A Saturated Autocatalytic Model with Phenomenological Thermal Activation: Continuous Dynamics, Diffusion-Driven Patterns, and Discretization Effects

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We formulate a two-intermediate model for precursor-fed autocatalysis with a bounded phenomenological thermokinetic factor. Positivity, dissipativity, existence of a positive equilibrium, an a priori equilibrium bound, and a conditional uniqueness criterion are established analytically. Generic saddle-node and Hopf conditions are stated with their nondegeneracy hypotheses. Numerical continuation identifies two transversal Hopf points with negative first Lyapunov coefficients under a specified eigenvector normalization. For the reaction–diffusion extension, global positivity and boundedness are established, and an exact finite-domain instability test is obtained by coupling the kinetic Jacobian to the Neumann spectrum. Fully specified grayscale simulations produce noise-selected spots and mode-selected stripe and square-symmetric structures. A positive semi-implicit discretization is then examined as a numerical dynamical system. The scheme preserves non-negative concentrations and equilibria, but the reference fixed point loses Schur stability at hc ≈ 0.617496, whereas the exact sampled flow remains stable. Step refinement further shows that h < hc is not an accuracy condition. Coarse-step periodic and positive-exponent responses are therefore attributed to the numerical map rather than to the planar chemical flow. The analysis separates chemical admissibility, homogeneous stability, spatial-mode stability, numerical fixed-point stability, and trajectory accuracy.

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