Dissensus is a modeling framework for networks of dynamic agents in competition for scarce resources. Originally inspired by biological cell behaviors, it also fits marketing, finance and many other application areas. Competition is often unstable in the sense that strong agents, those having access to large resources, gain more and more resources at the expenses of weak agents. Thus, strong agents duplicate when reaching a critical amount of resources, whereas weak agents die when losing all their resources. To capture all these phenomena we introduce discrete time gossip systems with unstable state dynamics interrupted by discrete events affecting the network topology. Invariancy of states, topologies, and network connectivity are explored.