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Function-valued adaptive dynamics and the calculus of variations

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Abstract

Adaptive dynamics has been widely used to study the evolution of scalar-valued, and occasionally vector-valued, strategies in ecologically realistic models. In many ecological situations, however, evolving strategies are best described as function-valued, and thus infinite-dimensional, traits. So far, such evolution has only been studied sporadically, mostly based on quantitative genetics models with limited ecological realism. In this article we show how to apply the calculus of variations to find evolutionarily singular strategies of function-valued adaptive dynamics: such a strategy has to satisfy Euler's equation with environmental feedback. We also demonstrate how second-order derivatives can be used to investigate whether or not a function-valued singular strategy is evolutionarily stable. We illustrate our approach by presenting several worked examples.

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References

  1. Beder, J.H., Gomulkiewicz, R.: Computing the selection gradient and evolutionary response of an infinite-dimensional trait. J. Math. Biol. 36, 299–319 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  2. Christiansen, F.B.: On conditions for evolutionary stability for a continuously varying character. Am. Nat. 138, 37–50 (1991)

    Article  Google Scholar 

  3. Dieckmann, U., Heino, M., Parvinen, K.: The adaptive dynamics of function-valued traits. (in prep.)

  4. Dieckmann, U., Law, R.: The dynamical theory of coevolution: A derivation from stochastic ecological processes. J. Math. Biol. 34, 579–612 (1996)

    MATH  MathSciNet  Google Scholar 

  5. Diekmann, O., Gyllenberg, M., Huang, H., Kirkilionis, M., Metz, J.A.J., Thieme, H.R.: On the formulation and analysis of general deterministic structured population models. II. Nonlinear theory. J. Math. Biol. 43, 157–189 (2001)

    MATH  MathSciNet  Google Scholar 

  6. Diekmann, O., Gyllenberg, M., Metz, J.A.J., Thieme, H.R.: On the formulation and analysis of general deterministic structured population models. I. Linear theory. J. Math. Biol. 36, 349–388 (1998)

    MATH  MathSciNet  Google Scholar 

  7. Ernande, B., Dieckmann, U., Heino, M.: Adaptive changes in harvested populations: Plasticity and evolution of age and size at maturation. Proc. Royal Soc. London B 271, 415–423 (2004)

    Article  Google Scholar 

  8. Ernande, B., Dieckmann, U.: The evolution of phenotypic plasticity in spatially structured environments: implications of intraspecific competition, plasticity costs and environmental characteristics. J. Evol. Biol 17, 613–628 (2004)

    Article  Google Scholar 

  9. Eshel, I.: Evolutionary and continuous stability. J. Theor. Biol. 103, 99–111 (1983)

    Article  MathSciNet  Google Scholar 

  10. 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)

    Article  Google Scholar 

  11. 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)

    Article  Google Scholar 

  12. Gomulkiewicz, R., Beder, J.H.: The selection gradient of an infinite-dimensional trait. SIAM J. Appl. Math. 56, 509–523 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  13. Gomulkiewicz, R., Kirkpatrick, M.: Quantitative genetics and the evolution of reaction norms. Evolution 46, 390–411 (1992)

    Article  Google Scholar 

  14. Gyllenberg, M., Metz, J.A.J.: On fitness in structured metapopulations. J. Math. Biol. 43, 545–560 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  15. Heino, M., Dieckmann, U., Parvinen, K.: Evolution of foraging strategies on resource gradients. (in prep.)

  16. Heino, M., Metz, J.A.J., Kaitala, V.: The enigma of frequency-dependent selection. Trends. Ecol. Evol. 13, 367–370 (1998)

    Article  Google Scholar 

  17. Jaffrézic, F., Pletcher, S.D.: Statistical models for estimating the genetic basis of repeated measures and other function-valued traits. Genetics 156, 913–922 (2000)

    Google Scholar 

  18. Kingsolver, J.G., Gomulkiewicz, R., Carter, P.A.: Variation, selection and evolution of function-valued traits. Genetica 112–113, 87–104 (2001)

    Google Scholar 

  19. Leimar, O.: Evolutionary change and Darwinian demons. Selection 2, 65–72 (2001)

    Article  Google Scholar 

  20. Levins, R.: Some demographic and genetic consequenses of environmental heterogeneity for biological control. Bull. Entomol. Soc. Am. 15, 237–240 (1969)

    Google Scholar 

  21. Levins, R.: Extinction. In: M. Gerstenhaber (ed.), Some Mathematical Problems in Biology. American Mathematical Society “Providence” RI, 1970, pp. 77–107

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

    MATH  Google Scholar 

  23. Maynard Smith, J.: Evolution and the theory of games. Am. Sci. 64, 41–45 (1976)

    Google Scholar 

  24. Meszéna, G., Kisdi, É., Dieckmann, U., Geritz, S.A.H., Metz, J.A.J.: Evolutionary optimisation models and matrix games in the unified perspective of adaptive dynamics. Selection 2, 193–210 (2001)

    Article  Google Scholar 

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

  26. Metz, J.A.J., Gyllenberg, M.: How should we define fitness in structured metapopulation models? Including an application to the calculation of ES dispersal strategies. Proc. Royal Soc. London B 268, 499–508 (2001)

    Article  Google Scholar 

  27. Metz, J.A.J., Mylius, S.D., Diekmann, O.: When does evolution optimize? On the relation between types of density dependence and evolutionarily stable life-history parameters. Working paper WP-96-004, IIASA, Laxenburg, Austria. http://www.iiasa.ac.at/cgi-bin/pubsrch?WP96004, 1996b

  28. 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)

    Google Scholar 

  29. Parvinen, K.: Evolutionary branching of dispersal strategies in structured metapopulations. J. Math. Biol. 45, 106–124 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  30. Parvinen, K.: On competitive exclusion in metapopulations. (in prep.)

  31. Wan, F.Y.M.: Introduction to the Calculus of Variations and its Applications. Chapman & Hall, 1993

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Authors and Affiliations

  1. Department of Mathematics, 20014 University of Turku, Finland

    Kalle Parvinen

  2. Adaptive Dynamics Network, International Institute for Applied Systems Analysis, 2361, Laxenburg, Austria

    Ulf Dieckmann

  3. Institute of Marine Research, Box 1870, Nordnes, 5817, Bergen, Norway

    Mikko Heino

Authors
  1. Kalle Parvinen
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  2. Ulf Dieckmann
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  3. Mikko Heino
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Correspondence to Kalle Parvinen.

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Parvinen, K., Dieckmann, U. & Heino, M. Function-valued adaptive dynamics and the calculus of variations. J. Math. Biol. 52, 1–26 (2006). https://doi.org/10.1007/s00285-005-0329-3

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  • DOI: https://doi.org/10.1007/s00285-005-0329-3

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