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Long-time behaviour of a nonlocal stochastic fractional reaction--diffusion equation arising in tumour dynamics

Nikos I. Kavallaris, Subramani Sankar, Manil T. Mohan, Christos V. Nikolopoulos, Shanmugasundaram Karthikeyan

TL;DR

A stochastic nonlocal reaction--diffusion model arising in tumour dynamics is introduced, clarifying how noise intensity can accelerate progression or, on favourable paths, enhance suppression consistent with extinction (loss of viability).

Abstract

We introduce a stochastic nonlocal reaction--diffusion model arising in tumour dynamics. Spatial dispersal is described by the fractional Laplacian, accounting for anomalous diffusion and long--range relocation events. The system is perturbed by multiplicative fractional Brownian motion (fBm) with Hurst parameter $H>1/2$, which we interpret as temporally correlated fluctuations in the tumour microenvironment and host response. We first establish well--posedness and identify parameter regimes leading to global--in--time solutions or finite--time blow--up under general multiplicative fractional noise. We then focus on linear multiplicative noise and, via a Doss--Sussmann transformation, derive sharper results: explicit lower and upper bounds for the blow--up time together with quantitative estimates of the blow--up probability, clarifying how noise intensity can accelerate progression or, on favourable paths, enhance suppression consistent with extinction (loss of viability). Finally, one--dimensional simulations illustrate the interplay between anomalous diffusion, fractional noise, and the nonlocal reaction mechanism in shaping the long--time dynamics.

Long-time behaviour of a nonlocal stochastic fractional reaction--diffusion equation arising in tumour dynamics

TL;DR

A stochastic nonlocal reaction--diffusion model arising in tumour dynamics is introduced, clarifying how noise intensity can accelerate progression or, on favourable paths, enhance suppression consistent with extinction (loss of viability).

Abstract

We introduce a stochastic nonlocal reaction--diffusion model arising in tumour dynamics. Spatial dispersal is described by the fractional Laplacian, accounting for anomalous diffusion and long--range relocation events. The system is perturbed by multiplicative fractional Brownian motion (fBm) with Hurst parameter , which we interpret as temporally correlated fluctuations in the tumour microenvironment and host response. We first establish well--posedness and identify parameter regimes leading to global--in--time solutions or finite--time blow--up under general multiplicative fractional noise. We then focus on linear multiplicative noise and, via a Doss--Sussmann transformation, derive sharper results: explicit lower and upper bounds for the blow--up time together with quantitative estimates of the blow--up probability, clarifying how noise intensity can accelerate progression or, on favourable paths, enhance suppression consistent with extinction (loss of viability). Finally, one--dimensional simulations illustrate the interplay between anomalous diffusion, fractional noise, and the nonlocal reaction mechanism in shaping the long--time dynamics.
Paper Structure (5 sections, 7 theorems, 111 equations, 1 figure)

This paper contains 5 sections, 7 theorems, 111 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Motivation
  3. Mathematical Preliminaries
  4. General Noise: global existence vs finite--time blow--up
  5. Linear Fractional Multiplicative Noise

Key Result

Theorem 3.1

Let $(B^{H}(t))_{t \geq 0}$ be an fBm with $H \in (1/2, 1),\ F \in C^{2}(\mathbb{R}).$ Then for any $t>0,$

Figures (1)

  • Figure 1: $\space(a)$ Metastatic cell movements: short bursts interspersed with long unidirectional journeys (local and nonlocal diffusion). $(b)$ Non-metastatic cell movements: more random or "jiggly" paths. Image adapted from Huda et al. H18.

Theorems & Definitions (19)

  • Theorem 3.1: Mishura2008
  • Lemma 3.2
  • Definition 3.3: Weak solution
  • Definition 3.4: Mild solution
  • Lemma 3.5
  • Theorem 4.1
  • proof
  • Remark 4.2
  • Remark 4.3: Biological interpretation: explosive tumour progression
  • Remark 4.4: Noise and (non-)eradication of blow--up: deterministic vs. stochastic dynamics
  • ...and 9 more