The branching bifurcation of Adaptive Dynamics

Della Rossa F, Dercole F, & Landi P (2015). The branching bifurcation of Adaptive Dynamics. International Journal of Bifurcation and Chaos 25 (7): p. 1540001. DOI:10.1142/S0218127415400015.

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Abstract

We unfold the bifurcation involving the loss of evolutionary stability of an equilibrium of the canoncal equation of Adaptive Dynamics (AD). The equation deterministically describes the expected long-term evolution of inheritable traits phenotypes or strategies-of coevolving populations, in the limit of rare and small mutations. In the vicinity of a stable equilibium of the AD canonical equation, a mutant type can invade and coexist with the present-resident-types, whereas the fittest always win far from equilibrium. After coexistence, residents and mutants effectively diversify, according to the enlarged canonical equation, only if natural selection favors outer rather than intermediate traits-the equilibrium being evolutionarily unstable, rather than stable. Though the conditions for evolutionary branching-the joint effect of resident-mutant coexistence and evolutionary instability- have been known for long, the unfolding of the bifurcation has remained a missing tile of AD, the reason being related to the nonsmoothness of the mutant invasion fitness after branching. In this paper, we develop a methodology that allows the approximation of the invasion fitness after branching in terms of the expansion of the (smooth) fitness before branching. We then derive a canonical model or the branching bifurcation and perform its unfolding around the loss of evolutionary stability. We cast our analysis in the simplest (but classical) setting of asexual, unstructured populations living in an isolated, homogeneous, and constant abiotic environment; individual traits are one-dimensional; intra-as well as inter-specific ecological interactions are described in the vicinity of a stationary regime.

Item Type: Article
Uncontrolled Keywords: Adaptive dynamics; bifurcation; evolutionary branching; evolutionary stability; invasion fitness; singular strategy
Research Programs: Evolution and Ecology (EEP)
Bibliographic Reference: International Journal of Bifurcation and Chaos; 25(7):1540001 (June 2015)
Depositing User: IIASA Import
Date Deposited: 15 Jan 2016 08:53
Last Modified: 25 Apr 2017 08:43
URI: http://pure.iiasa.ac.at/11413

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