1.B Mees Analysis (Batten, pp. 120-125)

Alistair Mees (Professor of Applied Mathematics, University of Western Australia) developed a qualitative nonlinear dynamic model in Mees (1975) illustrating how trading opportunities and employment dynamics might affect the relative size of urban and rural populations.

Batten argues (pp. 122-123) that the Mees model can be used to illustrate the Pirenne hypothesis. Mees posits the existence of two main economic groupings, farmers producing food and city merchants producing silk (say). Mees reasons that locally stable mixed population equilibria are more likely for weakly interconnected economies with isolated self-sufficient communities that have to provide for all of their own needs. But Mees also shows how mixed population equilibria can be destabilized by the opening up of new long-distance trading opportunities.

Following Batten, let f denote the percentage of the current population consisting of farmers, and let c denote the percentage of the current population consisting of city merchants. Consider, first, the case of a relatively isolated region in which farmers and city merchants must depend solely on each other for their needs. In this case, Mees reasoned that a mixed population of farmers and merchants would emerge rather than a specialized population consisting entirely of farmers or entirely of merchants.

This postulated system can be illustrated by means of the following phase portrait, where df/dt denotes the rate of change in f and dc/dt denotes the rate of change in c. Note that df/dt goes from negative to positive values as one ascends the left-vertical axis whereas dc/dt goes from negative to positive values as one descends the right-vertical axis.

                                                     f=100%
       df/dt                                         c=0%
        /|\                                           |
Positive |           -   -                            |
Values   |     -             -                        |  Negative
of df/dt |                                            |  Values
         | -                     -                    |  of dc/dt
      0  |____________________________________________|
         |Ec                       Em -            -  | Ef
         |                                      -     |
Negative |                               -   -        |  Positive
Values   |                                            |  Values
of df/dt |                                            |  of dc/dt
         |___________________________________________\|/
        f=0%                                        dc/dt
        c=100%                                       

Each point along the line between Ec and Ef corresponds to a particular population mix (f,c), which constitutes the state vector for the Mees dynamical system.

At point Ec=(0%,100%) the population (f,c) is specialized, consisting entirely of city merchants. At point Ef=(100%,0%) the population (f,c) is also specialized, consisting entirely of farmers. In contrast, at point Em=(fm,cm) the population (f,c) consists of a nontrivial mix of farmers and city merchants.

The three points Ec, Em, and Ef are all equilibrium points for the Mees dynamical system, in the classical sense that the system is at rest at each of these points. This is true because df/dt and dc/dt are both zero at each of these points, implying that the system state vector (f,c) is not changing.

The specialized population equilibria Ec and Ef are both unstable. That is, starting at either of these specialized population points, any sufficiently small perturbation in the population precipates a movement away from the specialized population point and towards the mixed population point Em.

For example, suppose the economy starts at Ec with f=0% and c=100%. Suppose a perturbation occurs that causes a slight increase in f (hence a slight decrease in c). This movement in the state vector (f,c) to the right of Ec results in a positive sign for df/dt and a negative sign for dc/dt, hence f continues to increase and c continues to decrease. This results in a further movement of (f,c) to the right. This rightward movement in (f,c) only ceases when the system converges to Em, where once again the state vector comes to rest. A similar analysis shows why any sufficiently small perturbation away from Ef also results in the convergence of the system to Em.

In contrast, Em = (fm,cm) is locally stable. Starting at Em, an analysis analogous to the preceding arguments shows that any perturbation in (fm,cm) either in the direction of Ec=(0%,100%) (fewer farmers, more city merchants) or in the direction of Ef=(100%,0%) (more farmers, fewer city merchants) that still leaves the population mixed to some degree will result in an eventual return to Em.

Thus, Em is the unique attractor for the Mees dynamical system as modelled above. Its basin of attraction is the whole range of state vectors (f,c) lying strictly between Ec=(0,100%) and Ef=(100%,0%).

How might Em be destabilized by the introduction of increased opportunities for long-distance trading?

These increased trading opportunities might increase the relative attractiveness (profitability) of being a city merchant, resulting in a flattening out of the shape of the "phase curve" between Ec and Em and a leftward movement in Em along the horizontal axis in the direction of Ec.

The leftward movement of Em -- hence of (f,c) -- implies that city merchants are becoming a larger portion of the mixed population in the locally stable equilibrium Em.

                                                     f=100%
       df/dt                                         c=0%
        /|\                                           |
Positive |                                            |
Values   |                                            |  Negative
of df/dt |    -  -                                    |  Values
         | -         -                                |  of dc/dt
      0  |____________________________________________|
         |Ec            Em -                       -  | Ef
         |                     -               -      |
Negative |                           -   -            |  Positive
Values   |                                            |  Values
of df/dt |                                            |  of dc/dt
         |___________________________________________\|/
        f=0%                                        dc/dt
        c=100%

If this process continues, eventually the phase curve could completely drop below the horizontal axis, implying the disappearance altogether of the mixed equilibrium point Em; see the diagram below.

In this case the only locally stable equilibrium point would be Ec, where the community consists entirely of city merchants who obtain all their food through trade with other regions.

                                                     f=100%
       df/dt                                         c=0%
        /|\                                           |
Positive |                                            |
Values   |                                            |  Negative
of df/dt |                                            |  Values
         |                                            |  of dc/dt
      0  |____________________________________________|
         |Ec                                     -    | Ef
         | -                               -          |
Negative |     -                     -                |  Positive
Values   |            -     -                         |  Values
of df/dt |                                            |  of dc/dt
         |___________________________________________\|/
        f=0%                                        dc/dt
        c=100%

Suppose the extent of long-distance trading opportunities in the Mees dynamical system can be increased from 0 by increasing a parameter q starting at q=0.

