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The dynamical attainability of ESS in evolutionary games. (English) Zbl 0734.92021

The attainability of ESS of the evolutionary game among n players under frequency independent selection is studied for a special concept of stability (SDS). It is assumed that natural selection and small mutations cause the phenotype to change gradually in the direction of increasing fitness. Strongly determined stability (SDS) is defined as follows.
Let \(F_ i(x_ 1,...,x_ n)\) denote the fitness of the i-th species in an n-species game, where the strategies of the various species are \(x_ 1,...,x_ n\). The above assumptions imply dynamics of the following form: \[ (1)\quad dx_ i/dt=k_ i(x_ 1,...,x_ n)\cdot \partial F_ i/\partial x_ i \] where \(k_ i(.)>0\) is a coefficient measuring the adjustment speed. An equilibrium point of (1) satisfies SDS if it is locally stable for any functional form of the \(k_ i\), and remains so under small perturbations of the \(F_ i.\)
Mathematically, this amounts to saying that all eigenvalues of the linearized system have negative real parts; this can be expressed in terms of certain conditions (Routh-Hurwicz) on the second derivatives of the \(F_ i\), independently of the \(k_ i\). The authors argue that an ESS can be considered as a goal of evolution only if it is SDS. It is shown that in the two-species competitive case, the Nash solution is always attainable, and that in general one of two species may attain minimum fitness as a result of evolution. The attainability of ESS is also examined in two examples on the sex ratio of wasps and aphids.
Reviewer: M.Nermuth (Wien)

MSC:

92D15 Problems related to evolution
91A40 Other game-theoretic models
91A80 Applications of game theory
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[1] Arrow, K. J., McManus, M.: A note on dynamic stability. Econometrica 26, 448-454 (1958) · Zbl 0107.37201 · doi:10.2307/1907624
[2] Bahl, C. A., Cain, B. E.: The inertia of diagonal multiples of 3 {\(\times\)} 3 real matrices. Linear Algebra Appl. 18, 267-280 (1977) · Zbl 0379.15004 · doi:10.1016/0024-3795(77)90056-8
[3] Cain, B. E.: Real, 3 {\(\times\)} 3, D-stable matrices. J. Res. NBS 80B, 75-77 (1976) · Zbl 0341.15009
[4] Charnov, E. L.: Optimal foraging, the marginal value theorem. Theor. Popul. Biol. 9, 129-136 (1976) · Zbl 0334.92009 · doi:10.1016/0040-5809(76)90040-X
[5] Cohen, D.: The optimal timing of reproduction. Am. Nat. 110, 801-807 (1976) · doi:10.1086/283103
[6] Eshel, L., Akin, E.: Coevolutionary instability of mixed Nash solutions. J. Math. Biology 18, 123-133 (1983) · Zbl 0531.92024
[7] Futuyma, D. J., Slatkin, M.: Coevolution. Sinauer Associates Inc., 1983
[8] Gantmacher, T. R.: Applications of theory of matrices. Interscience Publishers. (A division of) New York: John Wiley 1959 · Zbl 0085.01001
[9] Hines, W. G. S.: Multi-species population models and evolutionarily stable strategies. J. Appl. Probab. 18, 507-513 (1981) · Zbl 0453.92013 · doi:10.2307/3213297
[10] Hirose, T.: A graphical analysis of life history evolution in biennials with special reference to their distribution in a sand dune system. Bot. Mag. Tokyo 90, 37-47 (1983) · doi:10.1007/BF02489573
[11] Hofbauer, J., Schuster, P., Sigmund, K.: A note on evolutionary stable strategies and game dynamics. J. Theor. Biol. 81, 609-619 (1980) · doi:10.1016/0022-5193(79)90058-4
[12] 12.Hofbauer, J., Sigmund, K.: Dynamical systems and the theory of evolution. Cambridge: Univeristy Press 1987 · Zbl 0678.92010
[13] Jordan, D. W., Smith, P.: Nonlinear ordinary differential equations. Oxford: Clarendon Press 1977 · Zbl 0417.34002
[14] Lande, R.: Natural selection and random genetic drift in phenotypic evolution. Evolution 30, 314-334 (1976) · doi:10.2307/2407703
[15] Logofet, D. O.: On the hierarchy of subsets of stable matrices. Sov. Math., Dokl. 34, 247-250 (1987) · Zbl 0619.15013
[16] Maynard Smith, J., Price, G. R.: The logic of animal conflict. Nature 246, 15-18 (1973) · Zbl 1369.92134 · doi:10.1038/246015a0
[17] Nash, J. F.: Noncooperative games. Ann. Math. 54, 286-295 (1951) · Zbl 0045.08202 · doi:10.2307/1969529
[18] Roughgarden, J.: The theory of coevolution. In: Futuyma, D. J., Slatkin, M. (eds.) Coevolution. (pp. 33-64) Sinauer Associates Inc. 1983
[19] Shubik, M.: Game theory in the social sciences. London: MIT Press 1983 · Zbl 0903.90180
[20] Smith, C. C., Fretwell, S. D.: The optimal balance between size and number of offsprings, Am. Nat., 108, 499-506 (1974) · doi:10.1086/282929
[21] Suzuki, Y., Iwasa, Y.: A sex ratio theory of gregarious parasitoids. Res. Popul. Ecol. 22, 366-382 (1980) · doi:10.1007/BF02530857
[22] Taylor, P., Jonker, L.: Evolutionary stable strategies and game dynamics. Math. Biosci. 40, 145-156 (1978) · Zbl 0395.90118 · doi:10.1016/0025-5564(78)90077-9
[23] Winkler, D. W., Wallin, K.: Offspring size and number: A life history model linking effort per offspring and total effort. Amer. Nat. 129, 708-720 (1987) · doi:10.1086/284667
[24] Yamaguchi, Y.: Sex ratios of an aphid subject to local mate competition with variable maternal condition. Nature 318, 460-462 (1985) · doi:10.1038/318460a0
[25] Zeeman, C.: Population dynamics from game theory. In: Nitecki, Z., Robinson, C. (eds.) Global theory of dynamical systems. Proceedings, Evanston, Illinois 1979. (Lect. Notes Math., vol. 819) Berlin Heidelberg New York: Springer 1980 · Zbl 0409.58008
[26] Zeeman, C.: Dynamics of the evolution of animal conflicts. J. Theor. Biol. 89, 249-270 (1981) · doi:10.1016/0022-5193(81)90311-8
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