We propose and analyze a novel replicator-mutator type system to model the dynamics of market shares in a competitive duopoly, in which the effect of advertising is subject to delays in the availability of information as well as in the response of customer to the advertising effort. The proposed model describes the transitions of customers between the two competing firms. These transitions are dependent on the following parameters: customer preferences for each firm, the rate of mutation or exploration, also known as churn or infidelity, of customers with regard to the firms, and the value of the delays in advertising effects. Analysis and numerical results show the presence of oscillatory behaviour, Hopf bifurcations and chaos for certain ranges of parameter values.