This paper proposes a sixth-order compact finite difference framework to numerically solve nonlinear reaction-diffusion equations, with a particular focus on the FitzHugh-Nagumo (FHN) model. First, for the second-order spatial derivatives in the FHN equation, a five-point sixth-order compact difference scheme is used for internal points, and a asymmetric six-point compact difference scheme is used for boundary points to achieve spatial discretization, thereby transforming the problem into an ordinary differential equation; then, this is and then combined with the semi-implicit Crank-Nicholson method for the time discretization to obtain a numerical solution scheme for the FHN equation. We establish the stability and convergence of the method and validate it through numerical experiments. The feasibility and accuracy of the method were verified by conducting an error analysis on the numerical results and comparing them with other algorithms. It is proven that this method is an effective tool to solve the numerical solutions of nonlinear reaction-diffusion equations.

Tree growth is the consequence of developmental interactions between above- and below-ground compartments. However, a comprehensive view of the genetic architecture of growth as a cohesive whole is poorly understood. We propose a systems biology approach for mapping growth trajectories in genome-wide association studies viewing growth as a complex (phenotypic) system in which above- and below-ground components (or traits) interact with each other to mediate systems behavior. We further assume that trait–trait interactions are controlled by a genetic system composed of many different interactive genes and integrate the Lotka-Volterra predator–prey model to dissect phenotypic and genetic systems into pleiotropic and epistatic interaction components by which the detailed genetic mechanism of above- and below-ground co-growth can be charted. We apply the approach to analyze linkage mapping data of Populus euphratica, which is the only tree species that can grow in the desert, and characterize several loci that govern how above- and below-ground growth is cooperated or competed over development. We reconstruct multilayer and multiplex genetic interactome networks for the developmental trajectories of each trait and their developmental covariation. Many significant loci and epistatic effects detected can be annotated to candidate genes for growth and developmental processes. The results from our model may potentially be useful for marker-assisted selection and genetic editing in applied tree breeding programs. The model provides a general tool to characterize a complete picture of pleiotropic and epistatic genetic architecture in growth traits in forest trees and any other organisms.