Economics of spatial-dynamic processes.

The notion that we might think of human populations as represented by the reaction-diffusion equation was, rather surprisingly, first hypothesized by Harold Hotelling (1921), the great statistician/economist responsible for so many imaginative and important papers over a career that began in the mid-1920s. What is not generally known is that Hotelling actually began his education not as an economist or statistician, but as a journalism student, receiving a bachelor's degree from the University of Washington in Seattle in 1919. He then decided to undertake further study for a master's degree, but in mathematics rather than journalism. His M.S. Thesis submitted to the Department of Mathematics at the University of Washington in 1921 is entitled "A Mathematical Theory of Migration." The existence of this thesis is little known, but it foreshadows Hotelling's brilliant career and elegant modeling ability. Hotelling's thesis basically proposed that we think of the westward movement in the United States as a reaction-diffusion process. As he wrote:

(if we hypothesize) that the percentage rate of

natural increase at any place is proportional

to the difference between the density of population

and a fixed saturation point, ... we have

[partial derivative][pi]/[partial derivative]t = K[[nabla].sup.2][pi] + [alpha][pi]([sigma] - [pi]) (39)

This equation is not linear, and is difficult to handle. But as we have seen, we may be dealing with new and sparsely settled countries assuming a Malthusian principle that population increases in a geometric ratio; while for countries near the saturation point it is not unreasonable to assume that the number of births per unit of area is proportional to the difference between the density of population and the saturation point. For the first case, we combine equation (20) with (1); for the second we take (20) and (3). These give respectively

[partial derivative][pi]/[partial derivative]t = K[[nabla].sup.2] [pi] + r[pi] (40)

and

[partial derivative][pi]/[partial derivative]t = K[[nabla].sup.2] [pi] + b([sigma] - [pi]) (41)

The elegance of Hotelling's approach is admirable. He first proposes the general reaction-diffusion equation to model the westward movement (his notation for diffusion used the diffusion coefficient K, the more general gradient symbol [[nabla].sup.2] for the second partial derivative and the symbol [pi] for the population at any point). His justification for the simple Fick's Law type of diffusion process is, however, not the assumption that movement is random. Instead, he proposes that settlers move from high- to low-density areas for economic reasons (in search of low-cost land, for example). Then they will act as if they are diffusing randomly via Fick's Law, even though there is behavioral purpose driving movement. As he notes, the reaction-diffusion equation "is difficult to handle" (an understatement, since it was not really understood until sixteen years later) and hence he proposes solving two simpler problems, the solutions of which can actually be derived explicitly, and which qualitatively bracket the more general problem.

The remainder of Hotelling's thesis is empirical. He procures the U.S. census records for the 1790-1900 period and uses the data to compute the implied diffusion coefficients of the actual westward movement "wave front," thus calibrating the model to complete his analysis. Aside from the cleverness and elegance of thought that the thesis reveals in the young Hotelling, it also is the first depiction of a spatial-dynamic process driven by economic behavior depicted by a diffusion mechanism.

To summarize, the fundamental drivers of spatial-dynamic processes are diffusion or dispersal processes. These can be modeled with partial differential equation descriptions. As shown above, there is a rich variety of diffusion models available to fit circumstances ranging from simple random movement, to advection to joint diffusion/population growth. A few of these models can be solved analytically, but most cannot. This presents analysts of spatial-dynamic problems with challenges. There are essentially two ways to build understanding of these kinds of complex situations. One way is to construct numerical models of complex systems and attempt to synthesize some common understanding from the analysis of a range of cases. The other way is to start with very simple models, extracting what can be learned before adding detail. This is the method we illustrate next, using a familiar current policy problem.

