×

When do optimisation arguments make evolutionary sense? (English) Zbl 1379.92037

Chalub, Fabio A. C. C. (ed.) et al., The mathematics of Darwin’s legacy. Basel: Birkhäuser (ISBN 978-3-0348-0121-8/hbk; 978-3-0348-0122-5/ebook). Mathematics and Biosciences in Interaction, 233-268 (2011).
Summary: The simplest behaviour one can hope for when studying a mathematical model of evolution by natural selection is when evolution always maximises the value of some function of the trait under consideration, thus providing an absolute measure of fitness for the model. We survey the role of such models, known as optimisation models in the literature, and give some general results concerning the question of when a model turns out to be an optimisation model. The results presented vary from more abstract results with a game-theoretical flavour to more detailed considerations of life history models. We also give a number of concrete examples and discuss the role of optimisation models in the wider framework of adaptive dynamics.
For the entire collection see [Zbl 1220.92045].

MSC:

92D15 Problems related to evolution
91A30 Utility theory for games
PDFBibTeX XMLCite
Full Text: DOI Link

References:

[1] W. Ewens, What changes has mathematics made to the Darwinian theory? In F.A.C.C. Chalub and J.F. Rodrigues (eds.), The Mathematics of Darwin’s Legacy, 7-26, Birkh¨auser, Basel, 2011, This issue. · Zbl 1382.92201
[2] R. B¨urger, Some mathematical models in evolutionary genetics. In F.A.C.C. Chalub and J.F. Rodrigues (eds.), The Mathematics of Darwin’s Legacy, 67-89, Birkh¨auser, Basel, 2011, This issue. · Zbl 1379.92033
[3] J.A.J. Metz, Thoughts on the geometry of meso-evolution: Collecting mathematical elements for a post-modern synthesis. In F.A.C.C. Chalub and J.F. Rodrigues (eds.), The Mathematics of Darwin’s Legacy, 193-231, Birkh¨auser, Basel, 2011, This issue.
[4] J.A.J. Metz, Fitness. In S.E. J¨orgensen and B.D. Fath (eds.), Evolutionary Ecology, volume 2 of Encyclopedia of Ecology, 1599-1612, Elsevier, UK, 2008.
[5] Metz, JAJ; Diekmann, O., The dynamics of physiologically structured populations, volume 68 of Lecture Notes in Biomathematics (1986), Berlin: Springer-Verlag, Berlin · Zbl 0614.92014
[6] Metz, JAJ; de Roos, AM; DeAngelis, D.; Gross, L., The role of physiologically structured population models within a general individual-based modeling perspective, Individual-based models and approaches in ecology: Concepts and Models, 88-111 (1992), USA: Chapman & Hall, USA
[7] Metz, JAJ; Nisbet, RM; Geritz, SAH, How should we define “fitness” for general ecological scenarios?, Trends Ecol. Evol., 7, 198-202 (1992) · doi:10.1016/0169-5347(92)90073-K
[8] Ferri‘ere, R.; Gatto, M., Lyapunov exponents and the mathematics of invasion in oscillatory or chaotic populations, Theor. Popul. Biol., 48, 126-171 (1995) · Zbl 0863.92015
[9] Jagers, P., Branching Processes with Biological Applications (1975), London, UK: Wiley Series in Probability and Mathematical Statistics-Applied, London, UK · Zbl 0356.60039
[10] Athreya, KB; Karlin, S., Branching processes with random environments I - extinction probabilities, Ann. Math. Stat., 42, 1499 (1971) · Zbl 0228.60032 · doi:10.1214/aoms/1177693150
[11] Athreya, KB; Karlin, S., Branching processes with random environments II - limit theorems, Ann. Math. Stat., 42, 1843 (1971) · Zbl 0228.60033 · doi:10.1214/aoms/1177693051
[12] P. Haccou, P. Jagers, and V.A. Vatutin, Branching Processes. Variation, Growth, and Extinction of Populations, volume 5 of Cambridge Studies in Adaptive Dynamics. Cambridge University Press, UK, 2005. · Zbl 1118.92001
[13] Diekmann, O.; Heesterbeek, JAP; Metz, JAJ, On the definition and the computation of the basic reproduction ratio r0 in models for infectious-diseases in heterogeneous populations, J. Math. Biol., 28, 365-382 (1990) · Zbl 0726.92018 · doi:10.1007/BF00178324
[14] J.A.J. Metz and O. Leimar, A simple fitness proxy for structured populations with continuous traits, with case studies on the evolution of haplo-diploids and genetic dimorphisms (in press), J. Biol. Dyn. · Zbl 1403.92182
[15] Jacobs, FJA; Metz, JAJ, On the concept of attractor for community-dynamical processes I: the case of unstructured populations, J. Math. Biol., 47, 222-234 (2003) · Zbl 1023.92035 · doi:10.1007/s00285-003-0204-z
[16] Gyllenberg, M.; Jacobs, FJA; Metz, JAJ, On the concept of attractor for community-dynamical processes II: the case of structured populations, J. Math. Biol., 47, 235-248 (2003) · Zbl 1029.92024 · doi:10.1007/s00285-003-0213-y
[17] Eshel, I., Evolutionary and continuous stability, J. Theor. Biol., 103, 99-111 (1983) · doi:10.1016/0022-5193(83)90201-1
