Complexity_ A Guided Tour - Melanie Mitchell [148]
Chapter 1
“Ideas thus made up”: Locke, J., An Essay Concerning Human Understanding. Edited by P. H. Nidditch. Oxford: Clarendon Press, 1690/1975, p. 2.12.1.
“Half a million army ants”: This description of army ant behavior was gleaned from the following sources: Franks, N. R., Army ants: A collective intelligence. American Scientist, 77(2), 1989, pp. 138–145; and Hölldobler, B. and Wilson, E. O., The Ants. Cambridge, MA: Belknap Press, 1990.
“The solitary army ant”: Franks, N. R., Army ants: A collective intelligence. American Scientist, 77(2), 1989, pp. 138–145.
“what some have called a ‘superorganism’ ”: E.g., Hölldobler, B. and Wilson, E. O., The Ants. Cambridge, MA: Belknap Press, 1990, p. 107.
“I have studied E. burchelli”: Franks, N. R., Army ants: A collective intelligence. American Scientist, 77(2), 1989, p. 140.
“Douglas Hofstadter, in his book Gödel, Escher, Bach”: Hofstadter, D. R., Ant fugue. In Gödel, Escher, Bach: an Eternal Golden Braid. New York: Basic Books, 1979.
Chapter 2
“It makes me so happy”: Stoppard, T., Arcadia. New York: Faber & Faber, 1993, pp. 47–48.
“nature is exceedingly simple”: Quoted in Westfall, R. S., Never at Rest: A Biography of Isaac Newton. Cambridge: Cambridge University Press, 1983, p. 389.
“it was possible, in principle, to predict everything for all time”: Laplace, P. S., Essai Philosophique Sur Les Probabilites. Paris: Courcier, 1814.
“influences whose physical magnitude”: Liu, Huajie, A brief history of the concept of chaos, 1999 [http://members.tripod.com/~huajie/Paper/chaos.htm].
“If we knew exactly the laws of nature”: Poincaré, H., Science and Method. Translated by Francis Maitland. London: Nelson and Sons, 1914.
“Edward Lorenz found”: Lorenz, E. N., Deterministic nonperiodic flow. Journal of Atmospheric Science, 357, 1963, pp. 130–141.
“This is a linear system”: One could argue that this is not actually a linear system, since the population increases exponentially over time: nt = 2tn0. However, it is the map from nt to nt+1 that is the linear system being discussed here.
“an equation called the logistic model”: From [http://mathworld.wolfram.com/LogisticEquation.html]: “The logistic equation (sometimes called the Verhulst model, logistic map, or logistic growth curve) is a model of population growth first published by Pierre Verhulst (1845). The model is continuous in time, but a modification of the continuous equation to a discrete quadratic recurrence equation is also known as the logistic equation.” The logistic map is the name given to one particularly useful way of expressing the logistic model.
“I won’t give the actual equation”: Here is the logistic model:
where nt is the population at the current generation and k is the carrying capacity. To derive the logistic map from this model, let xt = nt/k, and R = (birthrate − deathrate). Note that xt is the “fraction of carrying capacity”: the ratio of the current population to the maximum possible population. Then
xt+1 = Rxt (1 − xt).
Because the population size nt is always between 0 and k, xt is always between 0 and 1.
“the logistic map”: The following references provide technical discussions of the logistic map, aimed at the general scientifically educated reader: Feigenbaum, M. J., Universal behavior in nonlinear systems. Los Alamos Science, 1 (1), 1980, pp. 4–27; Hofstadter, D. R., Mathematical chaos and strange attractors. In Metamagical Themas. New York: Basic Books, 1985; Kadanoff, Leo P., Chaos, A view of complexity in the physical sciences. In From Order to Chaos: Essays: Critical, Chaotic, and Otherwise. Singapore: World Scientific, 1993.
“a 1971 article by the mathematical biologist Robert May”: May, R. M., Simple mathematical models with very complicated dynamics. Nature, 261, pp. 459–467, 1976.
“Stanislaw Ulam, John von Neumann, Nicholas Metropolis, Paul Stein, and Myron Stein”: Ulam, S. M., and von Neumann, J., Bulletin of the American Mathematical Society, 53,