View More View Less
  • 1 Department of Biological Physics, Eötvös University H-1117 Budapest, Pázmány Péter sétány 1A, Hungary
  • | 2 Department of Mathematics, University of Turku Turku, Finland
  • | 3 Adaptive Dynamics Network, International Institute for Applied Systems Analysis Laxenburg, Austria
  • | 4 Department of Mathematics University of Turku Turku, Finland
  • | 5 Section Theoretical Evolutionary Biology, Institute of Evolutionary and Ecological Sciences (EEW), Leiden University Leiden, The Netherlands
Restricted access

Matrix game theory and optimisation models offer two radically different perspectives on the outcome of evolution. Optimisation models consider frequency-independent selection and envisage evolution as a hill-climbing process on a constant fitness landscape, with the optimal strategy corresponding to the fitness maximum. By contrast, in evolutionary matrix games selection is frequency-dependent and leads to fitness equality among alternative strategies once an evolutionarily stable strategy has been established. In this review we demonstrate that both optimisation models and matrix games represent limiting cases of the general framework of nonlinear frequency-dependent selection. Adaptive dynamics theory considers arbitrary nonlinear frequency and density dependence and envisages evolution as proceeding on an adaptive landscape that changes its shape according to which strategies are present in the population. In adaptive dynamics, evolutionarily stable strategies correspond to conditional fitness maxima: the ESS is characterised by the fact that it has the highest fitness if it is the established strategy. In this framework it can also be shown that dynamical attainability, evolutionary stability, and invading potential of strategies are pairwise independent properties. In optimisation models, on the other hand, these properties become linked such that the optimal strategy is always attracting, evolutionarily stable and can invade any other strategy. In matrix games fitness is a linear function of the potentially invading strategy and can thus never exhibit an interior maximum: Instead, the fitness landscape is a plane that becomes horizontal once the ESS is established. Due to this degeneracy, invading potential is part of the ESS definition for matrix games and dynamical attainability is a dependent property. We conclude that nonlinear frequency-dependent theory provides a unifying framework for overcoming the traditional divide between evolutionary optimisation models and matrix games.

  • Apaloo, J. (1997): Revisiting strategic models of evolution: The concept of neighborhood invader strategies. Theor. Popul. Biol.52: 52-71.

    'Revisiting strategic models of evolution: The concept of neighborhood invader strategies ' () 52 Theor. Popul. Biol. : 52 -71.

    • Search Google Scholar
  • Bishop, D. T. and Cannings, C. (1978): A generalised war of attrition. J. Theor. Biol.70: 85-124.

    'A generalised war of attrition ' () 70 J. Theor. Biol. : 85 -124.

  • Boots, M. and Haraguchi, Y. (1999): The evolution of costly resistance in host-parasite systems. Amer. Nat.153: 359-370.

    'The evolution of costly resistance in host-parasite systems ' () 153 Amer. Nat. : 359 -370.

    • Search Google Scholar
  • Charlesworth, B. and León, J. A. (1976): The relation of reproductive effort to age. Amer. Nat. 110: 449-459.

    'The relation of reproductive effort to age ' () 110 Amer. Nat. : 449 -59.

  • Charnov, E. L. (1976): Optimal foraging, the marginal value theorem. Theor. Popul. Biol.9: 129-136.

    'Optimal foraging, the marginal value theorem ' () 9 Theor. Popul. Biol. : 129 -136.

  • Charlesworth, B. (1980): Evolution in Age-structured Populations. Cambridge University Press, Cambridge.

    Evolution in Age-structured Populations , ().

  • Cheptou, P. O. and Mathias, A. (2001): Can varying inbreeding depression select for intermediary selfing rates? Amer. Nat.157: 361-373.

    'Can varying inbreeding depression select for intermediary selfing rates? ' () 157 Amer. Nat. : 361 -373.

    • Search Google Scholar
  • Cressman, R. (1996): Frequency-dependent stability for two-species interactions. Theor. Popul. Biol.49: 189-210.

