Looking across the linear programming models that have been presented here, we are able to observe several solution properties (in addition to general spatial sensitivity) that appear to capture well-recognized hydrologic and ecological...

9. REFERENCES
(pp. 235-250)
10. INDEX
(pp. 251-260)