In this assignment we assembled a land-use model to investigate the role of path-dependence and feedbacks in simulation modeling. Path dependence refers to the impact early development decision have on long-term trajectory: in a model with high path dependence, small changes in initial patterns have a large effect on outcome, while in a model with low path dependence, outcomes emerge regardless of initial patterns. Feedback refers to how a system’s changes influence its future changes: positive feedback occurs when changes in a certain direction predispose the system to further changes in that direction; negative feedback occurs when changes in a certain direction predispose the system to move in the opposite direction in the future. By building a model in which the degree of the impact of early decisions on later decisions is adjustable, we can investigate how these concepts play out in an actual model.

Model Components

The model we assembled simulated feedbacks between the land-use decision-making process and the landscape that it generates. The landscape, a 100 x 100 grid, consists of patches with three primary attributes: aesthetic quality, service center proximity and neighborhood density. Aesthetic quality is a simple function of distance from two attractions located at (25, 25) and (75, 75). In terms of path dependence, the aesthetic quality exhibits none, as it does not change from run to run, or during any given run. There is consequently no possibility for any sort of feedback: the aesthetic quality of any given patch is constant throughout.

The service center proximity attribute is calculated as a function of distance from service centers, whose location is more dynamic than the attractions. First, the model offers the option of starting with a service center at (50, 50) or with none. Then, the model adds service centers periodically as the model progresses; as each new service center is added the patches recalculate their service center proximity value. The service center proximity (SCP) does exhibit path dependency: service centers are placed in order to maximize proximity to recent development, which then lends that area superior SCP values, further incentivizing settlement. The SCP attribute thus has the potential to exhibit path dependency, and since proximity breeds proximity, it is an example of positive feedback.

The final attribute emerges only as a function of development: neighborhood density. Each patch sets its neighborhood density value as a function of how many of its neighboring patches have been developed as “homes.” This value begins at zero, of course, and only changes in response to settlement. It is therefore likely to exhibit some degree of path dependence.

In addition to the three primary attributes, each cell also has three other variables: status and utility. Status tracks whether a cell is unoccupied, a home, an attraction, or a service center. These states are mutually exclusive. Black patches are unoccupied, red are homes, yellow are attractions, and green are service centers. The utility variable synthesizes the relative importance of aesthetics, service centers, and neighborhood density into a single metric. This metric is the driver of site selection: at each step, the simulation picks a number of random sites, and places a home at the patch with the highest utility rating. The random selection of sites is the only stochastic element in this model, and the degree of stochasticity is inversely proportional to the number of potential sites: if only a single site is randomly selected, the siting is completely random; the larger the number the more power the selection submodel has to discriminate in favor of high utility patches.

The relative importance of each attribute is variable: for aesthetics and SCP, it ranges from overwhelmingly important to completely irrelevant. Density preference is calculated differently: preferences range from wanting only half the neighboring patches to be homes, to preferring that all patches are homes. Therefore neighborhood density can drive higher density or lower density, exhibiting positive or negative feedback.

Given the complexity of the utility function, it would be very easy to miss mistakes in the calculation of each submodel. That is why it is important to implement testing protocols such as those that generated the above images. By creating a visual reference for aesthetic quality, SCP, neighborhood density, and the utility variables, a variety of simple mistakes such as an incorrectly implemented utility calculation, or a SCP variable that is not updating, become immediately apparent. As can be seen from comparing images 2 and 3, the SCP is updating at each step. As can be seen from comparing images 5 and 6, the utility function is responding to re-weighted inputs as expected.

Initial Environmental Heterogeneity

Initial environmental heterogeneity is implemented in this model via the two attractions in the southwest and northeast, and the presence or absence of the initial service center. These provide a locus for early land-use, which then influence following land-use–depending on the relative weighting within the utility function, of course. If the aesthetic quality variable is given a weighting of zero, and the initial service center is turned off, the initial map has no utility gradient at all: any place is equally desirable until the first service center appears, and the location of that first service center is essentially entirely random. Given these conditions of low initial environmental heterogeneity, each run will exhibit a high degree of variability.

However, if the initial environmental heterogeneity is higher, i.e. with a higher weighting for the aesthetic quality attribute and an initial service center, the site-selection submodel will be constrained from the outset. Development is highly likely to cluster near the initial features every time, leading future service centers to be located there, yielding much less variability between runs.

Feedbacks and Path Dependence

The different parameters exhibit different kinds of feedback and path dependence. The aesthetic attraction attribute cannot be altered by further development; therefore it does not exhibit any kind of feedback. The service center proximity attribute changes as new service centers appear, and service centers appear near new homes; therefore new homes lead to service centers and service centers lead to new homes in a positive feedback loop. When the SCP attribute is heavily weighted, the initial location of service centers have a heavy influence on future centers; i.e. land use shows a high degree of path dependence. The neighborhood density is affected by land use change, and affects land use change, leading to positive feedback and path dependence. However, since it is possible for densities lower than 100% to be preferred, it is possible for high densities to deter future growth, leading to a negative feedback. Positive feedback, while it is not the same thing as path dependence, is a necessary element for path dependence to take effect. In this model, path dependence could be eliminated by setting the weighting of the SCP and neighborhood density attributes to zero.

Model Assumptions & Simplifications

The utility function in this model pulls together many different factors in land use decision making. Nonetheless, it relies on a number of simplifications that could influence our interpretation of model results. First, the model makes assumptions about what constitutes aesthetic quality in a very unidimensional way. The only relevant factor is raw distance to two attractions, with no consideration of more micro-scale factors that can influence aesthetics, such as neighborhood quality, location of potential eyesores, or the shape of local viewsheds. Nor is aesthetic quality given any way to change over time, despite the significant impact that changing land use can have on aesthetic quality of different locales. Another assumption the model makes is that all land users have identical set of preferences: while the global balance between the different elements of the utility function can be altered, there is no way to simulate individual users who have a greater or lesser preference for aesthetic quality, or a different ideal neighborhood density. Nonetheless, the model can demonstrate useful lessons about the role of different factors subject to differing levels or differing kinds of path-dependency and feedback. Under different conditions, the same model can generate land use patterns that are heavily constrained by initial environment, or develop in a very random direction. Alternately, the model can generate patterns where early decisions play a huge role in future decisions, or where they play little or no role at all. This demonstrates the importance of paying attention to elements of path dependence and feedback loops in model design.