Monomorphic ESS does not imply the stability of the corresponding polymorphic state in the replicator dynamics in matrix games under time constraints

Matrix games under time constraints are natural extensions of matrix games. They consider the fact that, in addition to the payoff, a pairwise interaction has a further consequence for the contestants. Namely, both players have to wait for some time before becoming fit to participate in a subsequent interaction. Every matrix game can be assigned a continuous dynamical system (the replicator equation) which describes how the frequencies of different phenotypes evolve in the population. One of the fundamental theorems of evolutionary matrix games asserts that the state corresponding to an evolutionarily stable strategy is an asymptotically stable rest point of the replicator equation (Taylor and Yonker 1978, Hofbauer et al. 1979, Zeeman 1980). Garay et al. (2018) and Varga et al. (2020) generalized the statement to two-strategy and, in some particular cases, three- or more strategy matrix games under time constraints. However, the question of whether the implication holds in general remained open. Here examples are provided demonstrating that the answer is no. Moreover, we point out through the rock-scissor-paper game that arbitrary small differences between waiting times can destabilize the rest point corresponding to an ESS. It is also shown that a stable limit cycle can arise around the unstable rest point in a supercritical Hopf bifurcation. Mathematics Subject Classification91A22, 92D15, 92D25, 91A80, 91A05, 91A10, 91A40, 92D40