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Complex adaptive systems: Exploring the known, the unknown and the unknowable

Author: Simon A. Levin
Journal: Bull. Amer. Math. Soc. 40 (2003), 3-19
MSC (2000): Primary 92B05, 92D15, 92D40
Published electronically: October 9, 2002
MathSciNet review: 1943129
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Abstract: The study of complex adaptive systems, from cells to societies, is a study of the interplay among processes operating at diverse scales of space, time and organizational complexity. The key to such a study is an understanding of the interrelationships between microscopic processes and macroscopic patterns, and the evolutionary forces that shape systems. In particular, for ecosystems and socioeconomic systems, much interest is focused on broad scale features such as diversity and resiliency, while evolution operates most powerfully at the level of individual agents. Understanding the evolution and development of complex adaptive systems thus involves understanding how cooperation, coalitions and networks of interaction emerge from individual behaviors and feed back to influence those behaviors. In this paper, some of the mathematical challenges are discussed.

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  • Viggo Andreasen, Juan Lin, and Simon A. Levin, The dynamics of cocirculating influenza strains conferring partial cross-immunity, J. Math. Biol. 35 (1997), no. 7, 825–842. MR 1479341, DOI
  • Bak P. Bak, How Nature Works, Springer-Verlag, New York, 1997. Bart N. H. Barton, M. Turelli, 1994. Genetic and statistical analyses of strong selection on polygenic traits: What, me normal? Genetics 138 (1994), 913–941. bolk B. M. Bolker, S. W. Pacala, S. A. Levin, Moment methods for stochastic processes in continuous space and time, The Geometry of Ecological Interactions: Simplifying Spatial Complexity, Cambridge University Press, Cambridge, 2000, pp. 388–411. Bowl S. Bowles, Cultural evolution and human social structure, unpublished manuscript. Butt L. A. Buttel, R. Durrett, S. A. Levin, Competition and species packing in patchy environments, Theoret. Population Biol. 61 (2002), 265–276. Chan S. Chandrasekhar, Stochastic problems in physics and astronomy, Rev. Modern Phys. 15 (1943), 1–89.
  • E. Teramoto and M. Yamaguti (eds.), Mathematical topics in population biology, morphogenesis and neurosciences, Lecture Notes in Biomathematics, vol. 71, Springer-Verlag, Berlin, 1987. MR 913318
  • Jonathan Roughgarden, Robert M. May, and Simon A. Levin (eds.), Perspectives in ecological theory, Princeton University Press, Princeton, NJ, 1989. MR 1055088
  • James F. Crow and Motoo Kimura, An introduction to population genetics theory, Harper & Row, Publishers, New York-London, 1970. MR 0274068
  • Durl S. Durlauf, Statistical mechanics approaches to socioeconomic behavior, The Economy as a Complex Evolving System II, Addison-Wesley, Redwood City, CA, 1997. DurLev94a R. Durrett, S. A. Levin, Stochastic spatial models: A user’s guide to ecological applications, Philos. Trans. Roy. Soc. London Ser. B 343 (1994), 329–350. DurLev94b R. Durrett, S. A. Levin. The importance of being discrete and (spatial), Theoret. Population Biol. 46 (1994), 363–394. DurLev97 R. Durrett, S. A. Levin. Allelopathy in spatially distributed populations, J. Theoret. Biol. 185 (1997), 165–171. ErhLev P. Ehrlich, S. A. Levin, In preparation.
  • Ilan Eshel, Evolutionary and continuous stability, J. Theoret. Biol. 103 (1983), no. 1, 99–111. MR 714279, DOI
