Crowding-modified Schnakenberg reaction–diffusion dynamics: exact equilibrium feasibility, Hopf/Turing bifurcations, Turing–Hopf interaction, and spatio-temporal complexity

Abstract

We study a crowding-modified Schnakenberg activator–inhibitor model in which the autocatalytic flux is saturated by a free-volume (volume-exclusion) factor, preserving the classical Schnakenberg mass-balance identity while preventing unrealistically large reaction rates at high concentrations. For the well-mixed kinetics we prove forward invariance of the nonnegative quadrant and global existence, derive the unique equilibrium in closed form together with a sharp feasibility condition, and give a complete linear stability classification. Treating the crowding parameter as a bifurcation parameter, we obtain an explicit Hopf threshold and verify transversality in closed form; moreover, using normal-form multilinear forms we provide an explicit evaluation-ready formula for the first Lyapunov coefficient and show that the Hopf bifurcation (when it occurs) is supercritical. Embedding the kinetics into a two-species reaction–diffusion system with Neumann boundary conditions, we derive the dispersion relation and sharp necessary and sufficient conditions for diffusion-driven (Turing) instability, including the unstable waveband and critical wavenumber. We further develop a weakly nonlinear steady Turing reduction in a tuned threshold setting, present modal Hopf thresholds for the PDE, and characterize codimension-two Turing–Hopf interaction points. Finally, numerical experiments illustrate the theoretical predictions and reveal parameter regimes with mixed-mode dynamics and spatio-temporal irregularity, quantified via spectral and Lyapunov-type diagnostics.