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Adaptive dynamics:

A framework for modelling long-term evolution

COSA-course at the Jyväskylä Summer School, 1999

Most ecologically motivated evolutionary problems imply frequency-dependent selection, where the fitness of a strategy depends on the kind and frequency of other strategies present in the population. The conventional tool to investigate frequency-dependent evolutionary models is to search for the so-called evolutionarily stable strategies (ESSs), i.e., strategies such that if already established in the population, then immune against invasion by any mutant strategy. However, it has long been recognised that the ESSs are not necessarily the attractors of evolution as a dynamic process. The emphasis in modelling adaptive evolution has been increasingly shifting towards models that consider the dynamic process itself.

This course gives an introduction to a novel modelling approach to study the dynamics of long-term evolution, including a few illustrative applications and a discussion of the relation between this framework and the more traditional methods.

Program:

    1. Introduction: Short- and long-term evolution. Invasion and fitness. Adaptive dynamics framework
    2. Evolutionary singularities and evolutionary branching
    3. Coevolution of coexisting strategies
    4. The canonical equation of directional evolution
    5. Generalisations, and an outlook on current research

Literature

Basic references

Adaptive dynamics framework

Convergence stability in polymorphic populations or with multiple traits

The canonical equation

Special topics Short- and long-term evolution

Invasion dynamics, nontrivial cases

Evolutionary stability and convergence stability

Adaptive dynamics and optimization

Evolutionary bifurcation theory

Adaptive dynamics with multiple population dynamical attractors

Adaptive dynamics in diploid populations

Evolutionary branching and sympatric speciation

Applications

Basic references

Adaptive dynamics framework of Geritz & Metz

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

Geritz, S. A. H., J. A. J. Metz, É. Kisdi, and G. Meszéna. 1997. Dynamics of adaptation and evolutionary branching. Phys. Rev. Letters 78:2024-2027.

Metz, J. A. J., S. A. H. Geritz, G. Meszéna, F. J. A. Jacobs, and J. S. van Heerwaarden. 1996. Adaptive dynamics, a geometrical study of the consequences of nearly faithful reproduction. Pp. 183-231 in S. J. van Strien, and S. M. Verduyn Lunel, eds. Stochastic and spatial structures of dynamical systems. North Holland, Amsterdam, The Netherlands.

Eshel, I., U. Motro, and E. Sansone. 1997. Continuous stability and evolutionary convergence. J. theor. Biol. 185:333-343.

The first paper gives a self-contained description of the adaptive dynamics framework and an illustrative example for how it can be used. The second paper is a short account specifically tailored for the interest of physicists; the third paper is more mathematical. Eshel et al. reached some of the results independently.
Convergence stability in polymorphic populations or with multiple traits

Matessi C. & Di Pasquale. 1996. Long-term evolution of multilocus traits. J. Math. Biol. 34:613-653.

The canonical equation

Dieckmann U. & R. Law. 1996. The dynamical theory of coevolution: A derivation from stochastic ecological processes. J. Math. Biol. 34:579-612.

Dieckmann U., P. Marrow & R. Law. 1995. Evolutionary cycling in predator-prey interactions: Population dynamics and the Red Queen. J. theor. Biol. 176:91-102.

The first paper provides the derivation of the canonical equation as an approximation for the mean path of the stochastic trajectories; higher-order approximations are also considered. The second paper describes an example and numerically compares the stochastic trajectories with the solution of the canonical equation.

Special topics

Short- and long-term evolution

Eshel, I. 1996. 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.

Hammerstein P. 1996. Darwinian adaptation, population genetics and the streetcar theory of evolution. J. Math. Biol. 34:511-532.

Invasion dynamics, nontrivial cases

Metz, J. A. J., R. M. Nisbet, S. A. H. Geritz. 1992. How should we define 'fitness' for general ecological scenarios? TREE 7:198-202.

stochastic environments: Kisdi E. & G. Meszena. 1993. Density dependent life history evolution in fluctuating environments. In: J. Yoshimura & C. Clark (eds): Adaptation in a stochastic environment. Lecture Notes in Biomathematics, Springer-Verlag, Vol. 98 pp. 26-62.

structured populations: Caswell H. 1989. Matrix population models. Sinauer Associates, Sunderland.

structured populations in stochastic environments: Tuljapurkar S. 1989. An uncertain life: Demography in random environments. Theor. Pop. Biol. 35:227-294.

chaotic populations: Ferriere R. & M. Gatto. 1995. Lyapunov exponents and the mathematics of invasion in oscillatory or chaotic populations. Theor.Pop. Biol. 48:126-171.

demographic stochasticity: Goel N. S. & N. Richter-Dyn. 1974. Stochastic models in biology. Academic Press, New York

Evolutionary stability and convergence stability

Maynard Smith J. 1982. Evolution and the theory of games. Cambridge University Press

The classic account on the ESS. The main part of the book, however, considers ESSs in matrix games; the general definition for an ESS in case of continuous traits and general frequency dependent selection is given in Appendix.