    'Frequency-dependent stability for two-species interactions ' () 49 Theor. Popul. Biol. : 189 -210.

    • Search Google Scholar
  • Christiansen, F. B. (1991): On conditions for evolutionary stability for a continuously varying character. Amer. Nat.138: 37-50.

    'On conditions for evolutionary stability for a continuously varying character ' () 138 Amer. Nat. : 37 -50.

    • Search Google Scholar
  • Day, T. (2000): Competition and the effect of spatial resource heterogeneity on evolutionary diversification. Amer. Nat.155: 790-803.

    'Competition and the effect of spatial resource heterogeneity on evolutionary diversification ' () 155 Amer. Nat. : 790 -803.

    • Search Google Scholar
  • Cressman, R. and Hines, W. G. S. (1984): Correction to the appendix of 'Three characterizations of population strategy stability'. J. Appl. Prob.21: 213-214.

    'Correction to the appendix of 'Three characterizations of population strategy stability' ' () 21 J. Appl. Prob. : 213 -214.

    • Search Google Scholar
  • Darwin, Ch. (1859): On the Origin of Species by Means of Natural Selection or the Preservation of Favoured Races in the Struggle for Life. Facsimile publication: Harvard University Press, 1964.

    On the Origin of Species by Means of Natural Selection or the Preservation of Favoured Races in the Struggle for Life , ().

    • Search Google Scholar
  • Dieckmann, U. (1994): Coevolutionary Dynamics of Stochastic Replicator Systems. Central Library of the Research Center Juelich, Germany.

    Coevolutionary Dynamics of Stochastic Replicator Systems , ().

  • Dieckmann, U. (1997): Can adaptive dynamics invade? Trends Ecol. Evol.12: 128-131.

    'Can adaptive dynamics invade? ' () 12 Trends Ecol. Evol. : 128 -31.

  • Dieckmann, U. and Doebeli, M. (1999): On the origin of species by sympatric speciation. Nature400: 354-357.

    'On the origin of species by sympatric speciation ' () 400 Nature : 354 -357.

  • Geritz, S. A. H. and Kisdi, É. (in press): Adaptive dynamics and evolutionary branching in mutation-limited evolution, In Dieckmann, U. and Metz, J. A. J. (eds): Elements of Adaptive Dynamics. Cambridge University Press, Cambridge.

    Elements of Adaptive Dynamics , ().

  • Dieckmann, U. and Law, R. (1996): The dynamical theory of coevolution: a derivation from stochastic ecological processes. J. Math. Biol.34: 579-612.

    'The dynamical theory of coevolution: a derivation from stochastic ecological processes ' () 34 J. Math. Biol. : 579 -612.

    • Search Google Scholar
  • Abrams, P. A., Matsuda, H. and Harada, Y. (1993): Evolutionarily unstable fitness maxima and stable fitness minima of continuous traits. Evol. Ecol.7: 465-487.

    'Evolutionarily unstable fitness maxima and stable fitness minima of continuous traits ' () 7 Evol. Ecol. : 465 -487.

    • Search Google Scholar
  • Abrams, P. A. and Matsuda, H. (1994): The evolution of traits that determine ability in competitive contests. Evol. Ecol.8: 667-686.

    'The evolution of traits that determine ability in competitive contests ' () 8 Evol. Ecol. : 667 -686.

    • Search Google Scholar
  • Geritz, S. A. H., Van Der Meijden, E. and Metz, J. A. J. (1999): Evolutionary dynamics of seed size and seedling competitive ability. Theor. Popul. Biol.55: 324-343.

    'Evolutionary dynamics of seed size and seedling competitive ability ' () 55 Theor. Popul. Biol. : 324 -343.

    • Search Google Scholar
  • Geritz, S. A. H., Gyllenberg, M., Jacobs, F. J. A. and Parvinen, K. (in press): Invasion dynamics and attractor inheritance. J. Math. Biol.

