I am new to evolutionary game theory so I can't figure out whether I'm looking at things correctly. I have the following payoff matrix:

$$ A = \begin{matrix} 3 & 0 \\ 5 & 1 \end{matrix} $$

My professor made mention in some previous exercises that we were dealing with symmetric two player games so I assume this is the case here (i.e. not a zero sum game). I need to find all evolutionary stable strategies (ESS), and I am unsure if my intuition is right. From what I see there is a unique nash equilibrium in pure strategies, $x^{*} = {(0,1), (0,1)}$ (there is a dominant strategy equilibrium, with the outcome at $a = 1$).

All examples I've done so far involved finding mixed strategy equilibrium so I guess my first question is whether a pure strategy equilibrium can classify as evolutionary stable (without getting into dynamics my intuition is that if the population starts to change and shifts a bit into the first strategy then you could show that this strategy can be made profitable, so that a pure strategy is not stable).

Since I have that $x*$ is a nash, and given that $u(x*,x*) > u(y,x*)$ always holds (for any $y \neq x^{*}$), technically I have an ESS but again but this feels wrong (not a very scientific argument, I know but I really have only started studying this topic today).

If anyone has any insights and/or guidance to solve this it would be very much appreaciated!


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