Pattern formation of the Holling–Tanner model with top-hat kernel functions on square domains

We investigate the effects of periodic boundary conditions on a nonlocal Holling–Tanner model defined on a square domain. A top-hat kernel function with finite support is employed to characterize nonlocal interactions, which we further adapt to accommodate periodic boundary conditions. Subsequently, we derive the Turing and spatiotemporal Hopf bifurcation curves and conduct numerical simulations across the parameter ranges delineated by these curves. Our findings demonstrate substantial differences in spatiotemporal patterns between the square domain and the one-dimensional case, including squares, stripes, mixed states, irregular spot-like hexagonal patterns, and coexistence states of three-stripes and spots. These observed patterns exhibit remarkable consistency with both chemical experimental results and the skin pigmentation patterns of fish, thereby offering valuable theoretical insights and predictive frameworks for understanding spatiotemporal patterns in chemical and biological systems.