  • Geritz, S. A. H., Gyllenberg, M. and Kisdi, É. (in prep.): Attractor inheritance and limiting similarity in evolution by small mutation steps.

  • Gross, M. R. (1985): Disruptive selection for alternative life histories in salmon. Nature313: 47-48.

    'Disruptive selection for alternative life histories in salmon ' () 313 Nature : 47 -48.

  • Hamilton, W. D. (1967): Extraordinary sex ratios. Science156: 477-488.

    'Extraordinary sex ratios ' () 156 Science : 477 -488.

  • Kisdi, É. (1999): Evolutionary branching under asymmetric competition. J. Theor. Biol.197: 149-162.

    'Evolutionary branching under asymmetric competition ' () 197 J. Theor. Biol. : 149 -162.

  • Lande, R. (1976): Natural selection and random genetic drift in phenotypic evolution. Evolution30: 314-334.

    'Natural selection and random genetic drift in phenotypic evolution ' () 30 Evolution : 314 -334.

    • Search Google Scholar
  • Heino, M., Metz, J. A. J. and Kaitala, V. (1998): The enigma of frequency-dependent selection. Trends Ecol. Evol.13: 367-370.

    'The enigma of frequency-dependent selection ' () 13 Trends Ecol. Evol. : 367 -370.

  • Hernandez, M. J. and León, J. A. (1995): Evolutionary perturbations of optimal life histories. Evol. Ecol.9: 478-494.

    'Evolutionary perturbations of optimal life histories ' () 9 Evol. Ecol. : 478 -494.

  • Hines, W. G. S. (1980): Three characterizations of population strategy stability. J. Appl. Prob.17: 333-340.

    'Three characterizations of population strategy stability ' () 17 J. Appl. Prob. : 333 -340.

    • Search Google Scholar
  • Hines, W. G. S. (1987): Evolutionary stable strategies: a review of basic theory. Theor. Popul. Biol.31: 195-272.

    'Evolutionary stable strategies: a review of basic theory ' () 31 Theor. Popul. Biol. : 195 -272.

    • Search Google Scholar
  • Hofbauer, J. and Sigmund, K. (1990): Adaptive dynamics and evolutionary stability. Appl. Math. Lett.3: 75-79.

    'Adaptive dynamics and evolutionary stability ' () 3 Appl. Math. Lett. : 75 -79.

  • Kisdi, É. and Geritz, S. A. H. (1999): Adaptive dynamics in allele space: evolution of genetic polymorphism by small mutations in a heterogeneous environment. Evolution53: 993-1008.

    'Adaptive dynamics in allele space: evolution of genetic polymorphism by small mutations in a heterogeneous environment ' () 53 Evolution : 993 -1008.

    • Search Google Scholar
  • Kisdi, É. and Geritz, S. A. H. (in press): Evolutionary branching and sympatric speciation in diploid populations. In Dieckmann, U. and Metz, J. A. J. (EDS): Elements of Adaptive Dynamics. Cambridge University Press, Cambridge.

    Elements of Adaptive Dynamics , ().

  • Kisdi, É. and Meszéna, G. (1993): Density dependent life history evolution in fluctuating environments. In Yoshimura, J. and Clark, C. (eds): Adaptation in a Stochastic Environment. Lecture Notes in Biomathematics Vol. 98, Springer Verlag, pp. 26-62.

    Adaptation in a Stochastic Environment. Lecture Notes in Biomathematics , () 26 -62.

    • Search Google Scholar
  • Kisdi, É. and Meszéna, G. (1995): Life histories with lottery competition in a stochastic environment: ESSs which do not prevail. Theor. Popul. Biol.47: 191-211.

    'Life histories with lottery competition in a stochastic environment: ESSs which do not prevail ' () 47 Theor. Popul. Biol. : 191 -211.

    • Search Google Scholar
  • Leimar, O. (in press): Multidimensional convergence stability and the canonical adaptive dynamics. In Dieckmann, U. and Metz, J. A. J. (eds): Elements of Adaptive Dynamics. Cambridge University Press, Cambridge.

