Spatial Dynamics in Populations

There has been a trend in disciplines that model aggregate populations to model components of the populations as storages, and model flows between the components (sectors, subpopulations, etc.) in an abstract sense, ignoring or drastically oversimplifying the spatial aspects. This was done presumably in the interest of parsimony, and very likely for limitations in computing resources at the time of the investigations. Such ground-breaking works as Forrester (1969) in socio-economic modeling and Odum (1974) in ecologic modeling concentrated primarily on populations and transfers between sectors, and tended to treat space as a giant unit (a single giant, relatively homogeneous population with perfect communication and interaction within its boundaries); this assumption has long been used in economic modeling as well.

Many researchers have discussed how spatial aspects of ecologic communities affect structure and dynamics (Holling, 1992; Pahl-Wostl, 1995), and evolutionary processes (Thompson, 1994). Explicit consideration of individuals and their spatial relationships in ecologic simulations has taken many varied forms. Holland (1992), for example, created the ECHO simulation package to adapt individual-based agents onto a one-dimensional space to explore population interactions, and was able to recreate many of the complexities of population dynamics. Schmitz and Booth (1997) found this spatial representation to be ``indefensible'' and adapted the model to a continuous two-dimensional space. Other research of note has been investigations into how modeling populations at the individual level has been successful in capturing ecosystem dynamics that top-down approaches tend to miss (DeAngelis et al., 1996). Cazoulat and Vitorri (1994) compared the famous Lotka-Volterra equations and ran simulations comparing the formulations through macroscopic differential equations and microscopic individual-based models, and found the two to complement each other nicely in their final outputs. Both systems were able to show behavior of the model centered around strange attractors. Booth (1997) noted similar behavior in the dynamics of individual-based representations. Individual-based modeling has shown promise as a reducdtionist method for capturing higher-level dynamics in ecosystems (omnicki, 1992).

Similar research has gone into studies of social systems. While research in this line is not immediately applicable to traffic studies, they are studies that shed light on how individual attitudes translate into aggregate group behaviors. Axelrod (1984) explored diffusion of cultural ideas over a two-dimensional space through two-dimensional lattice simulation. At least two studies (Cazoulat and Petit-Singeot, 1995; Lindgren and Nordahl, 1996) have investigated the dynamics of simulated populations that interact with each other by way of the well-studied Prisoner's Dilemma in a two-dimensional space, and reported very different dynamics than previous studies that had considered the players to be from a single, continuous pool.

Spatial expression of economic interaction has been explored via computer representations of individuals in a spatial context. Mulligan and Fik (1994) examined equilibria conditions for firms using the firm or enterprise as the basis of the simulation in one-dimensional space. Kohler (1993) modeled inter-household interactions including trade, intermarriage, and resource sharing to understand village formation and disintegration of Anasazi pueblos given historical records for climate and soil productivity. Both found distinct spatial patterns and sometimes unpredictable behavior when systems were modeled at the lower levels. Recent studies of economic organization from the bottom up that investigated organizational structures without taking space explicitly into consideration are Lin et al. (1996) and Weisbuch et al. (1996).

Allen and Sanglier (1981) applied principles described by Ilya Prigogine (Nichols and Prigogine, 1977; Prigogine and Stengers, 1984) in chemical systems and crystalline lattices to urban systems using lattice simulations that generalized human and capital flows. They found urban processes to show principles of self-organization. Straussfogel (1991) continued application and sought better methods of calibration of the models. There have been several articles discussing the applicability of Prigogine's work to geographical systems (Mazurkiewicz, 1987; Mizuno, 1994; Temkina, 1984). These works, like the Forrester models mentioned earlier, concentrate on regionalization and regional interaction, and are of limited applicability to the requirements of the boat traffic model mentioned at the beginning of this chapter.