Eshel I. 1983. Evolutionary and continuous stability. J. theor. Biol. 103:99-111.

Christiansen F. B. 1991. On conditions for evolutionary stability for a continuously varying character. Am. Nat. 138:37-50.

Abrams P. A., H. Matsuda & Y. Harada. 1993. Evolutionarily unstable fitness maxima and stable fitness minima of continuous traits. Evol. Ecol. 7:465-487.

The first of these three papers points out the difference between evolutionary and convergence stability. The name 'convergence stability' comes from the second paper; both the second and the third paper gives ecologically motivated examples where a convergence stable strategy is not an ESS. (Abrams et al. prefer to use the word 'evolutionary stability' instead of convergence stability (!) and refer to the classic ESS as 'fitness maximum'.)

Adaptive dynamics and optimization

Kisdi E. 1998. Frequency dependence versus optimization. TREE 13:508.

Mylius S. D. & J.A.J. Metz. When does evolution optimize? On the relationship between evolutionary stability, optimization and density dependence. In: U. Dieckmann, and J. A. J. Metz, eds. Elements of adaptive dynamics. Cambridge University Press, in press.

Evolutionary bifurcation theory

Geritz S. A. H., E. van der Meijden & J. A. J. Metz. 1999. Evolutionary dynamics of seed size and seedling competitive ability. Theor. Pop. Biol. 55:324-343.

This paper provides a detailed bifurcation analysis of adaptive dynamics in a specific model, and also describes some bifurcation structures as well as the connection points between isoclines and the boundary of the area of coexistence in general.

Jacobs F. & J. A. J. Metz. Bifurcation analysis for adaptive dynamics based on Lotka-Volterra competition models. in prep.

Adaptive dynamics with multiple population dynamical attractors

Rand D. A., H. B. Wilson & J. M. McGlade. 1994. Dynamics and evolution: Evolutionarily stable attractors, invasion exponents and phenotype dynamics. Phil. Trans. R. Soc. Lond. B 343:261-283.

Geritz S. A. H. Adaptive dynamics with multiple demographic attractors. in prep.

Adaptive dynamics in diploid populations

Kisdi E. & S. A. H. Geritz. 1999. Adaptive dynamics in allele space: Evolution of genetic polymorphism by small mutations in a heterogeneous environment. Evolution 53:993-1008.

The two-patch soft selection model with diploid genetics; full bifurcation analysis and a comparison to classic (short-term) models of multiple niche polymorphism are included.

Van Dooren T. J. M. 1999. The evolutionary ecology of dominance-recessivity. J. theor. Biol. 198:519-532.

The evolution of dominance in the two-patch soft selection model of Kisdi & Geritz (1999).

Evolutionary branching and sympatric speciation

Kisdi E. & S. A. H. Geritz. Evolutionary branching and sympatric speciation in diploid populations. In: U. Dieckmann, and J. A. J. Metz, eds. Elements of adaptive dynamics. Cambridge University Press, in press.

The evolution of assortative mating based on a separate single-locus mating trait in the two-patch soft selection model of Kisdi & Geritz (1999).

Dieckmann U. & M. Doebeli. 1999. On the origin of species by sympatric speciation. Nature 400:354-357.

Doebeli M. 1996. A quantitative genetic model for sympatric speciation. J. evol. Biol.9:893-909.

The first paper investigates a multilocus genetic model for evolutionary branching with the evolution of assortative mating. Mating may be based on the primary trait under selection or on a separate marker trait; in both cases, the propensity for mating assortatively ("choosiness") is also evolving. The second paper is an earlier account with assortative mating based on the primary trait only.

Applications

The following papers present specific applications of the adaptive dynamics framework of Geritz and Metz

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

This paper uses the clonal two-patch soft selection model as an illustrative example. The three-patch model can be found in Geritz et al. (1998; see among the basic references).

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

Doebeli, M., and G. D. Ruxton. 1997. Evolution of dispersal rates in metapopulation models: Branching and cyclic dynamics in phenotype space. Evolution 51:1730-1741.

Kisdi, É. 1999. Evolutionary branching under asymmetric competition. J. theor. Biol. 197:149-162.

Geritz, S. A. H., E. van der Meijden, and J. A. J. Metz. 1999. Evolutionary dynamics of seed size and seedling competitive ability. Theor. Pop. Biol. 55:324-343.

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

Boots M. & Y. Haraguchi. 1999. The evolution of costly resistance in host-parasite systems. Am. Nat. 153:359-370.

Mathias, A., and É. Kisdi. Evolutionary branching and coexistence of germination strategies. In: U. Dieckmann, and J. A. J. Metz (eds.): Elements of adaptive dynamics, Cambridge University Press, in press

Meszéna, G., and J. A. J. Metz. The role of effective environmental dimensionality. In: U. Dieckmann, and J. A. J. Metz (eds.): Elements of adaptive dynamics, Cambridge University Press, in press