    Elements of Adaptive Dynamics , ().

  • Levin, S. M. (1970): Community equilibria and stability, and an extension of the competitive exclusion principle. Amer. Nat.104: 413-423.

    'Community equilibria and stability, and an extension of the competitive exclusion principle ' () 104 Amer. Nat. : 413 -423.

    • Search Google Scholar
  • Loeschcke, V. and Christiansen, F. B. (1984): Evolution and intraspecific competition. II. A two-locus model for additive gene effects. Theor. Popul. Biol.26: 228-264.

    'Evolution and intraspecific competition. II. A two-locus model for additive gene effects ' () 26 Theor. Popul. Biol. : 228 -264.

    • Search Google Scholar
  • MacArthur, R. and Levins, R. (1964): Competition, habitat selection and character displacement in a patchy environment. Proc. Natl. Acad. Set51: 1207-1210.

    'Competition, habitat selection and character displacement in a patchy environment ' () 51 Proc. Natl. Acad. Set : 1207 -1210.

    • Search Google Scholar
  • Matsuda, H. and Abrams, P. A. (1994a): Timid consumers: self-extinction due to adaptive change in foraging and anti-predator effort. Theor. Popul. Biol.45: 76-91.

    'Timid consumers: self-extinction due to adaptive change in foraging and anti-predator effort ' () 45 Theor. Popul. Biol. : 76 -91.

    • Search Google Scholar
  • Matsuda, H. and Abrams, P.A. (1994b): Runaway evolution to self-extinction under asymmetrical competition. Evolution48: 1764-1772.

    'Runaway evolution to self-extinction under asymmetrical competition ' () 48 Evolution : 1764 -1772.

    • Search Google Scholar
  • Maynard Smith, J. (1982): Evolution and the Theory of Games. Cambridge University Press, Cambridge.

    Evolution and the Theory of Games , ().

  • Maynard Smith, J. (1989): Evolutionary Genetics. Oxford University Press.

    Evolutionary Genetics , ().

  • Maynard Smith, J. and Brown, R. L. (1986): Competition and body size. Theor. Popul. Biol.30: 166-179.

    'Competition and body size ' () 30 Theor. Popul. Biol. : 166 -179.

  • Maynard Smith, J. and Price, G. R. (1973): The logic of animal conflict. Nature246: 15-18.

    'The logic of animal conflict ' () 246 Nature : 15 -18.

  • Meszéna, G., Czibula, I. and Geritz, S. A. H. (1997): Adaptive dynamics in a 2-patch environment: a toy model for allopatric and parapatric speciation. J. Biol. Syst.5: 265-284.

    'Adaptive dynamics in a 2-patch environment: a toy model for allopatric and parapatric speciation ' () 5 J. Biol. Syst. : 265 -284.

    • Search Google Scholar
  • Meszéna, G. and Metz, J. A. J. (in press): Species diversity and population regulation: Importance of environmental feed-back dimensionality. In Dieckmann, U. and Metz, J. A. J. (eds): Elements of Adaptive Dynamics. Cambridge University Press, Cambridge.

    Elements of Adaptive Dynamics , ().

  • Marrow, P., Dieckmann, U. and Law, R. (1996): Evolutionary dynamics of predator-prey systems: an ecological perspective. J. Math. Biol.34: 556-578.

    'Evolutionary dynamics of predator-prey systems: an ecological perspective ' () 34 J. Math. Biol. : 556 -578.

    • Search Google Scholar
  • Mathias, A. and Kisdi, É. (in press): Evolutionary branching and coexistence of germination strategies. In Dieckmann, U. and Metz, J. A. J. (eds): Elements of Adaptive Dynamics. Cambridge University Press, Cambridge.

    Elements of Adaptive Dynamics , ().

  • Mathias, A., Kisdi, É. and Olivieri, I. (2001): Divergent evolution of dispersal in a heterogeneous landscape. Evolution,55: 246-259.

    'Divergent evolution of dispersal in a heterogeneous landscape ' () 55 Evolution : 246 -59.

