Beyond water limitation in vegetation–autotoxicity patterning: A cross-diffusion model

Many mathematical models describing vegetation patterns are based on biomass–water interactions, due to the impact of this limited resource in arid and semi-arid environments. However, in recent years, a novel biological factor called autotoxicity has proved to play a key role in vegetation spatiotemporal dynamics, particularly by inhibiting biomass growth and increasing its natural mortality rate. In a standard reaction-diffusion framework, biomass-toxicity dynamics alone are unable to support the emergence of stable spatial patterns. In this paper, we derive a cross-diffusion model for biomass and toxicity dynamics as the fast-reaction limit of a three-species system involving dichotomy and different time scales. Within this general framework, in addition to growth-inhibition and extra-mortality already considered in previous studies, the obtained cross-diffusion term accounts for the additional effect of “propagation reduction” (i.e., inhibition) induced by autotoxicity on vegetation spreading. By combining linearised analysis, simulations, and continuation, we investigate the formation of spatial patterns. Thanks to the cross-diffusion term, for the first time, a spatial model based solely on biomass–toxicity feedback without explicit water dynamics supports the formation of stable (Turing) vegetation patterns for a wide range of parameter values.