We study strategic party interaction in a spatial voting model where voters’ ideological positions may change. Building on a rich empirical and theoretical literature, we assume that voters align their ideology with others who are sufficiently close to them (social influence with bounded confidence) as well as with the party that they support (party attraction). We show that these changes have strong implications on the results of the party competition model by Laver (2005). Two strategies stand out in our simulations: Aggregators, who always follow the mean policy of their supporters, and predators, who always chase the strongest party. Aggregators are most likely to win in a large corridor of the parameter space. However, predators can outperform them if party attraction is strong. This is interesting because predators are on average the worst-performing parties in the static voter distribution benchmark. We argue that these results are connected to real-world debates about how mainstream parties should react to the rise of extremist parties, as the two strategies epitomize debates about focusing on own strengths and supporters (aggregators) vs. adapting towards successful extremists (predators). We also demonstrate that the level of polarization and fragmentation of parties and voters is strongly affected by social influence and party attraction. While medium-sized confidence bounds and party attraction increase the polarization of voters and parties, unconstrained social influence decreases it.