Ascent to the non-zero equilibrium in the general multitype case

Establish that the density-dependent multitype branching process with a type-1 soft carrying capacity reaches a neighborhood of its unique non-zero equilibrium $K\mathbf{Y}^*$ within order $\mathcal{O}(\log K)$ time with non-negligible probability under general multitype conditions, beyond the currently treated two-type asymptotically stable case.

Background

Theorem 5.1 establishes logarithmic-time ascent only to a type-1 population threshold aKaK, with $0

The paper explains that general multitype dynamics may include unstable equilibria or periodic orbits, so ascent to the equilibrium cannot be inferred without additional stability assumptions. A corresponding result is obtained only in the special case d=2d=2 when the non-zero equilibrium is asymptotically stable and the relevant monotonicity condition holds.

References

We cannot establish ascent of the process $\mathbf{x}K$ to the vicinity of the unique non-zero fixed point $K \mathbf{Y}*$ (with $Y_1=1$) in generality in the multitype case (see the discussion of periodic orbits and asymptotic stability in Section \ref{sec::FLLN_limiting_behaviour}).

— A multitype Markovian branching process with one-type population size dependence  (2609.12567 - Mehra et al., 11 Sep 2026) in Section 5, immediately before Theorem 5.1 (Theorem labeled logarithmic_ascent)