Separation of timescales for the seed bank diffusion and its jump-diffusion limit

Abstract We investigate scaling limits of the seed bank model when migration (to and from bank) is ‘slow’ compared to reproduction. This motivated by models for bacterial dormancy, where periods dormancy can be orders magnitude larger than reproductive times. Speeding up time, we encounter a separation timescales phenomenon which leads mathematically interesting observations, in particular providing prototypical example limit continuous diffusion will jump diffusion. For this situation, standard convergence results typically fail. While such situation could principle attacked sophisticated analytical scheme Kurtz (J Funct Anal 12:55–67, 1973), require significant technical efforts. Instead, our are able identify explicitly characterise well-defined via duality surprisingly non-technical way. Indeed, show that moment suitable sense stable under passage allows direct intuitive identification limiting semi-group while at same time probabilistic interpretation model. also obtain general strategy continuous-time Markov chains regime, independent interest.