**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

*Convergence stability in polymorphic
populations or with multiple traits*

*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 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.

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

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.

__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

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.

__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.

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.

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

__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.

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.

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*

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