Agent-based and analytical modeling to evaluate the effectiveness of greenbelts

Abstract

We present several models of residential development at the rural–urban fringe to evaluate the effectiveness of a greenbelt located beside a developed area, for delaying development outside the greenbelt. First, we develop a mathematical model, under two assumptions about the distributions of service centers, that represents the trade-off between greenbelt placement and width, their effects on the rate of development beyond the greenbelt, and how these interact with spatial patterns of aesthetic quality and the locations of services. Next, we present three agent-based models (ABMs) that include agents with the potential for heterogeneous preferences and a landscape with the potential for heterogeneous attributes. Results from experiments run with a one-dimensional ABM agree with the starkest of the results from the mathematical model, strengthening the support for both models. Further, we present two different two-dimensional ABMs and conduct a series of experiments to supplement our mathematical analysis. These include examining the effects of heterogeneous agent preferences, multiple landscape patterns, incomplete or imperfect information available to agents, and a positive aesthetic quality impact of the greenbelt on neighboring locations. These results suggest how width and location of the greenbelt could help determine the effectiveness of greenbelts for slowing sprawl, but that these relationships are sensitive to the patterns of landscape aesthetic quality and assumptions about service center locations.

Introduction

Population increase, decreasing household sizes (Liu et al., 2003), and increases in area developed per household (Vesterby and Heimlich, 1991) all contribute to increase in the amount of land converted for development in metropolitan areas throughout the world. Land development for residential, commercial and industrial uses at the urban–rural fringe can have a variety of negative ecosystem impacts, including habitat destruction and fragmentation, loss of biodiversity, and watershed degradation (Alberti, 2000). Landscape ecological theory (Turner et al., 2001) suggests that, in addition to how much development occurs, the extent of these impacts is determined by where the development occurs relative to ecological features and its overall spatial pattern.

A number of approaches have been proposed to minimize the ecological impacts of development, by manipulating the spatial patterns of development to minimize sprawl and excess land usage. These approaches include establishment of greenbelts of preserved lands around cities (Mortberg and Wallentinus, 2000), clustered or “new urbanism” designs (Arendt, 1991), which involve increased use of higher density development and mixtures of land uses within developments, purchase or transfer of development rights (Daniels, 1991), and alteration of tax or investment policies (Boyd and Simpson, 1999), among others. For each of these alternative strategies, the costs of implementation need to be considered (Boyd and Simpson, 1999) along with the long term conservation benefits obtained.

To evaluate the benefits of any given option, the dynamics of development at the urban–rural fringe and their linkages to ecological impacts need to be understood. Because the impacts are driven to a large extent by the location and spatial patterning of the development, this understanding needs to be spatially explicit. In order to understand the drivers of urban development and their possible future impacts on land development, and to develop scenarios that can be used to test alternative approaches to minimizing these impacts, a variety of spatial modeling approaches have been employed. The work of Landis and colleagues (Landis, 1994, Landis and Zhang, 1998a, Landis and Zhang, 1998b) illustrates a simulation approach based on discrete choice statistics that focuses on estimating the likely locations of development. Similarly, Pijanowski et al. (2002) used artificial neural networks to identify non-linear interactions between predictor variables and likely locations of development. Alternative modeling approaches have focused on how the patterns of development evolve through spatial interactions and, in many cases, have used analogies with physical systems (e.g. diffusion limited aggregation and correlated percolation) to represent processes of urban growth (Makse et al., 1998, Zanette and Manrubia, 1997). Cellular models (Clarke et al., 1997) represent an approach that is intermediate in realism between statistical location models and physical analog interaction models, combining some of the strengths of both.

These powerful simulation models have been used to evaluate the impacts of a variety of land-use policy instruments. Each of them represents the land-use state at each location and the variables and processes that determine that state. An important next step in the evolution of land-use models, and improving their utility for policy scenarios, is directly representing the heterogeneous set of actors in the land-use change process (Page, 1999), their decision making processes, and the physical manifestation of those changes on the landscape. Agent-based models (ABMs) serve as tools for this purpose. Otter et al. (2001) presented an ABM of land development that includes a reasonable representation of the different types of agents and that makes an initial contribution on which further developments in this area might build. Further, experimentation with this kind of model can improve our understanding of how the interaction between landscape characteristics and the preferences and behaviors of agents might influence ecological diversity and function.

A key challenge in modeling such multi-agent systems with agent-based models is providing confidence in the models’ results (Parker et al., 2003). Often establishing confidence in a computer model is divided into two steps: (1) verifying that the computer program is free of “bugs” and correctly implements the conceptual model and (2) validating the model by showing it generates output that matches the relevant aspects of the system being modeled (Kelton and Law, 1991). In practice, carrying out those procedures is not so straightforward. First, verification of program correctness cannot be guaranteed for any but the simplest of programs; thus in practice we can only increase confidence that a program is correct by a combination of software engineering and testing techniques (McConnell, 1993). Second, validation also is a non-trivial exercise, since it involves judgements about how well a particular model meets the modeller’s goals, which in turn depends on choices about what aspects of the real system to model and what aspects to ignore. Critical issues that must be considered include what level of detail to try to match (data resolution) and how to handle issues of “deep uncertainty” found in complex adapative systems (Bankes, 2002).

