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

TL;DR

This paper rigorously analyzes the limit behavior and convergence rate of a plant-growth model with autotoxicity, using projection methods and energy estimates, supported by numerical experiments.

Contribution

It provides a rigorous derivation of the cross-diffusion limit and establishes convergence rates for a reaction-diffusion system modeling plant autotoxicity.

Findings

Rigorous derivation of the cross-diffusion limit as the fast-reaction limit.

Establishment of well-posedness, stability, and regularity of the limiting system.

Numerical validation of the convergence rate.

Abstract

We investigate a fast-reaction--diffusion system modelling the effect of autotoxicity 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, forthcoming] to account for stable stationary spacial 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 a $L^\infty_{x,t}$-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, numerical experiments illustrate the analytical findings on the convergence rate.