Suppose, also, that the phase curve first drops completely below the horizontal line connecting Ec to Ef when q=q*, implying that the equilibrium Em is lost when the value of q hits q=q*.

The Mees dynamical system thus experiences a bifurcation at q=q*, since its set of equilibria changes abruptly from {Ec,Em,Ef} to {Ec,Ef} at q=q*.

2. Learning by Circulating (Batten, pp. 125-137)

Batten discusses how groups of merchants might come to form a crude mental model of a simple economic network (three linked cities) through the trial-and-error construction of a set of condition-action rules for profitable trade. Batten claims (p. 133) that this form of learning by circulating "contributed to the emergence of a new urban hierarchy in Europe." He relates his intuitive discussion to empirical data on the growth of cities in Europe from 1000 A.D. to 1400 A.D. (Table 4.2).

Batten then discusses how city growth in Europe during the eleventh and twelfth centuries exhibited the properties of an autocatalytic network, with the emergence of ever more nodes and links growing outward from previously established nodes and links. Two main commercial networks first emerged, one in the north on the shores of the Baltic and North Seas, and the other in the south on the shores of the Mediterranean and Adriatic Seas. Subsequently, traders operating in these two networks came into increasing contact with each other through a Middle European bridge: namely, the establishment of trading fairs, in particular the famous French fairs at Champagne in the twelfth and thirteenth centuries that attracted merchants from the whole of Europe.

Batten stresses (p. 136) that the Champagne fairs were important not only as a major market for international trade but also as the center of an embryonic foreign exchange market in which different currencies could be bought and sold at relative prices determined by supply and demand. Batten concludes (p. 137): "More than any other single activity, the fairs did most to bring about an end to the economic isolation that the West had suffered during the Middle Ages."

Nevertheless, the fairs did not evolve into manufacturing centers or cities. Rather, they eventually disappeared. Batten highlights four main factors contributing to this disappearance:

1. A new connection by sea between the north and the south, which meant that north and south traders could now directly exchange goods with each other without having to use the fairs as an intermediary;

2. The introduction of a more regular mail system;

3. The absorption of the Champagne fairs into the kingdom of France and the subsequent imposition of heavy royal taxation on fair trading;

4. The introduction of improved payment systems (central banking and credit systems established in Amsterdam and elsewhere), providing greater reliability and flexibility than the fairs could offer.

Batten (p. 137) identifies the final factor, improved payment systems, as being "perhaps the most influential in the long run" in promoting the development of a more fully connected European commercial trading network.

Glossary of Basic Concepts

Basic Concepts from Dynamical Systems Theory (Batten, p. 121; Clark, p. 100):

Batten (p. 121) uses "equilibrium" in the classical mathematical sense to refer to a possible resting point (or region) for a dynamical system -- that is, a point or region E in the space of all possible system states where all motion in the state of the dynamical system ceases.

An equilibrium E is globally stable if the state of the system converges to E from any initial state, and locally stable if the state of the system converges to E for all initial states in a sufficiently small neighborhood of E. Otherwise, E is unstable.

NOTE: See "Notes on Batten Chapter 1" for a more detailed discussion and illustrations of the meaning of "locally stable equilibria" for dynamical systems.

A locally stable equilibrium E for a dynamical system is also referred to as an attractor of the system. (Clark, Chapter 6, p. 100)

Let E denote an attractor for a dynamical system, and let B(E) denote the set of all possible initial states such that, starting anywhere in B(E), the state of the system necessarily converges to E. [Note that B(E) necessarily includes E, itself, and hence constitutes a "neighborhood" of E.] The set B(E) is called the basin of attraction for E.

A graphical depiction of the overall equilibria, attractors, and basins of attraction for a dynamical system is called a phase portrait (or phase diagram) for the system.

Suppose a system S depends on a parameter q, i.e., S = S(q). Suppose, starting at q=q*, a small change in q results in a change in the set of equilibria for system S and hence a significant structural change in its phase portrait. System S(q) is then said to undergo a bifurcation at q=q*.

Autocatalytic Network (Batten, p. 134):

Catalysis is any reaction brought about by a separate agent. An example of catalysis would be an increase in the rate of a chemical reaction induced by the presence of a material that itself remains chemically unchanged at the end of the reaction. Autocatalysis is the catalysis of a reaction by one of its products, implying that the reaction -- once set in motion -- tends to be self-reproducing.

Following Kauffman (1993, Chapter 7), an autocatalytic set is a set of member entities such that the formation of each member is catalyzed by other members of the set, implying that the set reproduces collectively. Kauffman has developed an origins of life theory based on the idea that life is an emergent self-organized property of collections of "catalytic polymers" (RNA sequences and/or peptides) that undergo a phase transition to an autocatalytic set. Kauffman refers to the "catalyzed reaction graph" associated with any autocatalytic set, in which each catalyzing agent is linked (in blue) with the reactions (depicted as red links) that it catalyzes.

A number of researchers extending Kauffman's ideas have used the phrase autocatalytic network to mean, in loose terms, a network in which the formation of new links and nodes encourages ("catalyzes") the formation of yet more links and nodes. Batten adopts this same usage.

Copyright © 2007 Leigh Tesfatsion. All Rights Reserved.