# An extension of the classification of evolutionarily singular strategies in Adaptive Dynamics

## Abstract

The existing classification of evolutionarily singular strategies in Adaptive Dynamics (Geritz et al. in Evol Ecol 12:35–57, 1998; Metz et al. in Stochastic and spatial structures of dynamical systems, pp 183–231, 1996) assumes an invasion exponent that is differentiable twice as a function of both the resident and the invading trait. Motivated by nested models for studying the evolution of infectious diseases, we consider an extended framework in which the selection gradient exists (so the definition of evolutionary singularities extends *verbatim*), but where the invasion fitness may lack the smoothness necessary for the classification à la Geritz et al. We derive the classification of singular strategies with respect to convergence stability and invadability and determine the condition for the existence of nearby dimorphisms. In addition to ESSs and invadable strategies, we observe what we call one-sided ESSs: singular strategies that are invadable from one side of the singularity but uninvadable from the other. Studying the regions of mutual invadability in the vicinity of a one-sided ESS, we discover that two isoclines spring in a tangent manner from the singular point at the diagonal of the mutual invadability plot. The way in which the isoclines unfold determines whether these one-sided ESSs act as ESSs or as branching points. We present a computable condition that allows one to determine the relative position of the isoclines (and thus dimorphic dynamics) from the dimorphic as well as from the monomorphic invasion exponent and illustrate our findings with an example from evolutionary epidemiology.

## Keywords

Adaptive dynamics Evolutionarily singular strategy One-sided ESS Evolutionary branching Evolutionary arms race Invasion exponent Superinfection## Mathematics Subject Classification (2000)

92D15 92D30 92D40## Notes

### Acknowledgments

We thank Hans Metz, Andrea Pugliese, Géza Meszéna, Mats Gyllenberg and Éva Kisdi for their valuable comments.

## References

- Abrams PA, Matsuda H (1994) The evolution of traits that determine ability in competitive contests. Evol Ecol 8(6):667–686CrossRefGoogle Scholar
- Boldin B, Diekmann O (2008) Superinfections can induce evolutionarily stable coexistence of pathogens. J Math Biol 56(5):635–672MathSciNetCrossRefzbMATHGoogle Scholar
- Dawkins R, Krebs JR (1979) Arms races between and within species. Proc R Soc Lond Ser B Biol Sci 205(1161):489–511CrossRefGoogle Scholar
- Dercole F, Rinaldi S (2008) Analysis of evolutionary processes: the adaptive dynamics approach and its applications. Princeton University Press, PrincetonGoogle Scholar
- Diekmann O, Heesterbeek H, Britton T (2012) Mathematical tools for understanding infectious disease dynamics. Princeton University Press, PrincetonGoogle Scholar
- Durinx M, Metz JAJ, Meszéna G (2008) Adaptive dynamics for physiologically structured population models. J Math Biol 56(5):673–742MathSciNetCrossRefzbMATHGoogle Scholar
- Geritz SAH (2005) Resident-invader dynamics and the coexistence of similar strategies. J Math Biol 50(1):67–82MathSciNetCrossRefzbMATHGoogle Scholar
- Geritz SAH, Kisdi E, Meszena G, Metz JAJ (1998) Evolutionary singular strategies and the adaptive growth and branching of the evolutionary tree. Evol Ecol 12:35–57CrossRefGoogle Scholar
- Kisdi É, Geritz SAH (2001) Evolutionary disarmament in interspecific competition. Proc R Soc Lond Ser B Biol Sci 268(1485):2589–2594CrossRefGoogle Scholar
- Law R, Marrow P, Dieckmann U (1997) On evolution under asymmetric competition. Evol Ecol 11(4):485–501CrossRefGoogle Scholar
- Meszéna G, Gyllenberg M, Jacobs FJ, Metz JAJ (2005) Link between population dynamics and dynamics of Darwinian evolution. Phys Rev Lett 95(7):078,105Google Scholar
- Metz JAJ, Geritz SAH, Meszéna G, Jacobs FJA, van Heerwaarden JS (1996) Adaptive dynamics: a geometrical study of the consequences of nearly faithful reproduction. In: van Strien SJ, Verduyn Lunel SM (eds) Stochastic and spatial structures of dynamical systems, pp 183–231Google Scholar
- Metz JAJ, Nisbet RM, Geritz SAH (1992) How should we define fitness for general ecological scenarios? Trends Ecol Evol 7(6):198–202CrossRefGoogle Scholar
- Mosquera J, Adler FR (1998) Evolution of virulence: a unified framework for coinfection and superinfection. J Theor Biol 195:293–313CrossRefGoogle Scholar
- Pugliese A Evolutionary dynamics of virulence. http://www.science.unitn.it/pugliese/lavori/puglieAd.pdf
- Rueffler C, Van Dooren TJM, Metz JAJ (2007) The interplay between behavior and morphology in the evolutionary dynamics of resource specialization. Am Nat 169(2):E34–E52CrossRefGoogle Scholar
- Smith J (2011) Superinfection drives virulence evolution in experimental populations of bacteria and plasmids. Evolution 65(3):831CrossRefGoogle Scholar