An Example: Bio-Invasions

To illustrate how we might model spatial-dynamic processes in order to gain understanding about policy options, I will now turn to an example, namely the example of bioinvasions. Bio-invasions are familiar to everyone, and there are numerous cases that come to mind, including kudzu, the gypsy moth, yellow starthistle, and zebra mussels. Bio-invasions are successful colonizations of alien species that have been either accidentally or purposefully introduced into ecosystems. They may generate damages (or benefits) as they spread, and costs of control depend upon when, how intensively, and where control activities are undertaken.

Bio-invasions are becoming increasingly prominent in the public eye, and the topic has also received considerable research attention, first by ecologists but more recently by economists. For the most part, however, economists have ignored the spatial-dynamic aspects of bio-invasions, often treating them as little more than a pest-control problem. This seems to miss the most interesting aspect of bio-invasions, namely their spatial-dynamic character. While simple characterizations that address questions such as whether to spray or not, or whether to quarantine or not are illuminating, the more important questions seem to be where to spray, when, and at what intensity in a landscape setting. (3) This requires a modeling apparatus that captures essential features of spatial-dynamic problems.

The economic issues that bio-invasions raise are several. How does an uncontrolled bioinvasion unfold? What governs its rate of spread, and how fast is that spread if no actions are taken? What are the consequences to humans of the invasion? What kinds of actions are available to control a bio-invasion? At what points over time and space are instruments available to manage the bio-invasion? What is the optimal way to use controls; when, where, and how intensively should they be implemented? How is the optimal control affected by bio-economic parameters? How do basic cost/benefit parameters affect the best choice? How about the discount rate and time horizon? Does the optimal control vary with the initial size of the invasion? If an invasion is uncertain, what kinds of monitoring and detection efforts should be implemented?

Answering these kinds of questions requires some kind of modeling effort. The modeling should first describe the biophysical diffusion mechanisms that drive the bio-invasion. Diffusion may occur in a smooth and homogeneous pattern over space, driven by short-term dispersal processes such as simple local movement. For example, the classic case in biology textbooks of muskrat spread in Europe reveals a stark radial pattern with a predictable velocity of the "wave front" over several decades (Murray 2002), much like predicted by the Fisher reaction-diffusion equation. Alternatively, dispersal may involve long-distance mechanisms driven by wind and currents, or by movement of hosts including humans, animals, and birds (Hastings et al. 2005). A model of a bio-invasion should also incorporate some assumptions about boundaries and processes that occur at boundaries, as well as the geometry of the relevant space. Finally, a model of bio-invasions needs to reflect humans and define the links between humans and biophysical environment. Are humans simply passive participants, or do they influence the spatial-dynamic patterns by their actions? How are humans impacted by the bio-invasion? Once these critical components, the biophysical systems with its dispersal mechanisms and spatial character and the human linkages, are specified, the system can be simulated and optimized to generate some predictions of the consequences of various policy options. Further refinements can then be added if necessary and policy conclusions expanded.

[FIGURE 3 OMITTED]

A Simple Bio-invasion Model

Consider the simplest possible model of a bio-invasion that incorporates the above elements, albeit in the most abstract fashion. Suppose that we consider the spatial geometry of our model system to be a corridor of fixed width, with the origin at the left-hand side, and an unbounded right-hand side. As in figure 3, we assume that there is an initial invasion that is discovered after having covered an area of amount [X.sub.0]. The invasion is assumed to spread along the length of the corridor with a velocity of v. (4) The area infested at any date t will be X(t) = [X.sub.0] + vt. Assume that damages are proportional to the total area infested in each period, where the marginal damage per unit space is d. I have in mind something like star thistle, a weed that spreads through pasture land and that grazing animals will not eat. Assume that control costs are incurred each period in a manner that influences the velocity of the wave front of the invasion (Sharov and Liebhold 1998). If [bar.v] is the natural velocity of the uncontrolled bioinvasion, let control costs be quadratic in the velocity reduction associated with control activities, or:

TC(v) = a([bar.v] - v) + (1/2)b[([bar.v] - v).sup.2]. (6)

[FIGURE 4 OMITTED]