Adaptive walks on changing landscapes: Levins’ approach extended. (English) Zbl 1106.92057

Summary: The assumption that trade-offs exist is fundamental in evolutionary theory. R. Levins [Am. Nat. 96, 361–372 (1962)] introduced a widely adopted graphical method for analyzing evolution towards an optimal combination of two quantitative traits, which are traded off. His approach explicitly excluded the possibility of density- and frequency-dependent selection. Here we extend Levins’ method towards models, which include these selection regimes and where therefore fitness landscapes change with population state. We employ the same kind of curves Levins used: trade-off curves and fitness contours. However, fitness contours are not fixed but a function of the resident traits and we only consider those that divide the trait space into potentially successful mutants and mutants which are not able to invade (‘invasion boundaries’). The developed approach allows to make a priori predictions about evolutionary endpoints and about their bifurcations. This is illustrated by applying the approach to several examples from the recent literature.


92D15 Problems related to evolution
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[1] Abrams, P.A.; Matsuda, H.; Harada, Y., Evolutionary unstable fitness maxima and stable fitness minima of continuous traits, Evol. ecol., 7, 465-487, (1993)
[2] Alexander, R.M., Optima for animals, (1996), Princeton University Press Princeton, NJ
[3] Bell, G., The costs of reproduction and their consequences, Am. nat., 116, 45-76, (1980)
[4] Benkman, C.W., Adaptation to single resources and the evolution of crossbill (loxia) diversity, Ecol. monogr., 63, 305-325, (1993)
[5] Brown, J.S.; Vincent, T.L., Coevolution as an evolutionary game, Evolution, 41, 66-79, (1987)
[6] Brown, J.S.; Vincent, T.L., A theory for the evolutionary game, Theor. popul. biol., 31, 140-166, (1987) · Zbl 0618.92014
[7] Charlesworth, B., 1994. Evolution in Age-Structured Populations, 2nd Edition, in Cambridge Studies in Mathematical Biology, Vol. 13. Cambridge University Press, Cambridge. · Zbl 0811.92016
[8] Charnov, E.L., The theory of sex allocation, monographs in population biology, (1982), Princeton University Press Princeton, NJ
[9] Charnov, E.L., 1993. Life History Invariants: Some Explorations of Symmetry in Evolutionary Ecology, Oxford Series in Ecology and Evolution. Oxford University Press, Oxford, UK.
[10] Charnov, E.L.; Maynard Smith, J.; Bull, J.J., Why be an hermaphrodite?, Nature, 263, 125-126, (1976)
[11] Christiansen, F.B., On conditions for evolutionary stability for a continuously varying character, Am. nat., 138, 37-50, (1991)
[12] Christiansen, F.B.; Loeschke, V., Evolution and intraspecific exploitation competition I. one-locus theory for small additve gene effects, Theor. popul. biol., 18, 297-313, (1980) · Zbl 0468.92013
[13] Claessen, D.; Dieckmann, U., Ontogenetic niche shifts and evolutionary branching in size-structured populations, Evol. ecol. res., 4, 189-217, (2002)
[14] Day, T.; Taylor, P.D., Evolutionarily stable versus fitness maximizing life histories under frequency-dependent selection, Proc. R. soc. London B, 263, 333-338, (1996)
[15] Day, T.; Abrams, P.A.; Chase, J.M., The role of size-specific predation in the evolution and diversification of prey life histories, Evolution, 56, 877-887, (2002)
[16] Dieckmann, U., Can adaptive dynamics invade?, Trends ecol. evol., 12, 128-131, (1997)
[17] Dieckmann, U.; Law, R., The dynamical theory of coevolutiona derivation from stochastic ecological processes, J. math. biol., 34, 579-612, (1996) · Zbl 0845.92013
[18] Doebeli, M.; Dieckmann, U., Evolutionary branching and sympatric speciation caused by different types of ecological interactions, Am. nat., 156, S77-S101, (2000)
[19] Ebenman, B.; Johansson, A.; Jonsson, T., Evolution of stable population dynamics through natural selection, Proc. R. soc. London B, 263, 1145-1151, (1996)
[20] Egas, M., Dieckmann, U., Sabelis, M.W., Evolution restricts the coexistence of specialists and generalists—the role of trade-off structure. Am. Nat., in press.
