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Generalization of the main convergence theorem to broader parameter regimes (including λ0 ≤ 0)

Establish whether the exponential convergence of the expectation semigroup of the telomere-driven branching process to a limiting eigenfunction–measure pair, as stated in Theorem \ref{thm:mainresult}, extends to the full set of parameter regimes used in the numerical simulations, including subcritical and critical cases where the Malthusian parameter λ0 ≤ 0. Specifically, determine if the convergence e^{-λ0 t} ψ_t[g](c,x) → η(c,x) ν[g] with an exponential rate continues to hold for those parameter choices beyond the restrictive assumptions under which the theorem was proved.

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Background

Theorem \ref{thm:mainresult} proves an exponential convergence of the mean semigroup of the branching model to a limiting quasi-stationary distribution under a set of restrictive assumptions, notably requiring a supercritical regime with λ0 > 0, as well as moment and decay conditions on the telomerase action distribution and probability. This yields both a growth rate characterization and stabilization of the average cell’s telomere-length distribution.

In the simulations section, the authors explore a wider range of parameters, including regimes where λ0 ≤ 0. Numerical evidence suggests that the same convergence behavior may persist beyond the theorem’s stated assumptions. The authors explicitly note that formalizing this extension remains unresolved and leave the generalization question open.

References

Despite the fact that Theorem \ref{thm:mainresult} was proved under restrictive assumptions (including $\lambda_0 > 0$), numerical simulations suggest that the convergence also holds true for all the parameters choices of Section~\ref{sec:simulations}. In particular, we consider certain parameter regimes where $\lambda_0 \le 0$, in order to illustrate that our result should hold under less restrictive assumptions, and to provide the reader with a more complete picture. We leave the question of the possible generalization of Theorem~\ref{thm:mainresult} to these parameters open.

— Stochastic branching models for the telomeres dynamics in a model including telomerase activity (2407.11453 - Benetos et al., 16 Jul 2024) in Remark rem:generalization, Appendix A (Section Algorithm)