Past exam of the mathematics course of the University of Cambridge / 2010 / iii / Paper 35 / 6 / ii

For $V=D>0$, choose pure Hawk. Every mutant $p<1$ ties it against a resident Hawk population, so the second condition must be checked. The Hawk-Dove game formula givesConsequentlyThe actual invasion advantage is $ϵV(1-p{)}^2/2>0$, despite the tie at invasion frequency zero.

If the equal parameters are negative, pure Dove is instead an evolutionarily stable strategy: for a mutant $p>0$, $e(0,0)-e(p,0)=-pV/2>0$. If $V=D=0$, every payoff is zero and no resident has a strict advantage over any mutant. Thus the full equal-parameter answer is Hawk for $V=D>0$, Dove for $V=D<0$, and no ESS for $V=D=0$.