Long-term behaviour of a cyclic catalytic branching system

We investigate the long-term behaviour of a system of SDEs for d>=2 types, involving catalytic branching and mutation between types. In particular, we show that the overall sum of masses converges to zero but does not hit zero in finite time a.s. We shall then focus on the relative behaviour of types in the limit. We prove weak convergence to a unique stationary distribution that does not put mass on the set where at least one of the coordinates is zero. Finally, we provide a complete analysis of the case d=2.