    • Search Google Scholar
  • Meszéna, G. and Pásztor, L. (1990): Population regulation and life-history strategies. In Maynard smith, J. and Vida, G. (eds): Proceeding in Nonlinear Science. Organizational constraints on the dynamics of evolution. Manchester University Press, Manchester and New York.

    Population regulation and life-history strategies

  • Metz, J. A. J., Geritz, S. A. H., Meszéna, G., Jacobs, F. J. A. and van Heerwaarden, J. S. (1996a): Adaptive dynamics, a geometrical study of the consequences of nearly faithful reproduction. In van Strien, S. J. and Verduyn Lunel, S. M. (eds): Stochastic and Spatial Structures of Dynamical Systems. North Holland, Amsterdam, pp. 183- 231.

    Stochastic and Spatial Structures of Dynamical Systems , () 183 -231.

  • Metz, J. A. J., Mylius, S. D. and Diekmann, O. (1996b): When does evolution optimize? On the relation between types of density dependence and evolutionarily stable life history parameters. Working paper WP-96-004, International Institute for Applied Systems Analysis, Laxenburg, Austria.

  • Metz, J. A. J., Nisbet, R. M. and Geritz, S. A. H. (1992): How should we define 'fitness' for general ecological scenarios? Trends Ecol. Evol.7: 198-202.

    'How should we define 'fitness' for general ecological scenarios? ' () 7 Trends Ecol. Evol. : 198 -202.

    • Search Google Scholar
  • Michod, R. E. (1979): Evolution of life histories in response to age-specific mortality factors. Amer. Nat.113: 531-550.

    'Evolution of life histories in response to age-specific mortality factors ' () 113 Amer. Nat. : 531 -550.

    • Search Google Scholar
  • Mylius, S. D. and Diekmann, O. (1995): On evolutionarily stable life histories, optimization and the need to be specific about density dependence. Oikos74: 218-224.

    'On evolutionarily stable life histories, optimization and the need to be specific about density dependence ' () 74 Oikos : 218 -224.

    • Search Google Scholar
  • Mylius, S. D. and Metz, J. A. J. (in press): When does evolution optimize? On the relationship between evolutionary stability, optimization and density dependence. In Dieckmann, U. and Metz, J. A. J. (eds): Elements of Adaptive Dynamics. Cambridge University Press, Cambridge.

    Elements of Adaptive Dynamics , ().

  • Nowak, M. A. (1990): An evolutionary stable strategy may be inaccessible. J. Theor. Biol.142: 237-241.

    'An evolutionary stable strategy may be inaccessible ' () 142 J. Theor. Biol. : 237 -241.

  • Nowak, M. A. and May, R. M. (1992): Evolutionary games and spatial chaos. Nature246: 15-18.

    'Evolutionary games and spatial chaos ' () 246 Nature : 15 -18.

  • Parvinen, K. (1999): Evolution of migration in a metapopulation. Bull. Math. Biol.61: 531-550.

    'Evolution of migration in a metapopulation ' () 61 Bull. Math. Biol. : 531 -550.

  • Brauchli, K., Killingback, T. and Doebeli, M. (1999): Evolution of cooperation in spatially structured populations. J. Theor. Biol.200: 405-417.

    'Evolution of cooperation in spatially structured populations ' () 200 J. Theor. Biol. : 405 -417.

    • Search Google Scholar
  • Brown, J. S. and Pavlovic, N. B. (1992): Evolution in heterogeneous environments: effects of migration on habitat specialization. Evol. Ecol.6: 360-382.

    'Evolution in heterogeneous environments: effects of migration on habitat specialization ' () 6 Evol. Ecol. : 360 -382.

    • Search Google Scholar
  • Brown, J. S. and Vincent, T. L. (1987a): A theory for the evolutionary game. Theor. Popul. Biol.31: 140-166.

    'A theory for the evolutionary game ' () 31 Theor. Popul. Biol. : 140 -166.

