Binary branching processes with Moran type interactions

The aim of this paper is to study the large population limit of a new collection of interacting particle systems (IPS) that encompasses branching models and fixed size Moran type IPS (see (J. Phys. A 29 (1996) 2633–2642); (Comm. Math. Phys. 214 (2000) 679–703); (Feynman–Kac Formulae: Genealogical and Interacting Particle Systems with Applications (2004) Springer-Verlag); (Mean Field Simulation for Monte Carlo Integration (2013) CRC Press); (ESAIM Probab. Stat. 18 (2014) 441–467)). We define an IPS where particles evolve, reproduce and die independently and, with a probability that may depend on the configuration of the whole system, the death of a particle may trigger the reproduction of another particle, while a branching event may trigger the death of an other one. We study the occupation measure of the new model, explicitly relating it to the Feynman–Kac semigroup of the underlying Markov evolution and quantifying the L ² distance between their normalisations. We show that our model outperforms the fixed size Moran type IPS when used to approximate a birth and death process. We discuss several other applications of our model including the neutron transport equation (Cox et al. (2020); Stochastic Neutron Transport and Non-local Branching Markov Processes (2023) Birkhäuser) and population size dynamics.