Evolution of populations with strategy-dependent time delays

The paper defines F(x) for two-strategy games with strategy-dependent delays and proves that, under a<c, d<b and either a<d<c or d<a<b, F is monotone: decreasing if τC>τD and increasing if τC<τD (Appendix Theorem A.6). This is derived by an explicit (but algebraically heavy) expression for F′(x) and sign analysis, grounded in the definition of F in Eq. (13) and the payoff parametrization ŪC=ax+b(1−x), ŪD=cx+d(1−x) (see the Methods and Appendix where F is introduced) . The candidate solution proves the same monotonicity by a cleaner change of variables R(x)=ŪD/ŪC, rewriting F in terms of R and showing F′ has a definite sign depending on τC−τD. It also correctly concludes uniqueness of a zero when endpoint signs differ (matching the paper’s use of monotonicity to guarantee a unique interior solution) . Aside from a minor algebraic slip (omitting the positive denominator ŪC^2 when writing R′), the candidate’s proof is sound and uses assumptions consistent with the paper (notably that F is real-valued on (0,1), implying ŪC,ŪD>0 there). Hence both are correct, with the model offering a different, simpler proof.