[21] Eshel, I., Evolutionary and continuous stability, J. theor. biol., 103, 99-111, (1983)
[22] Eshel, I.; Motro, U., Kin selection and strong evolutionary stability of mutual help, Theor. popul. biol., 19, 420-433, (1981) · Zbl 0473.92014
[23] Ferrière, R.; Gatto, M., Lyapunov exponents and the mathematics of invasion in oscillatory of chaotic populations, Theor. popul. biol., 48, 126-171, (1995) · Zbl 0863.92015
[24] Fisher, R.A., The genetical theory of natural selection, (1930), Clarendon Press Oxford · JFM 56.1106.13
[25] Gatto, M., The evolutionary optimality of oscillatory and chaotic dynamics in simple population models, Theor. popul. biol., 43, 310-336, (1993) · Zbl 0773.92010
[26] Geritz, S.A.H.; Metz, J.A.J.; Kisdi, É.; Meszéna, G., Dynamics of adaptation and evolutionary branching, Phys. rev. lett., 78, 2024-2027, (1997)
[27] Geritz, S.A.H.; Kisdi, É.; Meszéna, G.; Metz, J.A.J., Evolutionarily singular strategies and the adaptive growth and branching of the evolutionary tree, Evol. ecol., 12, 35-57, (1998)
[28] Geritz, S.A.H.; Gyllenberg, M.; Jacobs, F.J.A.; Parvinen, K., Invasion dynamics and attractor inheritance, J. math. biol., 44, 548-560, (2002) · Zbl 0990.92029
[29] Hamilton, W.D., Extraordinary sex ratios, Science, 156, 477-488, (1967)
[30] Heino, M.; Metz, J.A.J.; Kaitala, V., Evolution of mixed maturation strategies in semelparous life-historiesthe crucial role of dimensionality of feedback environment, Proc. R. soc. London B, 352, 1647-1655, (1997)
[31] Heino, M.; Metz, J.A.J.; Kaitala, V., The enigma of frequency-dependent selection, Trends ecol. evol., 13, 367-370, (1998)
[32] Hofbauer, J.; Sigmund, K., Adaptive dynamics and evolutionary stability, Appl. math. lett., 3, 75-79, (1990) · Zbl 0709.92015
[33] Huisman, J.; Weissing, F.J., Biodiversity of plankton by species oscillations and chaos, Nature, 402, 407-410, (1999)
[34] Kawecki, T.J., Age and size at maturation in a patchy environmentfitness maximization versus evolutionary stability, Oikos, 66, 309-317, (1993)
[35] Kisdi, É.; Meszéna, G., Density dependent life history evolution in fluctuating environments, () · Zbl 0803.92020
[36] Kisdi, É.; Jacobs, F.J.A.; Geritz, S.A.H., Red queen evolution by cycles of evolutionary branching and extinction, Selection, 2, 161-176, (2000)
[37] Koch, A.L., Competitive coexistance of two predators utilizing the same prey under constant environmental conditions, J. theor. biol., 44, 387-395, (1974)
[38] Lande, R., Natural selection and random genetic drift in phenotypic evolution, Evolution, 30, 314-334, (1976)
[39] Lawlor, L.R.; Maynard Smith, J., The coevolution and stability of competing species, Am. nat., 110, 79-99, (1976)
[40] Leimar, O., Evolutionary change and Darwinian demons, Selection, 2, 65-72, (2001)
[41] Lessells, C.M., 1991. Behavioural Ecology, 3rd Edition. Blackwell Scientific Publications, Oxford, pp. 32-68 (Chapter: The Evolution of Life Histories).
[42] Levins, R., Theory of fitness in a heterogeneous environment. 1. the fitness set and the adaptive function, Am. nat., 96, 361-373, (1962)
[43] Levins, R., Evolution in changing environments, Monographs in population biology, Vol. 2, (1968), Princeton University Press Princeton, NJ
[44] Levins, R., Coexistence in a variable environment, Am. nat., 114, 765-783, (1979)
[45] Marrow, P.; Law, R.; Cannings, C., The coevolution of predator-prey interactionsess and red queen dynamics, Proc. R. soc. London B, 250, 133-141, (1992)
[46] Matessi, C.; Di Pasquale, C., Long-term evolution of multilocus traits, J. math. biol., 34, 613-653, (1996) · Zbl 0851.92009
[47] Mathias, A., Kisdi, É., 1999. Evolutionary branching and coexistence of germination strategies. IIASA Interim Report IR-99-014, available at http://www.iiasa.ac.at/Research/ADN/Series.html.