Because of these difficulties, typical practice is to establish confidence in the results of a model through a mix of techniques, most of which contribute to both verifying and validating the model. Sensitivity analysis and other “parameter sweeping” technique can provide support for computer program correctness and model plausibility, by improving understanding of the behavior of a model under a range of plausible conditions (Kelton and Law, 1991, Miller, 1998). In some cases model calibration is carried out, i.e. model parameters are adjusted (“tuned”) until the model output matches the real world data of interest. For the calibration to be convincing, we also must show those parameter values are “plausible,” e.g. by basing them on empirical data or by arguing that experts support the “face validity” of the parameters chosen. We also can “dock” models to other related models (Axtell et al., 1996), to show the results are common to more than just one model or implementation.

Beyond simple verification and validation of an ABM, we also want to be confident that we have a clear understanding of the agent-based model’s processes and of the behavior and results those processes produce. Because agent-based modeling is a new, potentially valuable approach to understanding complex phenomena like settlement patterns, much can be gained from understanding the models themselves. Further, such an understanding of an ABM is a necessary step in using the model to understand the fundamental processes in the (more complex) real world system that the model is meant to represent.

Because an ABM usually is itself a complex system, it can take considerable effort to understand even the simplest of models (Casti, 1997, Axelrod, 1997, Bankes, 2002). Axelrod (1997) argues that simulation is a third way of doing science, combining aspects of deduction (knowledge based on proofs from axioms) and induction (knowledge from observed regularities in empirical data). That is, the ABM can be viewed as a fully specified formal system (like the axiomatic basis for deducing theorem proofs) which, when run, generates data that requires careful analysis (induction) to understand and summarize. For instance, we can induce regularities by analyzing the model output in ways similar to those used on data from a real-world system¹.

In this paper we demonstrate another way to understand the basic processes in an agent-based model and, by extension, to help us understand processes that may be at play in the system being modeled. The approach we use in this paper involves comparing the behavior of an agent-based model to the behavior of a simpler mathematical model of land development. This comparison has a number of benefits, including:

• By having two separate “implementations” which both generate the same fundamental results, we increase our confidence in the veracity of both models;

• The results from the stark mathematical model can be shown to hold in more general contexts which an ABM can represent, e.g. spatial heterogeneity, discrete service center distributions and other extensions not amenable to mathematical analysis; and

• The theorems we are able to prove for the mathematical model give us deeper insights into the processes that generate the fundamental dynamics of the ABM.

In general, agent-based models may be constructed to serve as minimal realistic models of real-world complex adaptive systems. However, the fact that we often cannot prove theorems about the agent-based models makes for a shaky foundation. But, if we can both prove theorems about simplifications of the ABMs and show that the conclusions of those theorems hold in more general agent-based models, we enrich the scientific enterprise.

The comparison of an ABM to a simpler mathematical model can also be viewed as a kind of “docking” exercise (Axtell et al., 1996). In this case one model is computational and the other is mathematical (instead of comparing two computational models), but the basic goal is the same, i.e. to study the “…troublesome case in which two models incorporating distinctive mechanisms bear on the same class of social phenomena, …” (Axtell et al., 1996, Section 1.1), in part to carry out “…tests of whether one model can subsume another” (Axtell et al., 1996, abstract). As emphasized in Axtell et al. (1996), a key issue is how to assess the “equivalence” of two models. For this paper, we focus on “relational equivalence” between the models, showing that they both generate the same relationships between results, e.g. as analogous parameters are varied. If the models are relationally equivalent, we can be more confident that (1) the mathematical model helps us understand the key processes in the ABM, and (2) the ABM can be viewed as subsuming the mathematical model, allowing us to study a wide variety of cases that are mot mathematically tractable.

In summary, in this paper we present several models of residential development at the rural–urban fringe. In all models, the common conceptual model consists of agents choosing where to locate based on preferences for minimizing distance to services and maximizing aesthetic quality of the chosen location. We use the models to evaluate the effectiveness of a greenbelt, which is adjacent to a developing area, for delaying development outside of the greenbelt. Our one-dimensional mathematical model focuses on the interactions between greenbelt location and width, the spatial distribution of aesthetic quality, and the resultant amount and timing of development beyond the greenbelt. We explore the model under two different assumptions about the spatial pattern of service centers. Next, we implement the same basic mechanisms of the mathematical model in a one-dimensional discrete ABM setting. We then demonstrate the flexibility of the ABM framework by relaxing assumptions and extending the representation of the system to include (1) a two-dimensional landscape and (2) an effect of the greenbelt on the aesthetic quality of the nearby environment.