Singular limit and convergence rate via projection method in a model for plant-growth dynamics with autotoxicity

We investigate a fast-reaction-diffusion system modeling the autotoxicity effect on plant-growth dynamics, in which the fast-reaction terms are based on the dichotomy between healthy and exposed roots depending on the toxicity. The model was proposed in [Giannino-Iuorio-Soresina, 2025] to account for stable stationary spatial patterns considering only biomass and toxicity, and its fast-reaction (cross-diffusion) limit was formally derived and numerically investigated. In this paper, the cross-diffusion limiting system is rigorously obtained as the fast-reaction limit of the reaction–diffusion system with fast-reaction terms by performing a bootstrap argument involving energies. Then, a thorough well-posedness analysis of the cross-diffusion system is presented, including an essential bound, uniqueness, stability, and regularity of weak solutions. This analysis, in turn, becomes crucial to establish the convergence rate for the fast-reaction limit, thanks to the key idea of using an inverse Neumann Laplacian operator. Finally, a numerical experiment illustrates the analytical findings on the convergence rate.