New models and numerical methods for partial differential equations applied to spatial stoichiometric aquatic ecosystems



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Ecological stoichiometry is a framework to study population dynamics that incorporates energy flow in trophic levels as well as chemical imbalances. Spatial variation in an ecological interaction also has effects on the population dynamics. We develop and analyze numerically two stoichiometric producer-grazer models in a spatially heterogeneous environment: a model with two grazing genotypes to understand selection processes and a mechanistic model that tracks explicitly the nutrient contents. Both of our model equations are non-linear reaction-diffusion partial differential equations (PDEs). Simulations of these models produce large data sets that are difficult to analyze and interpret. We used a reduced-order modeling technique to interpret the simulation data in terms of an underlying low-dimensional dynamical system. The reaction-diffusion equations of our model can be viewed as a stiff system of equations and require an A-stable time-stepping method. We implement high-order Implicit Runge--Kutta (IRK) methods for our model PDEs. Although IRK methods offer A-stability and high-order accuracy, these methods are not widely used in PDE discretization due to the resulting complicated linear system. To solve these systems, we have developed a new preconditioner based on a block LDU factorization with algebraic multigrid subsolves for scalability. We demonstrate the effectiveness of our preconditioner on two test problems: a heat equation and a double-glazing advection-diffusion problem, and find that the new preconditioner outperforms the others currently in the literature.



Ecological Stoichiometry, Diffusion, Population Dynamics, Genotypic Selection, Reduced-Order Modeling, Preconditioning, Implicit Runge-Kutta, Parabolic PDEs