Boundary behavior of continuous-state interacting multi-type branching processes with immigration

In this paper, we study continuous-state interacting multi-type branching processes with immigration (CIMBI processes), where inter-specific interactions -- whether competitive, cooperative, or of a mixed type -- are proportional to the product of their type-population masses. We establish sufficient conditions for the CIMBI process to never hit the boundary $\partial\mathbb{R}_{+}^{d}$ when starting from the interior of $\mathbb{R}_{+}^{d}$. Additionally, we present two results concerning boundary attainment. In the first, we consider the diffusion case and prove that when the constant immigration rate is small and diffusion noise is present in each direction, the CIMBI process will almost surely hit the boundary $\partial\mathbb{R}_{+}^{d}$. In the second result, under similar conditions on the constant immigration rate and diffusion noise, but with jumps of finite activity, we show that the CIMBI process hits the boundary $\partial\mathbb{R}_{+}^{d}$ with positive probability.