[18] Taylor, PD, Evolutionary stability in one-parameter models under weak selection, Theor. Popul. Biol., 36, 125-143 (1989) · Zbl 0684.92014 · doi:10.1016/0040-5809(89)90025-7
[19] Eshel, I., On the changing concept of evolutionary population stability as a reflection of a changing point of view in the quantitative theory of evolution, J. Math. Biol., 34, 485-510 (1996) · Zbl 0851.92011 · doi:10.1007/BF02409747
[20] S.A.H. Geritz, ´E. Kisdi, G. Meszena, and J.A.J. Metz, Evolutionarily singular strategies and the adaptive growth and branching of the evolutionary tree. Evol. Ecol. 12 (1998), 35-57.
[21] Leimar, O., Evolutionary change and Darwinian demons. Selection, 2, 65-72 (2001)
[22] Leimar, O., The evolution of phenotypic polymorphism: Randomized strategies versus evolutionary branching, Am. Nat., 165, 669-681 (2005) · doi:10.1086/429566
[23] Leimar, O., Multidimensional convergence stability. Evol. Ecol. Res., 11, 191-208 (2009)
[24] Diekmann, O.; Gyllenberg, M.; Metz, JAJ, Steady-state analysis of structured population models, Theor. Popul. Biol., 63, 309-338 (2003) · Zbl 1098.92062 · doi:10.1016/S0040-5809(02)00058-8
[25] Eshel, I.; Feldman, MW, Initial increase of new mutants and some continuity properties of ESS in two locus systems, Am. Nat., 124, 631-640 (1984) · doi:10.1086/284303
[26] Liberman, U., External stability and ESS: criteria for initial increase of new mutant allele, J. Math. Biol., 26, 477-485 (1988) · Zbl 0713.92015 · doi:10.1007/BF00276375
[27] Otto, SP; Jarne, P., Evolution – haploids – hapless or happening?, Science, 292, 2441-2443 (2001) · doi:10.1126/science.1062890
[28] Hammerstein, P.; Selten, R.; Auman, R.; Hart, S., Game theory and evolutionary biology, Handbook of Game Theory With Economic Applications, Vol. II, volume 11 of Handbooks in Econom., 929-993 (1994), Amsterdam: North-Holland, Amsterdam · Zbl 0925.90088
[29] Hammerstein, P., Darwinian adaptation, population genetics and the streetcar theory of evolution, J. Math. Biol., 34, 511-532 (1996) · Zbl 0845.92014 · doi:10.1007/BF02409748
[30] Weissing, FJ, Genetic versus phenotypic models of selection: Can genetics be neglected in a long-term perspective?, J. Math. Biol., 34, 533-555 (1996) · Zbl 0845.92015 · doi:10.1007/BF02409749
[31] Eshel, I.; Feldman, MW; Sober, E.; Orzack, S., Optimization and evolutionary stability under shortterm and long-term selection, Adaptationism and Optimality, 161-190 (2001), Cambridge, UK: Cambridge University Press, Cambridge, UK
[32] I. Eshel, Short-term and long-term evolution. In U. Dieckmann and J.A.J. Metz (eds.), Elements of Adaptive Dynamics, Cambridge University Press, UK, In press. · Zbl 0733.92012
[33] Metz, JAJ; Mylius, SD; Diekmann, O., When does evolution optimize?, Evol. Ecol. Res., 10, 629-654 (2008)
[34] Gyllenberg, M.; Service, R., Necessary and sufficient conditions for the existence of an optimisation principle in evolution, J. Math. Biol., 62, 259-369 (2011) · Zbl 1232.92059 · doi:10.1007/s00285-010-0340-1
[35] J.A.J. Metz, S.D. Mylius, and O. Diekmann, When does evolution optimise? on the relation between types of density dependence and evolutionarily stable life history parameters (1996), IIASA Working Paper WP-96-04.
[36] R.G. Bowers, On the determination of evolutionary outcomes directly from the population dynamics of the resident (in press), J. Math. Biol. · Zbl 1230.92034
[37] C. Rueffler, J.A.J. Metz, and T.J.M. Van Dooren, What life cycle graphs can tell about the evolution of life histories (in revision), J. Math. Biol. · Zbl 1258.92028
[38] Heino, M.; Metz, JAJ; Kaitala, V., The enigma of frequency-dependent selection, Trends Ecol. Evol., 13, 367-370 (1998) · doi:10.1016/S0169-5347(98)01380-9
[39] Hassell, MP; Lawton, JH; May, RM, Patterns of dynamical behaviour in singlespecies populations, J. Anim. Ecol., 45, 471-486 (1976) · doi:10.2307/3886
[40] Kemeny, JG; Snell, JL, Finite Markov Chains (1960), Princeton, NJ: Van Nostrand, Princeton, NJ · Zbl 0089.13704
[41] Metz, JAJ; Klinkhamer, PGL; de Jong, TJ, A different model to explain delayed germination, Evol. Ecol. Res., 11, 177-190 (2009)
[42] Anderson, RM; May, RM, Coevolution of hosts and parasites. Parasitology, 85, 411-426 (1982)
[43] Anderson, RM; May, RM, Infectious Diseases of Humans: Dynamics and Control (1991), Oxford, UK: Oxford University Press, Oxford, UK
[44] Dieckmann, U.; Metz, JAJ, Surprising evolutionary predictions from enhanced ecological realism, Theor. Popul. Biol., 69, 263-281 (2006) · Zbl 1117.92044 · doi:10.1016/j.tpb.2005.12.001
[45] Dieckmann, U.; Dieckmann, U.; Metz, JAJ; Sabelis, MW; Sigmund, K., Adaptive dynamics of pathogens-host interaction, Adaptive Dynamics of Infectious Diseases: In Pursuit of Virulence Management, 39-59 (2002), UK: Cambridge University Press, UK · Zbl 1228.92050
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.