Origin of ε-stabilisation in quasispecies dynamics

Determine whether the ε-stabilisation of the leading eigenvalue observed for the Crow–Kimura quasispecies model (defined by (M + Q_N)p = \bar{m} p with permutation-invariant mutation matrix Q_N on binary sequence space; see Equations (1.33) and (1.36)–(1.38)) at finite mutation rate μ is a purely mathematical consequence of the associated spectral problem for M + Q_N or whether it reflects an underlying biological law governing living systems.

Background

Section 1.7 establishes that the leading eigenvalue of the Crow–Kimura model exhibits limiting stabilisation as the mutation rate μ tends to infinity, and Section 1.8 introduces ε-stabilisation to formalise stabilisation at finite μ. Analytical approximations (e.g., Equation (1.53)) and examples demonstrate that ε-stabilisation can occur, aligning with the error-threshold phenomenon.

However, examples also show that ε-stabilisation at finite μ does not always occur for all fitness landscapes, and the causes of this behaviour remain unclear. This motivates a fundamental question about whether the observed ε-stabilisation is inherent to the spectral properties of the model’s linear operator or is indicative of a deeper biological principle.