  • Falc D. S. Falconer, Introduction to Quantitative Genetics, The Ronald Press Co., New York, 1960. FBBC W. M. Fitch, R. M. Bush, C. A. Bender, N. J. Cox, Long term trends in the evolution of H(3) HA1 human influenza type A, Proc. Nat. Acad. Sci. U.S.A. 9 (1997), 7712–7718. FlGrLe G. Flierl, D. Grünbaum, S. Levin, D. Olson, From individuals to aggregations: the interplay between behavior and physics, J. Theoret. Biol. 196 (1999), 397–454. GMKM S. A. H. Geritz, J. A. J. Metz, E. Kisdi, G. Meszéna, Dynamics of adaptation and evolutionary branching, Phys. Rev. Lett. 78 (1997), 2024–2027. Gomo R. E. Gomory, The known, the unknown and the unknowable, Sci. Amer. 272 (1995), 120. Grun D. Grünbaum, Local processes and global patterns: Biomathematical models of bryozoan feeding currents and density dependent aggregations in Antarctic krill, Ph. D. thesis, Cornell University, Ithaca, NY, 1992. HamMay W. D. Hamilton, R. M. May, Dispersal in stable habitats, Nature 269 (1977), 578–581. IwaLev Y. Iwasa, S. A. Levin, The timing of life history events, J. Theoret. Biol. 172 (1995), 33–42. Jac1 F. Jacob, Evolution and tinkering, Science 196, (1977), 1161–1166. Jac2 F. Jacob, The Possible and the Actual, Pantheon, New York, 1982. Kauf S. Kauffman, The Origins of Order, Oxford University Press, New York, 1993.
  • Stuart Kauffman and Simon Levin, Towards a general theory of adaptive walks on rugged landscapes, J. Theoret. Biol. 128 (1987), no. 1, 11–45. MR 907587, DOI
  • KLDP A. P. Kinzig, S. A. Levin, J. Dushoff, S. Pacala, Limiting similarity and species packing in competition-colonization models, Amer. Natur. 153 1999, 371–383. Kirs J. Kirschner, Presentation for the epistemology panel discussion during the Chapman Conference on the Gaia Hypothesis, March 7-11, 1988, San Diego, CA. LenLove T. M. Lenton, J. E. Lovelock, Daisyworld is Darwinian: Constraints on adaptation are important for planetary self-regulation, J. Theoret. Biol. 206 (2000), 109–114. Lev78 S. A. Levin, On the evolution of ecological parameters, Ecological Genetics: The Interface Sciences, Springer-Verlag, New York, 1978, pp. 2–26. Lev99 S. A. Levin, Fragile Dominion: Complexity and the Commons, Perseus Books, Reading, MA, 1999. LevMorPow S. A. Levin, A. Morin, T. H. Powell, Patterns and processes in the distribution and dynamics of Antarctic krill, Scientific Committee for the Conservation of Antarctic Marine Living Resources Selected Scientific Papers Part 1, SC-CAMLR-SSP/5, CCAMLR. Hobart, Tasmania, Australia, 1989, pp. 281–299. LevMul S. A. Levin, H. C. Muller-Landau, The evolution of dispersal and seed size in plant communities, Evol. Ecology Res. 2 (2000), 409–435. LevPac S. A. Levin, S. W. Pacala, Theories of simplification and scaling of spatially distributed processes, Spatial Ecology: The Role of Space in Population Dynamics and Interspecific Interactions, Princeton University Press, Princeton, NJ, 1997, pp. 271–296. LevSeg76 S. A. Levin, L. A. Segel, An hypothesis for the origin of planktonic patchiness, Nature 259, no. 5545, (1976), 659. LevSeg82 S. A. Levin, L. A. Segel, Models of the influence of predation on aspect diversity in prey populations, J. Math. Biol. 14 (1982), 253–284. Lewo R. C. Lewontin, Evolution and the theory of games, J. Theoret. Biol. 1 (1961), 382–403.
  • S. Lin and B. W. Kernighan, An effective heuristic algorithm for the traveling-salesman problem, Operations Res. 21 (1973), 498–516. MR 359742, DOI
  • J. D. Murray, Mathematical biology, Biomathematics, vol. 19, Springer-Verlag, Berlin, 1989. MR 1007836
