Branching processes with interactions: the subcritical cooperative regime

In this paper, we introduce a family of processes with values on the nonnegative integers that describes the dynamics of populations where individuals are allowed to have different types of interactions. The types of interactions that we consider include pairwise interaction, such as competition, annihilation and cooperation; and interaction among several individuals that can be considered as catastrophes. We call such families of processes $\mathit{\text{branching}}\mathit{\text{processes}}\mathit{\text{with}}\mathit{\text{interactions}}$. Our aim is to study their long term behaviour under a specific regime of the pairwise interaction parameters that we introduce as {\it subcritical cooperative regime}. Under such regime, we prove that a process in this class comes down from infinity and has a moment dual which turns out to be a jump-diffusion that can be thought as the evolution of the frequency of a trait or phenotype and whose parameters have a classical interpretation in terms of population genetics. The moment dual is an important tool for characterizing the stationary distribution of branching processes with interactions whenever such distribution exists but it is an interesting object on its own right.