Pattern formation in a two-dimensional nonlocal Klausmeier model for vegetation–water dynamics

Abstract

Vegetation patterns in semi-arid regions provide information about ecosystem state and have been investigated as potential early-warning signals of desertification risk. To account for the spatially nonlocal characteristics of long-range seed dispersal, we construct a two-dimensional nonlocal vegetation–water model on flat terrain, in which the local diffusion term describing plant dispersal is replaced by a nonlocal convolution integral term. First, we prove that the nonlocal convolution operator converges to the local Laplacian diffusion term under a suitable coupling-strength scaling and in the local limit. Then, linear stability analysis gives the critical conditions for Turing instability, and weakly nonlinear analysis yields the amplitude equations near the Turing bifurcation point, from which the existence and stability regions of stationary patterns are obtained. Matched local–nonlocal analysis and numerical simulations show that finite-width nonlocal seed redistribution shifts the Turing threshold and characteristic wavelength relative to its leading-order-matched local-diffusion counterpart. Across separate fixed-rainfall simulations, the model recovers the established overall ordering from gaps through stripe-dominated states to spots, which serves as a consistency check rather than a new pattern-transition result. At fixed nonlocal coupling strength, shortening the mean seed-dispersal distance enlarges the pattern-forming region, whereas at fixed mean seed-dispersal distance, increasing the nonlocal coupling strength narrows it. Under a matched spatial second moment, the power-law kernel considered here is associated with a higher Turing-onset rainfall threshold and a lower-connectivity spot morphology over the explored parameter range. These results highlight the crucial role of long-range seed dispersal and kernel shape beyond the matched second moment in Turing instability and pattern selection.