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The dynamical theory of coevolution: A derivation from stochastic ecological processes. (English) Zbl 0845.92013

Summary: We develop a dynamical theory of coevolution in ecological communities. The derivation explicitly accounts for the stochastic components of evolutionary change and is based on ecological processes at the level of the individual. We show that the coevolutionary dynamic can be envisaged as a directed random walk in the community’s trait space. A quantitative description of this stochastic process in terms of a master equation is derived. By determining the first jump moment of this process we abstract the dynamics of the mean evolutionary path. To first order the resulting equation coincides with a dynamics that has frequently been assumed in evolutionary game theory. Apart from recovering this canonical equation we systematically establish the underlying assumptions. We provide higher order corrections and show that these can give rise to new, unexpected evolutionary effects including shifting evolutionary isoclines and evolutionary slowing down of mean paths as they approach evolutionary equilibria. Extensions of the derivation to more general ecological settings are discussed. In particular we allow for multi-trait coevolution and analyze coevolution under nonequilibrium population dynamics.

MSC:

92D15 Problems related to evolution
92D40 Ecology
60G50 Sums of independent random variables; random walks
60J20 Applications of Markov chains and discrete-time Markov processes on general state spaces (social mobility, learning theory, industrial processes, etc.)
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[1] Abrams, P. A.: Adaptive responses of predators to prey and prey to predators: the failure of the arms-race analogy. Evolution40, 1229–1247 (1986) · doi:10.2307/2408950
[2] Abrams, P. A., Matsuda, H., Harada, Y.: Evolutionarily unstable fitness maxima and stable fitness minima of continuous traits. Evol. Ecol.7, 465–487 (1993) · doi:10.1007/BF01237642
[3] Bailey, N. T. J.: The elements of stochastic processes. New York: John Wiley and Sons 1964 · Zbl 0127.11203
[4] Baker, G. L., Gollub, J. P.: Chaotic dynamics: an introduction. Cambridge: Cambridge University Press 1990 · Zbl 0712.58002
[5] Brown, J. S., Vincent, T. L.: Coevolution as an evolutionary game. Evolution41, 66–79 (1987a) · Zbl 0644.92013 · doi:10.2307/2408973
[6] Brown, J. S., Vincent, T. L.: Predator-prey coevolution as an evolutionary game. In: Cohen, Y. (ed.) Applications of Control Theory in Ecology, pp. 83–101. Lecture Notes in Biomathematics 73. Berlin: Springer Verlag 1987b
[7] Brown, J. S., Vincent, T. L.: Organization of predator-prey communities as an evolutionary game. Evolution46, 1269–1283 (1992) · doi:10.2307/2409936
[8] Christiansen, F. B.: On conditions for evolutionary stability for a continuously varying character. Amer. Natur.138, 37–50 (1991) · doi:10.1086/285203
[9] Dawkins, R.: The selfish gene. Oxford: Oxford University Press 1976
[10] Dawkins, R., Krebs, J. R.: Arms races between and within species. Proc. Roy. Soc. Lond. B205, 489–511 (1979) · doi:10.1098/rspb.1979.0081
[11] Dieckmann, U.: Coevolutionary dynamics of stochastic replicator systems. Jülich Germany: Berichte des Forschungszentrums Jülich (Jül-3018) 1994
[12] Dieckmann, U., Marrow, P., Law, R.: Evolutionary cycles in predator-prey interactions: population dynamics and the Red Queen. J. Theor. Biol.176, 91–102 (1995) · doi:10.1006/jtbi.1995.0179
[13] Ebeling, W., Feistel, R.: Physik der Selbstorganisation und Evolution. Berlin: Akademie-Verlag 1982
