Critical Markov Branching-Immigration Systems: Complete Classification of the State Space and Exact Rates of Convergence

We study a critical homogeneous continuous-time Markov branching-immigration system whose offspring and immigration generating functions regularly vary at point 1 of orders 1+ν and δ with ν,δ∈(0,1]. The slowly varying factors are controlled through their index functions, which causes regular variation of the second order, both factors tending to finite positive limits. Two numbers govern the classification: the difference γ=δ−ν of the two orders and the limit Λ∞ of the ratio of the immigration factor to the offspring one. We classify the state space completely: it is positive recurrent for γ>0, transient for γ<0, and on the boundary γ=0, null recurrent or transient accordingly as Λ∞≤ν or Λ∞>ν; at ν=δ=1, this is the classical threshold, read under the present second-order hypotheses. In each regime, we identify the limiting generating function of the normalised transition functions, together with the leading term of the deviation from it, uniformly in the argument on compact subsets of [0,1), the transient regime under two further inequalities on the parameters. Every rate is carried by a complex-analytic argument and with no monotonicity hypothesis to the individual transition probabilities as an upper bound, which for each fixed state is the exact order when γ≠0. An exact integral identity for the survival probability of the system without immigration makes the constants explicit and yields a three-term expansion of that probability. A numerical section on explicit models confirms the constants and the rates.