From classical phenological models to reaction–diffusion equations: A spatial bioclimatic framework for cherry phenology

Abstract

Thermal-time models are widely used to predict fruit-tree phenology, but most operational formulations describe only temporal development and assume that an orchard is spatially homogeneous. This assumption can conceal within-orchard asynchrony caused by microclimate, topography, soil properties, canopy structure, and management. We propose a spatial extension of the classical monomolecular phenological model for sweet cherry (Prunus avium L.) using a reaction–diffusion equation. The reaction term represents local physiological development as a function of accumulated growing degree days (GDD), whereas the diffusion term provides an effective spatial coupling between neighbouring orchard sectors. One- and two-dimensional formulations are considered, with a bounded space–time development rate that accommodates persistent gradients and management zones. The equations are solved by an implicit-explicit finite-difference scheme in which reaction is treated explicitly and diffusion implicitly. Simulations based on thermal conditions from the Maule Region of Chile reproduce smooth but persistent intrafield phenological gradients. Warming scenarios advance phenological development at the same calendar date and may amplify differences among orchard sectors, particularly at later stages. The framework connects established thermal-time models with spatial process modelling and provides a basis for zone-specific irrigation, crop protection, and harvest planning.