[14] Eckmann, J.-P., Ruelle, D.: Ergodic theory of chaos and strange attractors. Rev. Mod. Phys.57, 617–656 (1985) · Zbl 0989.37516 · doi:10.1103/RevModPhys.57.617
[15] Emlen, J. M.: Evolutionary ecology and the optimality assumption. In: Dupre, J. (ed.) The latest on the best, pp. 163–177. Cambridge: MIT Press 1987
[16] Eshel, I.: Evolutionary and continuous stability. J. Theor. Biol.103, 99–111 (1983) · doi:10.1016/0022-5193(83)90201-1
[17] Eshel, I., Motro, U.: Kin selection and strong stability of mutual help. Theor. Pop. Biol.19, 420–433 (1981) · Zbl 0473.92014 · doi:10.1016/0040-5809(81)90029-0
[18] Falconer, R. A.: Introduction to quantitative genetics. 3rd Edition. Harlow: Longman 1989
[19] Fisher, R. A.: The genetical theory of natural selection. New York: Dover Publications 1958 · JFM 56.1106.13
[20] Futuyma, D. J., Slatkin, M.: Introduction. In: Futuyma, D. J., Slatkin, M. (eds.) Coevolution, pp. 1–13. Sanderland Massachusetts: Sinauer Associates 1983
[21] Gillespie, D. T.: A general method for numerically simulating the stochastic time evolution of coupled chemical reactions. J. Comp. Phys.22, 403–434 (1976) · doi:10.1016/0021-9991(76)90041-3
[22] Goel, N. S., Richter-Dyn, N., Stochastic models in biology. New York: Academic Press 1974
[23] Hofbauer, J., Sigmund, K.: Theory of evolution and dynamical systems. New York: Cambridge University Press 1988 · Zbl 0678.92010
[24] Hofbauer, J., Sigmund, K.: Adaptive dynamics and evolutionary stability. Appl. Math. Lett.3, 75–79 (1990) · Zbl 0709.92015 · doi:10.1016/0893-9659(90)90051-C
[25] Iwasa, Y., Pomiankowski, A., Nee, S.: The evolution of costly mate preferences. II. The ”handicap” principle. Evolution45, 1431–1442 (1991) · doi:10.2307/2409890
[26] Kimura, M.: The neutral theory of molecular evolution. Cambridge: Cambridge University Press 1983
[27] Kubo, R., Matsuo, K., Kitahara, K.: Fluctuation and relaxation of macrovariables. J. Stat. Phys.9, 51–96 (1973) · doi:10.1007/BF01016797
[28] Lande, R.: Quantitative genetic analysis of multivariate evolution, applied to brain: body size allometry. Evolution33, 402–416 (1979) · doi:10.2307/2407630
[29] Lawlor, L. R., Maynard Smith, J.: The coevolution and stability of competing species. Amer. Natur.110, 79–99 (1976) · doi:10.1086/283049
[30] Lewontin, R. C.: Pitness, survival, and optimality. In: Horn, D. J., Stairs, G. R., Mitchell, R. D. (eds.) Analysis of Ecological Systems, pp. 3–21. Ohio State University Press 1979
[31] Lewontin, R. C.: Gene, organism and environment. In: Bendall, D. S. (ed.) Evolution from molecules to men, pp. 273–285. Cambridge: Cambridge University Press 1983
[32] Lewontin, R. C.: The shape of optimality. In: Dupre, J. (ed.) The latest on the best, pp. 151–159. Cambridge: MIT Press 1987
[33] Loeschke, V. (ed.): Genetic constraints on adaptive evolution. Berlin: Springer-Verlag 1987
[34] Mackay, T. F. C.: Distribution of effects of new mutations affecting quantitative traits. In: Hill, W. G., Thompson, R., Woolliams, J. A. (eds.) Proc. 4th world congress on genetics applied to livestock production, pp. 219–228. 1990
[35] Marrow, P., Cannings, C.: Evolutionary instability in predator-prey systems. J. Theor. Biol.160, 135–150 (1993) · doi:10.1006/jtbi.1993.1008
[36] Marrow, P., Dieckmann, U., Law, R.: Evolutionary dynamics of predator-prey systems: an ecological perspective. J. Math. Biol.34, 556–578 (1996) · Zbl 0845.92018 · doi:10.1007/BF02409750
[37] Marrow, P., Law, R., Cannings, C.: The coevolution of predator-prey interactions: ESSs and Red Queen dynamics. Proc. Roy. Soc. Lond. B250, 133–141 (1992) · doi:10.1098/rspb.1992.0141
[38] Maynard Smith, J.: Evolution and the theory of games. Cambridge: Cambridge University Press 1982 · Zbl 0526.90102
