Determination of cross-repression strength and coexistence structure

Determine the cross-repression strength in the two-trait adaptive-dynamics model of membrane-permeability evolution and establish whether it produces a single stable coexistence node or two off-diagonal winner-take-all attractors at equal environmental frequencies.

Background

At equal environmental frequencies, the adaptive-dynamics model can exhibit qualitatively different phase portraits depending on the strength of cross-repression between the two permeability traits. Weak repression yields a single stable fixed point with both traits maintained, whereas strong repression would produce two off-diagonal fixed points corresponding to winner-take-all outcomes. The simulations favor the single-node structure, but the empirical data do not independently identify the repression parameter, leaving the mechanistic strength of cross-repression unresolved.

References

The repression strength is not determined by the data (Sec.~XI of the Supplementary Material), so the model alone does not establish the single-node structure.

— Stochasticity and Environmental Switching Shape Quorum Sensing Evolution in Bacterial Populations  (2610.10492 - Kumar et al., 7 Oct 2026) in Section 3.4, “Symmetric environments produce stochastic coexistence”; Section 4.5, “Coexistence at the symmetric point”; Supplementary Material, Section XI