We study a dynamic model of elections where many parties may enter or exit political competition. At each election a new political leadership arrives for each party. The leadership cannot choose the party's platform (ideological identities are fixed) but must decide whether or not to contest the election. Contesting elections is costly and this cost is higher if the party has recently been inactive. The distribution of voters' ideal policies, or public opinion, changes over time via a Markov process with a state independent persistence parameter. We characterise stable party systems where the set of contestants is invariant to the recent most observed opinion. We show that stable party systems exist only when public opinion is sufficiently volatile, while highly persistent moods lead to instability and change in the party system whenever public opinion changes.