----------------- HES POSTING ----------------- Allow me to add a further note to my discussion of treatment of the equilibrium concept in biology/ ecology. There is another figure in here who had a well developed idea of equilibrium, even arguably of general equilibrium to some extent, and more interestingly of dynamic equilibrium (and still more interestingly of dynamic non-equilibria), although curiously enough he was largely drawing on physics ideas. This was Alfred J. Lotka, whose magnum opus was _Elements of Physical Biology_, 1925, Baltimore: Williams and Wilkins (reprinted in 1945 as _Elements of Mathematical Biology_). Lotka is most famous as the originator of the idea of the predator-prey cycle, initially in 1920, but fully discussed in his 1925 book. It was taken up and advanced further in 1931 by Volterra and more later by Kolmogorov, with both Richard Goodwin and Paul Samuelson applying it to economics in 1967. Samuelson was very heavily influenced by Lotka in his analysis of the relationship between statics and dynamics using the Correspondence Principle, and cites Lotka in his _Foundations of Economic Analysis_. Lotka was aware of different concepts of equilibrium. Thus, in Chap. 9 he drew on Pareto as well as Le Chatelier and Maxwell and Poincare for his discussion of static equilibrium. He then distinguished moving equilibria from stationary states (and analyzed their local stability). He noted a "kinematic" equilibrium" in which "velocities vanish," also a "dynamic" equilibrium in which "forces are balanced and the resultant force vanishes," an "energetic" concept in "virtual work done in any small displacement compatible with constraints vanishes," which is a minimization of potential energy concept. Furthermore he saw evolution as an irreversible process driven by the Second Law of Thermodynamics and he labeled as "quasi-equilibria" states maintained by a "dissipation or degradation of available energy." He also consciously used combined human-animal examples (mosquitos and humans, for example) and said that he was analyzing "the biological basis for economics." I note that he studied most of the kinds of dynamic patterns of fluctuations that we now know of with the major exception of chaotic dynamics. He well understood the important concept of the bifurcation of dynamic equilibria, which he got from Poincare. His work was enormously influential on another physicist who went into ecology, Robert May, whose 1976 article in Nature was the first to consciously suggest the application of chaos theory to economics, although some others had observed such patterns in economic models without understanding what they were observing (Strotz et al, 1953). References: Alfred J. Lotka, 1920. "Analytical Notes on Certain Rhythmic Relations in Organic Systems," Proceedings of the National Academy of Sciences of the United States, 6, 410-415. Alfred J. Lotka, 1925. Elements of Physical Biology. Baltimore: Williams and Wilkns (reprinted in 1945 as Elements of Mathematical Biology) Robert H. Strotz, J.C. McAnulty, and Joseph B. Naines, Jr. 1953. "Goodwin's Nonlinear Theory of the Business Cycle: An Electro-Analog Solution," Econometrica, 21, 390-411. [speaking of engineering approaches] Richard M. Goodwin, 1967. "A Growth Cycle," in C.H. Feinstein, ed. Socialism, Capitalism and Economic Growth. Cambridge: Cambridge University Press. Paul A. Samuelson, 1967. "A Universal Cycle?" in R. Henn, ed. Methods of Operations Research III. Muhlgasse: Verlag Anton Hain. Robert M. May, 1976. "Simple mathematical models with very complicated dynamics," Nature, 269, 471-477. Barkley Rosser ------------ FOOTER TO HES POSTING ------------ For information, send the message "info HES" to [log in to unmask]