On the mathematical modelling and control of leaf movements

Abstract

To contribute to the fight against forest fires, which cause enormous damage every year, we propose in this paper a model of the movements of dry grass in a forest, since the occurrence and magnitude of the fire in an area are proportional to the amount of grass in it. Inspired by the works of Rachik et al. [(Soukaina et al. in Math Biol Neurosci 2020, 2020), (Zakary et al. in Int J Dyn Control 1–14, 2016a, Adv Differ Equ 2016:169, 2016b)]. We subdividing the total area under consideration into an appropriate number of hexagonal patches, we propose a discrete spatiotemporal mathematical model, taking into account the intensity and direction of the wind vector, to describe the transfer of dry grasses from one patch to neighbouring patches. After stating that a patch is at high risk if the amount of grass found exceeds a predefined tolerance threshold, we adopt the control variable “gathering of grass” to minimize the risk of a possible fire. To illustrate the theoretical results obtained, we propose numerical simulations for several scenarios applying the forward-backwards sweep method (FBSM) to solve our optimality system in an iterative process.