  • Brown, J. S. and Vincent, T. L. (1987b): Coevolution as an evolutionary game. Evolution41: 66-79.

    'Coevolution as an evolutionary game ' () 41 Evolution : 66 -79.

  • Brown, J. S. and Vincent, T. L. (1992): Organization of predator-prey communities as an evolutionary game. Evolution46: 1269-1283.

    'Organization of predator-prey communities as an evolutionary game ' () 46 Evolution : 1269 -1283.

    • Search Google Scholar
  • Christiansen, F. B. and Loeschcke, V. (1980): Evolution and intraspecific exploitative competition I. One locus theory for small additive gene effects. Theor. Popul. Biol.18: 297-313.

    'Evolution and intraspecific exploitative competition I. One locus theory for small additive gene effects ' () 18 Theor. Popul. Biol. : 297 -313.

    • Search Google Scholar
  • Christiansen, F. B. and Loeschcke, V. (1984): Evolution and intraspecific competition. III. One-locus theory for small additive gene effects and multidimensional resource qualities. Theor. Popul. Biol.31: 33-416.

    'Evolution and intraspecific competition. III. One-locus theory for small additive gene effects and multidimensional resource qualities ' () 31 Theor. Popul. Biol. : 33 -416.

    • Search Google Scholar
  • Christiansen, F. B. (1988): Frequency dependence and competition. Phil. Trans. R. Soc. Lond. B319: 587-600

    'Frequency dependence and competition ' () 319 Phil. Trans. R. Soc. Lond. B : 587 -00.

  • Christiansen, F. B. (1991): On conditions for evolutionary stability for a continuously varying character. Amer. Nat.138 138-37.

    'On conditions for evolutionary stability for a continuously varying character ' () 138 Amer. Nat. : 37 -50.

    • Search Google Scholar
  • Cohen, D. and Levin, S. A. (1991): Dispersal in patchy environments: The effects of temporal and spatial structure. Theor. Popul. Biol.39: 63-99.

    'Dispersal in patchy environments: The effects of temporal and spatial structure ' () 39 Theor. Popul. Biol. : 63 -99.

    • Search Google Scholar
  • Dieckmann, U., Marrow, P. and Law, R. (1995): Evolutionary cycling of predator-prey interactions: population dynamics and the Red Queen. J. Theor. Biol.176: 91-102.

    'Evolutionary cycling of predator-prey interactions: population dynamics and the Red Queen ' () 176 J. Theor. Biol. : 91 -102.

    • Search Google Scholar
  • Dieckmann, U. and Metz, J. A. J. (in prep.): Unfolding the degeneracy of evolutionary game theory.

  • Dieckmann, U. and Metz, J. A. J. (in press): Scales and limits in adaptive dynamics. In Dieckmann, U. and Metz, J. A. J. (eds): Elements of Adaptive Dynamics. Cambridge University Press, Cambridge.

    Elements of Adaptive Dynamics , ().

  • Dieckmann, U. et al. (in prep.): Adaptive dynamics in two and more dimensions: a classification of evolutionary singularities.

  • Doebeli, M. (1996): A quantitative genetic competition model for sympatric speciation. J. Evol. Biol.9: 893-909

    'A quantitative genetic competition model for sympatric speciation ' () 9 J. Evol. Biol. : 893 -09.

    • Search Google Scholar
  • Doebeli, M. and Dieckmann, U. (2000): Evolutionary branching and sympatric speciation caused by different types of ecological interactions. Amer. Nat.156: S77-S101

    'Evolutionary branching and sympatric speciation caused by different types of ecological interactions ' () 156 Amer. Nat. : S77 -101.

    • Search Google Scholar
  • Doebeli, M. and Ruxton, G. D. (1997): Evolution of dispersal rates in metapopulation models: Branching and cyclic dynamics inphenotype space. Evolution51: 1730-1741.

    'Evolution of dispersal rates in metapopulation models: Branching and cyclic dynamics inphenotype space ' () 51 Evolution : 1730 -1741.

