Skip to Main Content

Bulletin of the American Mathematical Society

The Bulletin publishes expository articles on contemporary mathematical research, written in a way that gives insight to mathematicians who may not be experts in the particular topic. The Bulletin also publishes reviews of selected books in mathematics and short articles in the Mathematical Perspectives section, both by invitation only.

ISSN 1088-9485 (online) ISSN 0273-0979 (print)

The 2020 MCQ for Bulletin of the American Mathematical Society is 0.84.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

Complex adaptive systems: Exploring the known, the unknown and the unknowable
HTML articles powered by AMS MathViewer

by Simon A. Levin PDF
Bull. Amer. Math. Soc. 40 (2003), 3-19 Request permission

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.
References
  • 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 10.1007/s002850050079
    • 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.
  • J. J. Corliss, Upper limits to the real roots of a real algebraic equation, Amer. Math. Monthly 46 (1939), 334–338. MR 4, DOI 10.1080/00029890.1939.11998880
  • 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, DOI 10.1007/978-3-642-93360-8
  • Jonathan Roughgarden, Robert M. May, and Simon A. Levin (eds.), Perspectives in ecological theory, Princeton University Press, Princeton, NJ, 1989. MR 1055088, DOI 10.1515/9781400860180
  • 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 10.1016/0022-5193(83)90201-1
    • 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 10.1016/S0022-5193(87)80029-2
    • 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 10.1287/opre.21.2.498
  • J. D. Murray, Mathematical biology, Biomathematics, vol. 19, Springer-Verlag, Berlin, 1989. MR 1007836, DOI 10.1007/978-3-662-08539-4
  • Morgan Ward and R. P. Dilworth, The lattice theory of ova, Ann. of Math. (2) 40 (1939), 600–608. MR 11, DOI 10.2307/1968944
  • C. J. Everett Jr., Annihilator ideals and representation iteration for abstract rings, Duke Math. J. 5 (1939), 623–627. MR 13
    • 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, DOI 10.1515/9780691188331
    • 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.
Similar Articles
  • Retrieve articles in Bulletin of the American Mathematical Society with MSC (2000): 92B05, 92D15, 92D40
  • Retrieve articles in all journals with MSC (2000): 92B05, 92D15, 92D40
Additional Information
  • Simon A. Levin
  • Affiliation: Department of Ecology and Evolutionary Biology, Princeton University, Princeton, New Jersey 08544-1003
  • Email: slevin@princeton.edu
  • 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.
  • © Copyright 2002 American Mathematical Society
  • Journal: Bull. Amer. Math. Soc. 40 (2003), 3-19
  • MSC (2000): Primary 92B05, 92D15, 92D40
  • DOI: https://doi.org/10.1090/S0273-0979-02-00965-5
  • MathSciNet review: 1943129