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Branching random walks with ageing

Daniela Bertacchi, Elena Montanaro, Fabio Zucca

Abstract

Branching processes are models used to describe populations that reproduce and die over time. In the classical setting, an individual's reproductive capacity remains constant throughout its lifetime. However, in real-world situations, reproductive capacity typically undergoes ageing - that is, after reaching a peak, it decreases over time. In this work, we study the influence of ageing on the behaviour of the process and how modifying its parameters, along with reproduction rates, affects the destiny of the process.

Branching random walks with ageing

Abstract

Branching processes are models used to describe populations that reproduce and die over time. In the classical setting, an individual's reproductive capacity remains constant throughout its lifetime. However, in real-world situations, reproductive capacity typically undergoes ageing - that is, after reaching a peak, it decreases over time. In this work, we study the influence of ageing on the behaviour of the process and how modifying its parameters, along with reproduction rates, affects the destiny of the process.
Paper Structure (10 sections, 8 theorems, 38 equations, 1 figure, 1 table)

This paper contains 10 sections, 8 theorems, 38 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Branching random walks: with or without ageing
  3. Continuous-time Branching Random Walks
  4. Discrete-time Branching Random Walks
  5. The discrete-time counterpart of a continuous-time BRW
  6. BRW with ageing
  7. Survival and extinction of a BRW with ageing
  8. Critical parameters under local modifications
  9. The expected number of individuals in an ageing BRW
  10. Conclusions

Key Result

Theorem 3.1

Let $(X,\mathcal{R})$ be an ageing BRW. Let $K= (k_{xy})_{x,y \in X}$ be the matrix defined by Denote by $k^{(n)}_{xy}$ the entries of its $n$-th power. Fix $\lambda>0$ and $x\in X$. The process survives locally at $x$ if and only if Equivalently, $\lambda_s(x)=1/ \limsup_{n \to +\infty} \sqrt[n]{k^{(n)}_{xx}}$ and

Figures (1)

  • Figure 1: The expected value of individuals of the ageing branching process with breeding intensity $\lambda e^{-\alpha t}$ (blue, solid line) and of the continuous-time branching process with intensity $\lambda/(1+\alpha)$ (red, dashed line), as functions of the time variable $t$.

Theorems & Definitions (29)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • proof
  • Theorem 3.3
  • ...and 19 more