[39] Maynard Smith, J., Burian, R., Kauffman, S. Alberch, P., Campbell, J., Goodwin, B., Lande, R., Raup, D., Wolpert, L.: Developmental constraints and evolution. Q. Rev. Biol.60, 265–287 (1985) · doi:10.1086/414425
[40] Maynard Smith, J., Price, G. R.: The logic of animal conflict. Nature Lond.246, 15–18 (1973) · Zbl 1369.92134 · doi:10.1038/246015a0
[41] Metz, J. A. J., Nisbet, R. M., Geritz, S. A. H.: How should we define ”fitness” for general ecological scenarios? Trends Ecol. Evol.7, 198–202 (1992) · doi:10.1016/0169-5347(92)90073-K
[42] Metz, J. A. J, Geritz, S. A. H., Iwasa, Y.: On the dynamical classification of evolutionarily singular strategies. University of Leiden Preprint (1994)
[43] Nicolis, J. S.: Dynamics of hierarchical systems. Berlin: Springer-Verlag 1986 · Zbl 0588.93005
[44] Ott, E.: Chaos in dynamical systems. Cambridge: Cambridge University Press 1993 · Zbl 0792.58014
[45] Rand, D. A., Wilson, H. B.: Evolutionary catastrophes, punctuated equilibria and gradualism in ecosystem evolution. Proc. Roy. Soc. Lond. B253, 239–244 (1993) · doi:10.1098/rspb.1993.0093
[46] Rand, D. A., Wilson, H. B., McGlade, J. M.: Dynamics and evolution: evolutionarily stable attractors, invasion exponents and phenotype dynamics. Phil. Trans. Roy. Soc. Lond. B343, 261–283 (1994) · doi:10.1098/rstb.1994.0025
[47] Reed, J., Stenseth, N. C.: On evolutionarily stable strategies. J. Theor. Biol.108, 491–508 (1984) · doi:10.1016/S0022-5193(84)80075-2
[48] Rosenzweig, M. L., Brown, J. S., Vincent, T. L.: Red Queens and ESS: the coevolution of evolutionary rates. Evol. Ecol.1, 59–94 (1987) · doi:10.1007/BF02067269
[49] Roughgarden, J.: The theory of coevolution. In: Futuyma, D. J., Slatkin, M. (eds.) Coevolution, pp. 33–64. Sunderland Massachusetts: Sinauer Associates 1983.
[50] Saloniemi, I.: A coevolutionary predator-prey model with quantitative characters. Amer. Natur.141, 880–896 (1993) · doi:10.1086/285514
[51] Schuster, H. G.: Deterministic chaos: an introduction. Weinheim: VCH Verlagsgesellschaft 1989 · Zbl 0709.58002
[52] Serra, R., Andretta, M., Compiani, M., Zanarini, G.: Introduction to the physics of complex systems. Oxford: Pergamon Press 1986 · Zbl 0667.58044
[53] Stearns, S. C.: The evolution of life histories. Oxford: Oxford University Press 1992
[54] Takada, T., Kigami, J.: The dynamical attainability of ESS in evolutionary games. J. Math. Biol.29, 513–529 (1991) · Zbl 0734.92021 · doi:10.1007/BF00164049
[55] Taper, M. L., Case, T. J.: Models of character displacement and the theoretical robustness of taxon cycles. Evolution46, 317–333 (1992) · doi:10.2307/2409853
[56] Taylor, P. D.: Evolutionary stability in one-parameter models under weak selection. Theor. Pop. Biol.36, 125–143 (1989) · Zbl 0684.92014 · doi:10.1016/0040-5809(89)90025-7
[57] van Kampen, N. G.: Fundamental problems in statistical mechanics of irreversible processes. In: Cohen, E. G. D. (ed.) Fundamental problems in statistical mechanics, pp. 173–202. Amsterdam: North Holland 1962 · Zbl 0111.44203
[58] van Kampen, N. G.: Stochastic processes in physics and chemistry. Amsterdam: North Holland 1981 · Zbl 0511.60038
[59] Vincent, T. L.: Strategy dynamics and the ESS. In: Vincent, T. L., Mees, A. I., Jennings, L. S. (eds.) Dynamics of complex interconnected biological systems, pp. 236–262. Basel: Birkhäuser 1990 · Zbl 0825.92096
[60] Vincent, T. L., Cohen, Y., Brown, J. S.: Evolution via strategy dynamics. Theor. Pop. Biol.44, 149–176 (1993) · Zbl 0788.92018 · doi:10.1006/tpbi.1993.1023
[61] Wissel, C., Stöcker, S.: Extinction of populations by random influences. Theor. Pop. Biol.39, 315–328 (1991) · Zbl 0725.92024 · doi:10.1016/0040-5809(91)90026-C
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