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

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## 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.

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