Numerical Methods for a Class of Reaction-Diffusion Equations With Free Boundaries

The spreading behavior of new or invasive species is a central topic in ecology. The modelings of free boundary problems are widely studied to better understand the nature of spreading behavior of new species. From mathematical modeling point of view, it is a challenge to perform numerical simulations of free boundary problems, due to the moving boundary, the stiffness of the system and topological changes. In this work, we design numerical methods to investigate the spreading behavior of new species for a diffusive logistic model with a free boundary and a diffusive competition system with free boundaries. We develop a front-tracking method, which explicitly tracks the location of the moving boundary, in one dimension and higher dimensions with spherical symmetry. In higher dimensional cases, we introduce level set method to handle topological bifurcations. Various numerical simulations in one and two dimensional spaces are presented to validate the accuracy, and stability of the proposed numerical methods. To efficiently solve stiff reaction-diffusion equations, we also develop implicit integration factor (IIF) method combining Krylov subspace to solve the diffusive logistic model with a free boundary in one dimension. Compared with different numerical schemes, it can be observed that Krylov IIF is advantageous to other approaches in terms of stability and efficiency by direct comparison through numerical examples

TILLAGE AND FERTILIZATION INFLUENCES ON AUTOTROPHIC NITRIFIERS IN AGRICULTURAL SOIL

Nitrification is a biological oxidation of NH3 to NO2- and then to NO3-. Understanding how the nitrifier community responds to agricultural management is essential because the community composition is complex and functional distinction of subgroups occurs. Better managing nitrifiers could benefit the environment by increasing nitrogen (N) fertilizer use efficiency, decreasing NO3- leaching, and reducing NO and N2O emissions. This study examined how long-term N fertilization and tillage influenced nitrifier density, ratios, nitrification rates, and the community structure of ammonia-oxidizing bacteria (AOB), ammonia-oxidizing archaea (AOA), and nitrite-oxidizing bacteria (NOB). The study site was a long-term (\u3e40 years) continuous maize (Zea mays L.) experiment with three N fertilization rates (0, 168, and 336 kg ha-1) and either no-tillage (NT) or plow tillage (PT). Most Probable Number method was used to estimate the density of AOB and NOB; the shaken slurry method was used to measure potential nitrification rates; PCR-denaturing gradient gel electrophoresis (DGGE) was used to analyze nitrifier communities. Tillage, fertilization, and their interaction all significantly influenced the AOB and NOB densities, the ratio of AOB to NOB, and potential nitrification rate. Nitrifier densities and potential nitrification rates increased with increased N fertilization; NOB density increased faster than AOB density with fertilization. The influence of tillage on nitrification was different for different fertilization rates. The trends for nitrifier density and potential nitrification rate were not consistent. Nitrifier community structure was influenced by sample season, N fertilization rates, tillage, and their interaction. Different nitrifier groups had different responses to the treatments. The AOB became more diverse with increasing N input; tillage rather than N fertilizer played a dominant role affecting the AOA community; two NOB genera had different responses to N fertilization rates: Nitrobacter diversity increased with more N applied; Nitrospira was the opposite. Unique bands/members were discovered in different treatments, manifesting environmental selection. Long-term field trials were useful in better understanding how soil management influenced the relationship between nitrifier densities, nitrification rates, and community structure, which may facilitate new approaches to optimize nitrification and provide new clues to discover which environmental factors most influence the nitrifier community in agroecosystems