[48] Mathias, A.; Kisdi, É., Adaptive diversification of germination strategies, Proc. R. soc. London B, 269, 151-155, (2002)
[49] Matsuda, H., Evolutionary stable strategies for predator switching, J. theor. biol., 115, 351-366, (1985)
[50] Maynard Smith, J., Optimization theory in evolution, Annu. rev. ecol. syst., 9, 31-56, (1978)
[51] Maynard Smith, J., Evolution and the theory of games, (1982), Cambridge University Press Cambridge, UK · Zbl 0526.90102
[52] Maynard Smith, J.; Price, G.R., The logic of animal conflict, Nature, 246, 15-18, (1973) · Zbl 1369.92134
[53] de Mazancourt, C.; Loreau, M.; Dieckmann, U., Can the evolution of plant defense lead to plant-herbivore mutualism?, Am. nat., 158, 109-123, (2001)
[54] McNamara, J.M., Implicit frequency-dependence and kin selection in fluctuating environments, Evol. ecol., 9, 185-203, (1995)
[55] McNamara, J.M.; Houston, A.I.; Collins, E.J., Optimality models in behavioral biology, SIAM rev., 43, 413-466, (2001) · Zbl 0973.92027
[56] Meszéna, G.; Kisdi, É.; Dieckmann, U.; Geritz, S.A.H.; Metz, J.A.J., Evolutionary optimization models and matrix games in the unified perspective of adaptive dynamics, Selection, 2, 193-210, (2001)
[57] 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)
[58] Metz, J.A.J., Geritz, S.A.H., Meszéna, G., Jacobs, F.J.A., S., V.H.J., 1996a. Adaptive dynamics: a geometrical study of the consequences of nearly faithful reproduction. In: van Strien, S.J., Verduyn Lunel, S. (Eds.), Stochastic and Spatial Structures of Dynamical Systems. Proceedings of the Royal Dutch Academey of Science, North-Holland, Elsevier, Amsterdam, pp. 183-231. · Zbl 0972.92024
[59] Metz, J.A.J., Mylius, S.D., Diekmann, O., 1996b. When does evolution optimize? on the relation between types of density dependence and evolutionarily stable life history parameters. IIASA Working Paper WP-96-04, available at http://www.iiasa.ac.at/Research/ADN/Series.html.
[60] Michod, R.E., Evolution of life histories in response to age-specific mortality factors, Am. nat., 113, 531-550, (1979)
[61] Motro, U., Optimal rates of dispersal I. haploid populations, Theor. popul. biol., 21, 394-411, (1982) · Zbl 0513.92009
[62] Motro, U., Evolutionary and continuous stability in asymmetric games with continuous strategy setsthe parental investment conflict an example, Am. nat., 144, 229-241, (1994)
[63] Mylius, S.D.; Diekmann, O., On evolutionary stable life histories, optimization and the need to be specific about density dependence, Oikos, 74, 218-224, (1995)
[64] Nowak, M., An evolutionarily stable strategy may be inaccessible, J. theor. biol., 142, 237-241, (1990)
[65] Philippi, T.; Seger, J., Hedging One’s evolutionary bets revisited, Trends ecol. evol., 4, 41-44, (1989)
[66] Pianka, E.R.; Parker, W.S., Age-specific reproductive tactics, Am. nat., 109, 45-464, (1975)
[67] Reed, J.; Stenseth, N.C., On evolutionary stable strategies, J. theor. biol., 108, 491-508, (1984)
[68] Roff, D., The evolution of life histories, (1992), Chapman & Hall New York
[69] Roff, D., 2002. Life History Evolution, Sinauer, Sunderland, Massachusetts, USA.
[70] Schaffer, W.M., Selection for optimal life histories, the effects of age structure, Ecology, 55, 291-303, (1974)
[71] Schluter, D., Adaptive radiation in sticklebackssize, shape, and habitat use efficiency, Ecology, 74, 699-709, (1993)
[72] Stearns, S.C., The evolution of life histories, (1992), Oxford University Press Oxford, UK
[73] Stearns, S.C., Life history evolutionsuccesses, limitations, and prospects, Naturwissenschaften, 87, 476-486, (2000)
[74] Stephens, D.W.; Krebs, J.R., Foraging theory, (1986), Princeton University Press Princeton, NJ
[75] Svensson, E.; Sheldon, B.C., The social context of life history evolution, Oikos, 83, 466-477, (1998)
[76] Takada, T., Evolution of semelparous and iteroparous perennial plantscomparison between the density-independent and the density-dependent dynamics, J. theor. biol., 173, 51-60, (1995)
[77] Taylor, P.D., Evolutionary stability of one-parameter models under weak selection, Theor. popul. biol., 36, 125-143, (1989) · Zbl 0684.92014
[78] van Tienderen, P.H.; de Jong, G., Sex ratio under the haystack modelpolymorphism may occur, J. theor. biol., 122, 69-81, (1986)
[79] Vincent, T.L.; Brown, J.S., The evolution of ess theory, Annu. rev. ecol. syst., 19, 423-443, (1988)
[80] Wilson, D.S.; Yoshimura, J., On the coexistence of specialists and generalsits, Am. nat., 144, 692-707, (1994)
[81] Wright, S., Evolution in Mendelian populations, Genetics, 16, 97-159, (1931)
[82] Yodzis, P., Introduction to theoretical ecology, (1989), Harper & Row NY · Zbl 0763.92012
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