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

Fabio Dercole,Fabio Della Rossa,Pietro Landi

doi:10.1038/srep26310

Abstract

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.

Highlights

Evolutionary branching takes off when a resident and a similar mutant type coexist in the same environment and natural selection is disruptive, i.e., favors outer rather than intermediate phenotypes
In the restricted formulation in which resident individuals are characterized by the same value x of a one-dimensional strategy, Geritz, Metz et al.1–3 derived explicit conditions for evolutionary branching in terms of the invasion fitness5—the exponential rate of growth sx(y) initially shown by a mutant strategy y appeared when the resident is at its ecological regime
Evolutionary branching requires, first, the resident strategy x to be in the vicinity of a convergence stable singular strategy x*—a strategy making neutral the selection pressure measured by the fitness gradient ∂ysx(y)|y=x, i.e

Summary

We consider two similar populations, with densities n1 and n2 and one-dimensional strategies x1 and x2 ≈x1. C1′: sx1,x2(y) = sx1(y), along the extinction boundary 2 on which only strategy x1 is present, we derive the expansion [4, 6] under the coexistence condition [2] C1′: sx ⁎+ε cos θ2(ε),x ⁎+ε sin θ2(ε)(x⁎ + ∆y) = sx ⁎+ε cos θ2(ε)(x⁎ + ∆y), to be imposed together with its (ε, Δy)-derivatives at (ε, Δy) =(0, 0) up to order 3 This involves the angle θ2(0)—the tangent direction to the extinction boundary 2 at (x*, x*)—and the first derivative θ2′(0)—the local curvature of the boundary (whether θ increases or decreases while moving away from (x*, x*), see Fig. 1e,f). The derivation and the analysis of the canonical model [8] are detailed in the Supplementary Information (Sects. 4 and 5)

An example

Additional Information