Economics of spatial-dynamic processes.

For the remainder of my talk I would like to "deconstruct" spatial-dynamic processes by discussing features that define these kinds of problems and ways that they might be modeled. Then, I will illustrate using an example of bio-invasions. There are several features of spatial-dynamic problems with which economists are not particularly familiar. Figure 1 is an abstract representation that captures the nature of these kinds of problems. The heart of the process is the diffusion or dispersal process that governs the way something spreads over space and creates patterns. Is movement random or purposeful and behavioral? Are patterns self-generated, or are they also influenced by directional forces such as winds and currents? Does the front spread uniformly, or does it follow transportation corridors and nodes of populations? What mechanism drives movement--animal transport, human transport, spores, rhizomes, swimming, walking?

[FIGURE 1 OMITTED]

The other important aspect of spatial-dynamic problems concerns the nature of space, and in particular boundaries, geometry, and heterogeneity. What happens at boundaries, and how should we characterize it? Mathematicians use terms such as absorbing or reflecting boundaries or zero flux boundaries. When lemmings migrate to breeding grounds near the sea at normal population levels, they reach the barrier, mill around, and begin the process of breeding. But when populations are high, they reach the sea and jump in, illustrating the difference between reflecting and absorbing boundaries. Some boundaries exist for biophysical reasons, such as the transition between land and sea. Others exist because of habitat quality, and these boundaries may be species specific. Yet other boundaries exist for geopolitical reasons. For example, a forest pest spreading north from New England into Canada might spread across the border but be treated as if it stopped at the border when the U.S. Forest Service decides how much control to initiate. Another important aspect is the geometry of space. Is the relevant spatial unit like a featureless plain, or does it have corridors or choke points that influence density and flux? Is space homogeneous, or does it exhibit differentiated features that influence either the diffusion pattern or damages? Soils often exhibit widely varying character even within small scales, complicating the depiction of the flows of water, oil, or contaminants. A landscape contains a mosaic of human uses so that a bio-invasion might warrant more focused control efforts at particular points that are located in advance of certain areas with high potential damages.

Diffusion Processes

In light of the central role played by diffusion processes in spatial-dynamic problems, it is useful to elaborate how they are typically modeled (see Okubo and Levin 2002). What makes these kinds of problems interesting is the fact that patterns are generated by integrated dynamic and spatial forces. The simplest kind of spatial-dynamics process can be represented by a diffusion equation

[partial derivative]C(X, t)/[partial derivative]t = [partial derivative]/ [partial derivative]X [D [partial derivative]C(X, t)/[partial derivative]X]

= D [[partial derivative].sup.2]C/[partial derivative][X.sup.2] (1)

This is a partial differential equation that expresses a process in terms of derivatives in both time and space. This particular equation represents the most basic type of diffusion, namely random diffusion. Consider measuring the concentration C(X, t) of something like a group of particles at a point on a line X at time t. Suppose that any particle can move either right or left on the line with equal probability. Then, it can be shown that the concentration of the particle will be governed by Fick's Law, which states that the spatial diffusion at a point will be proportional to the spatial gradient at that point (Murray 2002). Here, D is the diffusion coefficient, (assumed constant) which indicates the rate of spatial flow. The essence of this idea is that particles will flow on net from high to low density areas simply because high-density areas have more particles that have a chance of ending up in low-density areas than vice versa.

A partial differential equation like equation (1) is not something with which economists are particularly familiar, and solving these kinds of equations is difficult and somewhat of an art form in mathematics (Holmes et al. 1994). As with ordinary differential equations that are expressed only as a function of time, the explicit solution must incorporate boundary conditions and initial conditions, in numbers equal to the number of derivatives in the equation. Assume that this process is started with a "point-source" injection of m particles at the origin and at time zero, and that the one-dimensional line depicting space is of infinite length in both directions away from zero. Then, this particular equation can be explicitly solved to yield

C(X, t) = m/[square root of 4[pi]Dt] exp (-[X.sup.2]/4Dt). (2)

This describes how the concentration changes over space and time. As figure 2 shows, the initial concentration spreads over space and time. Recall that this process is completely driven by random movement. At the origin initially, particles may move right or left with equal probability. But because there is a point concentration, the gradient is steep around the origin, and hence, more particles will move from the origin to adjacent low-density points in space. This is Fick's Law in action, causing the flow at a point to be proportional to the spatial gradient at that point, and causing, at the global level, particles to spread out over space.

[FIGURE 2 OMITTED]

Another useful representation of a diffusion process that fits many examples found in resource problems is one described by

[partial derivative]C(X, t)/[partial derivative]t = D [[partial derivative].sup.2]C/[partial derivative][X.sup.2] - V [partial derivative]C/[partial derivative]X. (3)

This equation is simply the random diffusion representation characterizing Fick's Law in equation (1), but modified to include what is called an advection term. The advection term contains the constant V that depicts drift of the process (Murray 2002). Advection applies whenever a diffusion process is influenced by external forcing factors such as wind or currents. If the advection process is strong enough, the process depicted in figure 2 will not only dissipate the initial infusion, but also shift the concentration over time as suggested by the solution

C(X, t) = m/[square root of 4[pi]Dt] exp (-[(X - Vt).sup.2]/4Dt). (4)

The solution to the diffusion with advection equation shows that the maximum concentration shifts over time according to the term in the exponent that acts as an axis shifter. (2)

A final representation of a diffusion process is the famous Fisher reaction-diffusion equation

[partial derivative]P(X, t)/[partial derivative]t = D [[partial derivative].sup.2]P/[partial derivative][X.sub.2] + [alpha][X.sup.2] + [alpha]P(1 - P). (5)

This equation was examined by R.A. Fisher (1937) and it depicts a process that is especially suitable to examining biological organisms. It contains the random diffusion term but also a term that represents logistic growth at a point in space. The logistic equation is a popular and useful representation of density-dependent growth of populations. At each point in space, then, a population P(X, t) will grow according to two forces: (net) diffusion and local growth. The percentage rate of local growth is highest when the population is smallest, and other things equal the population approaches a carrying capacity level at each point, where the carrying capacity is normalized at one in the above equation.

As it turns out, it is impossible to derive a closed-form solution similar to equations (2) and (4) for this equation. However, Fisher was able to show that as time gets large, the solution exhibits what is called a "traveling wave" property. A traveling wave can be envisioned as similar to a tsunami, with the sea surface height behind the wave at the same height as the crest. Kolmogorov, Petrovsky, and Piscounov (1937) provided a conjecture about the velocity of the wave front, namely that it is [square root of 4[alpha]D], or a value associated with the product of the diffusion coefficient and the growth rate of the logistic process at small population levels. This conjecture has recently been proven correct (Uchiyama 1978). The reaction-diffusion equation is a useful way of thinking about populations of plants, or animals or even humans.

A Digression: The Genius of Harold Hotelling