There is an extensive literature on the entropy behavior of polynomial interval maps as they vary in families. In particular, the monotonicity problem (due to Milnor) which asks about the connectedness of the level sets of the entropy function (the isentropes) has been of immense interest and is very well studied in the context of polynomial interval maps. In contrast, the entropy behavior of real rational maps is a much less studied albeit much more general setting. After introducing a real entropy function on a moduli space of real rational maps, we focus on the case of real quadratic rational maps where, supported by experimental evidence, we show that the isentropes are connected in certain dynamically defined regions of the moduli space while in general, they become disconnected due to a non-polynomial behavior. Speaker(s): Khashayar Filom (U(M))