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Convergence of a Critical Multitype Bellman-Harris Process

E. T. Kolkovska, J. A. López-Mimbela, J. H. Ramírez-González

Abstract

In this work, we study a two-type critical branching particle system in $\mathbb{R}^{N}$, where particles follow symmetric stable motions, with type-dependent lifetimes and offspring distributions. Our main result is the convergence as $t\to\infty$ of the particle system to a non-trivial limiting population, focusing on two cases: (1) all particle lifetimes have finite mean, or (2) one type has a lifetime distribution with a heavy tail, while the others have finite mean. This complements previous results on extinction \cite{Kevei}. Using the Extended Final Value Theorem, we prove the existence of a limiting distribution for the particle system. The non extinction of the limiting population is demonstrated using a technique inspired in \cite{Fino}. These results describe the long-term behavior of the particle system, highlighting the interaction between mobility, longevity, and offspring variability. Additionally, the study of a particle system with a finite number of types would follow analogously with the techniques presented here. Our approach introduces new techniques for the asymptotic study of critical multitype branching processes.

Convergence of a Critical Multitype Bellman-Harris Process

Abstract

In this work, we study a two-type critical branching particle system in , where particles follow symmetric stable motions, with type-dependent lifetimes and offspring distributions. Our main result is the convergence as of the particle system to a non-trivial limiting population, focusing on two cases: (1) all particle lifetimes have finite mean, or (2) one type has a lifetime distribution with a heavy tail, while the others have finite mean. This complements previous results on extinction \cite{Kevei}. Using the Extended Final Value Theorem, we prove the existence of a limiting distribution for the particle system. The non extinction of the limiting population is demonstrated using a technique inspired in \cite{Fino}. These results describe the long-term behavior of the particle system, highlighting the interaction between mobility, longevity, and offspring variability. Additionally, the study of a particle system with a finite number of types would follow analogously with the techniques presented here. Our approach introduces new techniques for the asymptotic study of critical multitype branching processes.

Paper Structure

This paper contains 5 sections, 14 theorems, 150 equations.

Table of Contents

  1. Introduction and previous work
  2. Main Results
  3. Lifetimes distribution wit finite mean and critical branching in the domain of a $(1+\beta_i)$-stable law
  4. Lifetimes with the domain of attraction of a $\gamma-$stable law and tail distribution with polynomial decay $(\eta > 1)$, with critical branching in the domain of a $1+\beta_i$ stable law
  5. Proofs of main results

Key Result

Theorem 2.1

Under Condition Ac, (i) Suppose $q\geq \frac{N}{\alpha_1\wedge\alpha_2}>\frac{1}{\beta_1\wedge\beta_2}$. Then, where $U_{\phi}(x,i):=(V_0(x,i)-\mu_ic_i\int_{0}^{\infty}V_{s}^{1+\beta_i}(x,i)ds)$. Additionally, let $\epsilon > 0$ and define $\phi_{\epsilon} := -\log(1 - \epsilon(1 - e^{-\phi}))$. Then, (ii) Suppose $\frac{N}{\alpha_1\wedge\alpha_2}>q> \frac{1}{\beta_1\wedge\beta_2}$. Then, the li

Theorems & Definitions (19)

  • Theorem 2.1
  • Theorem 2.2
  • Example 2.3
  • Theorem 2.4
  • Remark 2.5
  • Remark 2.6
  • Remark 2.7
  • Lemma 3.1
  • Definition 3.2
  • Theorem 3.3
  • ...and 9 more