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

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.

💡 Research Summary

In this paper the authors develop and analyze a spatially heterogeneous reaction‑diffusion model for tumor growth that incorporates both a nonlocal dispersal term and a maturation delay, together with a spatially varying therapy term. The governing equation reads

∂ₜu = d Δu + F(u,∫ΩS(x,y)u(y,t‑τ)dy) u – (β q(u)+r(x)) u,

where d>0 is the diffusion coefficient, S(x,y)≥0 is a normalized kernel describing the probability that a newly born cell at location y survives the delay τ and relocates to x, τ≥0 is the maturation delay, β>0 measures the overall intensity of therapy, q(u) is a decreasing function of the local tumor density, and r(x) captures spatial heterogeneity in therapeutic efficacy. The nonlinear function F(·,·) models proliferation and may include logistic‑type saturation and competition with the delayed nonlocal term.

The authors first study the existence of nontrivial steady states. By treating β as a bifurcation parameter they identify a principal eigenvalue β* of the linearized operator Lβ = d Δ + F(0,0) – r(x) – β acting on H²∩H₀¹(Ω). The corresponding positive eigenfunction φ* is used to decompose the solution space via Lyapunov‑Schmidt reduction: X = Ker(Lβ*) ⊕ X₁, Y = Ker(Lβ*) ⊕ Y₁. Projecting the steady‑state equation onto the kernel and its complement yields a scalar bifurcation equation

f(ν,β) = ν