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A generalised spatial branching process with ancestral branching to model the growth of a filamentous fungus

Lena Kuwata

TL;DR

This work introduces a spatial branching process to model the growth of the mycelial network of a filamentous fungus, and characterises the limiting process as the weak solution to a system of partial differential equations.

Abstract

In this work, we introduce a spatial branching process to model the growth of the mycelial network of a filamentous fungus. In this model, each filament is described by the position of its tip, the trajectory of which is solution to a stochastic differential equation with a drift term which depends on all the other trajectories. Each filament can branch either at its tip or along its length, that is to say at some past position of its tip, at some time- and space-dependent rates. It can stop growing at some rate which also depends on the positions of the other tips. We first construct the measure-valued process corresponding to this dynamics, then we study its large population limit and we characterise the limiting process as the weak solution to a system of partial differential equations.

A generalised spatial branching process with ancestral branching to model the growth of a filamentous fungus

TL;DR

This work introduces a spatial branching process to model the growth of the mycelial network of a filamentous fungus, and characterises the limiting process as the weak solution to a system of partial differential equations.

Abstract

In this work, we introduce a spatial branching process to model the growth of the mycelial network of a filamentous fungus. In this model, each filament is described by the position of its tip, the trajectory of which is solution to a stochastic differential equation with a drift term which depends on all the other trajectories. Each filament can branch either at its tip or along its length, that is to say at some past position of its tip, at some time- and space-dependent rates. It can stop growing at some rate which also depends on the positions of the other tips. We first construct the measure-valued process corresponding to this dynamics, then we study its large population limit and we characterise the limiting process as the weak solution to a system of partial differential equations.
Paper Structure (12 sections, 7 theorems, 171 equations)

This paper contains 12 sections, 7 theorems, 171 equations.

Table of Contents

  1. Introduction
  2. Definition of the spatial birth-death process and statement of the main results
  3. Overview of the model and main results
  4. Construction of the process
  5. Martingale problem
  6. Large population limit
  7. Tightness
  8. Tightness on the compactified space
  9. Uniform control on the spatial extent of the population
  10. Characterisation of the limit
  11. Uniqueness of the limit
  12. Absolute continuity of the limiting measures

Key Result

Proposition 2.3

For every $f \in \mathcal{C}_b^{1,2}([0,T]\times \mathbb{R}^2)$, the process is a martingale, where for $i\in \{1,2\}$, $L_{s-r}^i$ is the $i$-th coordinate of the function $L_{s-r}$.

Theorems & Definitions (17)

  • Remark 2.1
  • Remark 2.2
  • Proposition 2.3
  • Theorem 2.4
  • Remark 2.5
  • Proposition 2.6
  • Proposition 2.7
  • proof : Proof of Proposition \ref{['Prop_eq_Z']}
  • proof : Proof of Proposition \ref{['Prop_Tk']}
  • Lemma 3.1
  • ...and 7 more