A cyclically catalytic super-Brownian motion

In generalization of the mutually catalytic super-Brownian motion in R of D awson and Perkins and Mytnik, a function-valued cyclically catalytic model X is constructed as a strong Markov solution to a martingale problem. Start ing with a finite population X-0, each pair of neighboring types will globa lly segregate in the long-term limit (noncoexistence of neighboring types). Also finer extinction-survival properties depending on X-0 are studied in the spirit of Mueller and Perkins. In fact, X-0 can be chosen in such a way that all types survive for all finite times. On the other hand, sufficient conditions on X-0 are stated for the following situation: given a type k a nd a positive time t, the kth subpopulation X-k dies by time t with a large probability, provided that its initial value X-0(k) was sufficiently small .