Evolutionary dynamics for κ ≠ 0

Rearranging terms, the gain function 𝒢(z) given by Eq. (¹¹) can be alternatively written as

where

is a polynomial in Bernstein form⁵³ of degree n − 1 with coefficients given by

(ζ ₀, ζ ₁) = (−(1 + κ), 1 + κ) if n = 2.

(ζ ₀, ζ ₁, ζ ₂) = (−1, 0, 1) if n = 3.

(ζ ₀, ζ ₁, ζ ₂, ζ ₃) = (−1, κ, −κ, 1) if n = 4.

(ζ ₀, ζ ₁, ζ ₂, …, ζ _n−3, ζ _n−2, ζ _n−1) = (−1, κ, 0, …, 0, −κ, 1) if n ≥ 5.

The number of sign changes (and hence of singular points) of 𝒢(z) is bounded from above by the number of sign changes of 𝒫(z). Moreover, and by the variation-diminishing property of polynomials in Bernstein form⁵³, the number of sign changes of 𝒫(z) is equal to the number of sign changes of the sequence of coefficients (ζ ₀, …, ζ _n−1) minus an even integer. It then follows that the number of singular points is at most one if n ≤ 3 or if scaled relatedness is nonpositive, κ ≤ 0. In this case, the unique singular point z ^* is convergence unstable. However, if n ≥ 4 and κ > 0, there could be up to three singular points z _L, z _M, and z _R satisfying 0 < z _L < z _M < z _R < 1 such that z _L and z _R are convergence unstable and z _M is convergence stable.