  • Nash1 J. F. Nash, Equilibrium points in n-person games, Proc. Nat. Acad. Sci. U.S.A. 36 (1950), 48–49. Nash2 J. F. Nash, Non-cooperative games, Ann. of Math. 54 (1951), 286–295. NewEbl M. E. J. Newman, G. J. Eble, Decline in extinction rates and scale invariance in the fossil record, Paleobiology 25 (1999), 434–439. NSDLCL J. Norberg, D. P. Swaney, J. Dushoff. J. Lin, R. Casagrandi, S. A. Levin, Phenotypic diversity and ecosystem functioning in changing environments: A theoretical framework, Proc. Nat. Acad. Sci. U.S.A. 98, no. 20, (2001), 11376–11381. NoBoMa M. A. Nowak, S. Bonhoeffer, R. M. May, Spatial games and the maintenance of cooperation, Proc. Nat. Acad. Sci. U.S.A. 91 (1994), 4877–4881. NoKrak M. Nowak, D. Krakauer, The evolution of language, Proc. Nat. Acad. Sci. U.S.A. 96 (1999), 8028–8033. Oku A. Okubo, Diffusion-induced instability in model ecosystems, Chesapeake Bay Institute Technical Report 86, Johns Hopkins University, Baltimore, MD, 1974, 17 pp. PCSSKRR S. W. Pacala, C. D. Canham, J. Saponara, J. A. Silander, R. K. Kobe, E. Ribbens, Forest models defined by field measurements: II. Estimation, error analysis and dynamics, Ecol. Mono. 66 (1996), 1–43. PacLev97 S. W. Pacala, S. A. Levin, Biologically generated spatial pattern and the coexistence of competing species, Spatial Ecology: The Role of Space in Population Dynamics and Interspecific Interactions. Princeton University Press, Princeton, NJ, 1997, pp. 204–232. Sak S. Sakai, A model for group structure and its behavior, Biophysics (Japan), 13 (1973), 82–90. SegLev L. A. Segel, S. A. Levin, Applications of nonlinear stability theory to the study of the effects of diffusion on predator prey interactions, Topics in Statistical Mechanics and Biophysics: A Memorial to Julius L. Jackson. American Institute of Physics, New York, 1976, pp. 123–152. Sky B. Skyrms, Evolution of the Social Contract, Cambridge University Press, New York, 1996, 146 pp. Slob L. B. Slobodkin, The strategy of evolution, Amer. Sci. 52 (1964), 342–357. MSmi J. Maynard Smith, Evolution and the Theory of Games, Cambridge University Press, Cambridge, 1982. Tur A. M. Turing, The chemical basis of morphogenesis, Philo. Trans. Roy. Soc. London Ser. B 237 (1952), 37–72. Vau K. J. Vaughan, Invisible hand, The New Palgrave: A Dictionary of Economics, Stockton Press, New York, 1987, pp. 997–999.
  • Duncan J. Watts, Small worlds, Princeton Studies in Complexity, Princeton University Press, Princeton, NJ, 1999. The dynamics of networks between order and randomness. MR 1716136
  • WatStr D. J. Watts, S. H. Strogatz, Collective dynamics of ‘small-world’ networks, Nature 393 (1998), 440–442. WebElHam L. H. Weber, S. Z. El-Sayed, I. Hampton, The variance spectra of phytoplankton, krill and water temperature in the Antarctic Ocean south of Africa, Deep-Sea Res. 33 (1986), 1327–1343.

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Additional Information

Simon A. Levin
Affiliation: Department of Ecology and Evolutionary Biology, Princeton University, Princeton, New Jersey 08544-1003

Keywords: Complex adaptive systems, self-organization, natural selection, ecosystems
Received by editor(s): December 12, 2000
Received by editor(s) in revised form: February 21, 2002
Published electronically: October 9, 2002
Additional Notes: It is a pleasure to acknowledge the support of the Alfred P. Sloan Foundation, grant award 97-3-5, and of the National Science Foundation, grant award DEB-0083566. Jonathan Dushoff and Helene Muller-Landau provided useful comments.
Article copyright: © Copyright 2002 American Mathematical Society