The branching bifurcation of Adaptive Dynamics

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

The social diversification of fashion

We propose a model to investigate the dynamics of fashion traits purely driven by social interactions. We assume that people adapt their style to maximize social success, and we describe the interaction as a repeated group game in which the payoffs reflect the social norms dictated by fashion. On one hand, the tendency to imitate the trendy stereotypes opposed to the tendency to diverge from them to proclaim identity; on the other hand, the exploitation of sex appeal for dating success opposed to the moral principles of the society. These opposing forces promote diversity in fashion traits, as predicted by the modeling framework of adaptive dynamics. Our results link the so-called horizontal dynamics—the primary driver of fashion evolution, compared with the vertical dynamics accounting for interclass and economic drivers—to style variety

The transition from evolutionary stability to branching: A catastrophic evolutionary shift

Evolutionary branching—resident-mutant coexistence under disruptive selection—is one of the main contributions of Adaptive Dynamics (AD), the mathematical framework introduced by S.A.H. Geritz, J.A.J. Metz, and coauthors to model the long-term evolution of coevolving multi-species communities. It has been shown to be the basic mechanism for sympatric and parapatric speciation, despite the essential asexual nature of AD. After 20 years from its introduction, we unfold the transition from evolutionary stability (ESS) to branching, along with gradual change in environmental, control, or exploitation parameters. The transition is a catastrophic evolutionary shift, the branching dynamics driving the system to a nonlocal evolutionary attractor that is viable before the transition, but unreachable from the ESS. Weak evolutionary stability hence qualifies as an early-warning signal for branching and a testable measure of the community’s resilience against biodiversity. We clarify a controversial theoretical question about the smoothness of the mutant invasion fitness at incipient branching. While a supposed nonsmoothness at third order long prevented the analysis of the ESS-branching transition, we argue that smoothness is generally expected and derive a local canonical model in terms of the geometry of the invasion fitness before branching. Any generic AD model undergoing the transition qualitatively behaves like our canonical model