  Motivated by the stochastic Lotka-Volterra model, we introduce discrete-state
interacting multitype branching processes. We show that they can be obtained as
the sum of a multidimensional random walk with a Lamperti-type change
proportional to the population size; and a multidimensional Poisson process
with a time-change proportional to the pairwise interactions. We define the
analogous continuous-state process as the unique strong solution of a
multidimensional SDE. We prove that the scaling limits of the discrete-state
process correspond to its continuous counterpart. In addition, we show that the
continuous-state model can be constructed as a generalized Lamperti-type
transformation of multidimensional L\'evy processes.
