Vegetation Pattern Formation and Transition Caused by Cross-Diffusion in a Modified Vegetation-Sand Model

Vegetation pattern generated by aeolian sand movements is a typical type of vegetation pattern in arid and semi-arid areas. Considering the effects of sand movements on vegetation diffusion as a cross-diffusion term, an improved vegetation-sand is studied to explore the effects of cross-diffusion on vegetation pattern formation and transition. Turing bifurcation analysis and amplitude analysis are performed. Theoretical analysis shows that three types of vegetation patterns can be formed: stripes, hexagons and mixture of stripes and hexagons. Numerical simulations on the vegetation pattern formation are consistent with theoretical results, which also show hysteresis in pattern transition process. Moreover, when striped vegetation is formed, with the increase of cross-diffusion coefficient, vegetation stripes break into spots. And when spotted vegetation is formed, with the decrease of cross-diffusion coefficient, spots persist but the statistics of vegetation pattern vary.