Lions and hyenas coexist in the Savanna. Because hyenas have neither the speed nor the agility, they stand no chance against a lion in a one-on-one duel. The lion emerges victorious in such an encounter every time and gets a payoff of 3. Hyenas, however, can be more or less harmed or benefited by fighting a lion, ending up with a payoff of x. When lions encounter other lions, they fight and end up with zero payoffs each. When hyenas encounter other hyenas they do not fight and end up with 1 each. 

(a) Assume x = 0. Is having a population of all lions evolutionarily stable with respect to hyenas? Is having a population of all hyenas evolutionarily stable with respect to lions? 

(b) For the rest of this question, assume x takes on some value in the interval (-1,2). If there are twice as many hyenas as lions, for which values of x are lions fitter than hyenas? 

(c) If there are 20 times as many lions as hyenas, for which values of x are hyenas fitter than lions? 

(d) In terms of the proportion of lions (α) in the population, how big should the payoff hyenas get when facing lions (x) be for hyenas to be fitter than lions? 

(e) Are there any polymorphic equilibria in the Savanna?