Dynamical analysis in a nonlocal delayed reaction-diffusion tumor model with therapy

TL;DR

This paper analyzes a complex tumor-therapy reaction-diffusion model with nonlocal delays, establishing steady-state solutions, stability conditions, and bifurcation regimes through theoretical and numerical methods.

Contribution

It introduces a novel analysis of a nonlocal delayed reaction-diffusion tumor model, including bifurcation and stability analysis with explicit criteria.

Findings

Existence of nontrivial steady states bifurcating from trivial solutions

Explicit stability criteria depending on delay and parameters

Numerical demonstration of bifurcation behavior and treatment effects

Abstract

In this work, we investigate the dynamical properties of a reaction-diffusion system arising from tumor-therapy modelling that features both nonlinear interactions and nonlocal delay. By applying the Lyapunov-Schmidt reduction, we establish the existence of a nontrivial steady-state solution bifurcating from the trivial solution. In particular, we derive an approximate expression for a spatially nonhomogeneous steady-state solution. Then, we provide a detailed spectral characterization of the linearized operator and explicit stability criteria and identify the delay-dependent Hopf bifurcation regimes. To illustrate the theoretical results, we include a concrete example that verifies the claims in our theorems and numerically demonstrates how changes in treatment parameters alter stability and bifurcation behaviour.