A multitype Markovian branching process with one-type population size dependence

Motivated by a within-host framework of immunity-modulated parasitic disease, we formulate a multitype Markovian branching process with reproductive parameters dependent on the number of individuals of a single type only. We impose a soft carrying capacity $K$ with respect to the controlling type, serving as a threshold between subcritical and supercritical dynamics. We prove that extinction occurs almost surely when the offspring batch of each non-controlling type includes an individual of the controlling type with positive probability, with the time to extinction possessing finite moments of all orders. We then construct a sequence of density-dependent processes indexed by $K$ and study scaling limits of the process using the canonical functional law of large numbers (FLLN) and central limit theorem, with a view towards characterising the emergence of metastable behaviour in the vicinity of asymptotically stable equilibria, or stable limit cycles of the FLLN.