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Normal Forms and Pattern Selection for Reaction-Diffusion Equations with Memory and Nonlocal Averaging

  • Hao Shen [1] ; Hao Wang [1] ; Hao Wang [2]
    1. [1] University of Alberta

      University of Alberta

      Canadá

    2. [2] East China Normal University

      East China Normal University

      China

  • Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 25, Nº 4, 2026
  • Idioma: inglés
  • Enlaces
  • Resumen

    • Spatially extended biological systems often exhibit history dependence and finiterange interactions, yet rigorous tools for how distributed memory and spatial nonlocality shape pattern formation remain limited. We study a class of reactiondiffusion systems in which local kinetics are coupled to (i) spatiotemporally distributed memory implemented by temporal convolution and (ii) spatially nonlocal interactions given by averaging kernels. For this class, we develop a normal-form reduction at critical spectra corresponding to Turing and Hopf bifurcations and investigate codimension-two Turing-Hopf instabilities. The reduction is obtained via a centermanifold construction adapted to memory terms and nonlocal operators, yielding explicit leading-order amplitude equations with coefficients expressed in terms of the linearized resolvent and the memory/nonlocal kernels. These formulas determine the direction and stability of primary bifurcations and the cross-coupling structure at Turing-Hopf points, enabling rigorous pattern-selection predictions. As an application, we analyze a Holling-Tanner consumer-resource model with both distributed memory and spatial averaging.We show that the nonlocal interaction, when combined with memory, generates Turing-Hopf bifurcation in parameter regimes where memory alone does not, and we characterize the parameter geometry separating steady Turing patterns, temporal oscillations, and mixed spatiotemporal states. Numerical simulations of the full system corroborate the normal-form predictions (existence and stability of patterned equilibria and oscillatory patterns) and illustrate the role of kernel shape in shifting instability thresholds. The results provide a general framework for bifurcation analysis in reaction-diffusion equations with memory and nonlocality and clarify how these effects interact to produce spatiotemporal complexity.

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