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

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 $H>1\mathrm{/}2$, 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.