    • Search Google Scholar
  • Ellner, S. (1985): ESS germination strategies in randomly varying environments I. Logistic-type models. Theor. Popul. Biol.28: 50-79.

    'ESS germination strategies in randomly varying environments I. Logistic-type models ' () 28 Theor. Popul. Biol. : 50 -79.

    • Search Google Scholar
  • Eshel, I. (1983): Evolutionary and continuous stability. J. Theor. Biol.103: 99-111.

    'Evolutionary and continuous stability ' () 103 J. Theor. Biol. : 99 -111.

  • Ferriere, R. and Gatto, M. (1995): Lyapunov exponents and the mathematics of invasion in oscillatory, or chaotic populations. Theor. Popul. Biol.48: 126-171

    'Lyapunov exponents and the mathematics of invasion in oscillatory, or chaotic populations ' () 48 Theor. Popul. Biol. : 126 -71.

    • Search Google Scholar
  • Garay, J. (1999): Relative advantage: a substitute for mean fitness in Fisher's fundamental theorem? J. Theor. Biol.201: 215-218.

    'Relative advantage: a substitute for mean fitness in Fisher's fundamental theorem? ' () 201 J. Theor. Biol. : 215 -218.

    • Search Google Scholar
  • Geritz, S. A. H., Kisdi, É., Meszéna, G. and Metz, J. A. J. (1998): Evolutionarily singular strategies and the adaptive growth and branching of the evolutionary tree. Evol. Ecol.12: 35-57.

    'Evolutionarily singular strategies and the adaptive growth and branching of the evolutionary tree ' () 12 Evol. Ecol. : 35 -57.

    • Search Google Scholar
  • Geritz, S. A. H., Metz, J. A. J., Kisdi, E. and Meszéna, G. (1997): Dynamics of adaptation and evolutionary branching. Phys. Rev. Letters78: 2024-2027.

    'Dynamics of adaptation and evolutionary branching ' () 78 Phys. Rev. Letters : 2024 -2027.

    • Search Google Scholar
  • Lande, R. (1979): Quantitative genetic analysis of multivariate evolution, applied to brain: body size allometry. Evolution33: 402-4116.

    'Quantitative genetic analysis of multivariate evolution, applied to brain: body size allometry ' () 33 Evolution : 402 -4116.

    • Search Google Scholar
  • Law, R. and Dieckmann, U. (1998): Symbiosis without exploitation and the merger of lineages in evolution. Proc. R. Soc. Lond. B265: 1245-1253.

    'Symbiosis without exploitation and the merger of lineages in evolution ' () 265 Proc. R. Soc. Lond. B : 1245 -1253.

    • Search Google Scholar
  • Law, R., Marrow, P. and Dieckmann, U. (1997): On evolution under asymmetric competition. Evol. Ecol.11: 485-501.

    'On evolution under asymmetric competition ' () 11 Evol. Ecol. : 485 -501.

  • Tilman, D. (1982): Resource Competition and Community Structure. Princeton University Press.

    Resource Competition and Community Structure , ().

  • Van Tienderen, P. H. and De Jong, G. (1986): Sex ratio under the haystack model: Polymorphism may occur. J. Theor. Biol.122: 69-81.

    'Sex ratio under the haystack model: Polymorphism may occur ' () 122 J. Theor. Biol. : 69 -81.

    • Search Google Scholar
  • Jacobs, F. J. A., Metz, J. A. J., Geritz, S. A. H. and Meszéna, G. (in prep.): Invasion implies fixation.

  • Kisdi, É. (1998): Frequency dependence versus optimization. Trends Ecol. Evol.13: 508.

    'Frequency dependence versus optimization ' () 13 Trends Ecol. Evol. : 508.

  • Hofbauer, J. and Sigmund, K. (1998): Evolutionary Games and Population Dynamics. Cambridge University Press, Cambridge.

    Evolutionary Games and Population Dynamics , ().

  • Geritz, S. A. H. and Kisdi, É. (2000): Adaptive dynamics in diploid sexual populations and the evolution of reproductive isolation. Proc. R. Soc. Lond. B267: 1671-1678

    'Adaptive dynamics in diploid sexual populations and the evolution of reproductive isolation ' () 267 Proc. R. Soc. Lond. B : 1671 -678.

    • Search Google Scholar
  • Pettifor, R. A., Perrins, C. M. and McCleery, R. H. (1988): Individual optimization of clutch size in great tits. Nature336: 160-162.

    'Individual optimization of clutch size in great tits ' () 336 Nature : 160 -162.

  • Pohley, H. J. and Thomas, B. (1983): Non-linear ESS models and frequency dependent selection. BioSystems16: 87-100.

    'Non-linear ESS models and frequency dependent selection ' () 16 BioSystems : 87 -100.

  • PÁSZTOR, L., MESZÉNA, G. and KISDI, É. (1996): R0 or r: A matter of taste? J. Evol. Biol.9: 511-518.

    'R0 or r: A matter of taste? ' () 9 J. Evol. Biol. : 511 -518.

  • Rand, D. A., Wilson, H. B. and McGlade, J. M. (1994): Dynamics and evolution: Evolutionary stable attractors, invasion exponents and phenotype dynamics. Phil. Trans. R. Soc. Lond. B243: 261-283.

    'Dynamics and evolution: Evolutionary stable attractors, invasion exponents and phenotype dynamics ' () 243 Phil. Trans. R. Soc. Lond. B : 261 -283.

    • Search Google Scholar
  • Stephen, D. W. and Krebs, J. R. (1986): Foraging Theory. Princeton University Press, Princeton.

    Foraging Theory , ().

  • Szabó, G. and Töke, C. (1998): Evolutionary prisoner's dilemma game on a square lattice. Phys. Rev. E.58: 69.

    'Evolutionary prisoner's dilemma game on a square lattice ' () 58 Phys. Rev. E. : 69.

  • Szabó, G., Antal, T., Szabó, P. and Droz, M. (2000): Spatial evolutionary prisoner's dilemma game with three strategies and external constraints. Phys. Rev. E.62: 1095.

    'Spatial evolutionary prisoner's dilemma game with three strategies and external constraints ' () 62 Phys. Rev. E. : 1095.

    • Search Google Scholar
  • Taper, M. L. and Case, T. J. (1992): Models of character displacement and the theoretical robustness of taxon cycles. Evolution46: 317-333.

    'Models of character displacement and the theoretical robustness of taxon cycles ' () 46 Evolution : 317 -333.

    • Search Google Scholar
  • Taylor, P. D. (1989): Evolutionary stability in one-parameter models under weak selection. Theor. Popul. Biol.36: 125-143.

    'Evolutionary stability in one-parameter models under weak selection ' () 36 Theor. Popul. Biol. : 125 -143.

    • Search Google Scholar
  • Taylor, P. D. and Jonker, L. B. (1978): Evolutionary stable strategies and game dynamics. Math. Biosci.40: 145-156.

    'Evolutionary stable strategies and game dynamics ' () 40 Math. Biosci. : 145 -156.

  • Thomas, B. (1984): Evolutionary stability: States and strategies. Theor. Popul. Biol.26: 49-67.

    'Evolutionary stability: States and strategies ' () 26 Theor. Popul. Biol. : 49 -67.

  • Vincent, T. L., Cohen, Y. and Brown, J. S. (1993): Evolution via strategy dynamics. Theor. Popul. Biol.44: 149-176.

    'Evolution via strategy dynamics ' () 44 Theor. Popul. Biol. : 149 -176.

  • Wright, S. (1931): Evolution in Mendelian populations. Genetics16: 97-159.

    'Evolution in Mendelian populations ' () 16 Genetics : 97 -159.

  • Zeeman, E. C. (1980): Population dynamics from game theory. In Global Theory of Dynamical Systems. Lecture Notes in Mathematics 819. Springer, New York.

    Global Theory of Dynamical Systems , ().