Mathematics

Previous research on simulated ecological networks has often focused on issues such as the distribution of the number of links between species, without generally categorizing the types of inter-species relationships that develop, unless those relationships are of some predesigned form (e.g., food webs). In this work we use a model system to examine general, dynamically-evolved ecological networks that are subject to occasional invasion by novel species. Keeping track of the specific types and numbers of interactions between species over time leads to insight on how these affect network stability, fragility and evolution. We discuss the role that assembly rules play on the evolutionary trajectories of randomly initialized communities. We also investigate the occurrence of certain types of interactions (e.g., cyclic) and quantify their destabilizing effect on the network. In particular, extinctions and secondary extinctions ("avalanches") are related to these issues.

Species are faced with a multitude of risk factors for extinction including small carrying capacity, large environmental variation, habitat loss, climate change, and invasive species (Drake 2006; Kuussaari et al. 2009). These factors naturally occur at certain baseline levels, but modern humans have contributed to an astonishing increase in the degree to which they exacerbate risks of biodiversity loss.

In this paper we describe and test a sub-model that integrates the cycling of carbon (C), nitrogen (N) and phosphorus (P) in the Soil Water Assessment Tool (SWAT) watershed model. The core of the sub-model is a multi-layer, one-pool soil organic carbon (SC) algorithm, in which the decomposition rate of SC and input rate to SC (through decomposition and humification of residues) depend on the current size of SC. The organic N and P fluxes are coupled to that of C and depend on the available mineral N and P, and the C:N and N:P ratios of the decomposing pools. Tillage explicitly affects the soil organic matter turnover rate through tool-specific coefficients. Unlike most models, the turnover of soil organic matter does not follow first order kinetics. Each soil layer has a specific maximum capacity to accumulate C or C saturation (Sx) that depends on texture and controls the turnover rate. It is shown in an analytical solution that Sx is a parameter with major influence in the model C dynamics. Testing with a 65-yr data set from the dryland wheat growing region in Oregon shows that the model adequately simulates the SC dynamics in the topsoil (top 0.3 m) for three different treatments. Three key model parameters, the optimal decomposition and humification rates and a factor controlling the effect of soil moisture and temperature on the decomposition rate, showed low uncertainty as determined by generalized likelihood uncertainty estimation. Nonetheless, the parameter set that provided accurate simulations in the topsoil tended to overestimate SC in the subsoil, suggesting that a mechanism that expresses at depth might not be represented in the current sub-model structure. The explicit integration of C, N, and P fluxes allows for a more cohesive simulation of nutrient cycling in the SWAT model. The sub-model has to be tested in forestland and rangeland in addition to agricultural land, and in diverse soils with extreme properties such high or low pH, an organic horizon, or volcanic soils.► A model coupling the cycling of carbon (SC), nitrogen and phosphorus in the Soil Water Assessment Tool (SWAT) watershed model is presented. ► The SC cycling kinetic is controlled by the ratio of the current SC to that of a preset C saturation level (Sx), which is a function of soil texture. ► An analytical solution shows that Sx is a critical parameter of the model. ► Three key parameters controlling the SC turnover and stabilization rate showed low uncertainty as determined by generalized likelihood uncertainty estimation.

A mathematical model of a valve-regulated lead-acid battery under discharge is presented as simplified from a standard electrodynamics model. This nonlinear reaction-diffusion model of a battery cell is solved using an operator splitting method to quickly and accurately simulate sulfuric acid concentration. This splitting method incorporates one-sided approximation schemes to preserve continuity over material interfaces encompassing discontinuous parameters. Numerical results are compared with measured data by calculating battery voltage from modeled acid concentration as derived from the Nernst equation.

The Belousov-Zhabotinskii reaction has been modelled by Field and Noyes [5] as a pair of nonlinear parabolic equations. Previous studies of these, both theoretical and numerical, have assumed wave solutions travelling with constant velocity leading to a simplification of the mathematical model in the form of a system of ordinary differential equations. In the present study a finite element Galerkin method is used directly on the original parabolic system for a range of parameter values.

In multigrid methods, it is preferred to employ smoothing techniques which are convergent. In practice, the standard Jacobi and Gauss-Seidel methods are the choices for “smoothers” However, it is known that if the spectral radius condition is violated, then convergence is not guaranteed for these methods. In this paper we develop a simple two-step Jacobi-type method which has better convergence properties and which can be employed as a convergent “smoother” wherever the standard iterative methods fail. We provide the convergence proofs and demonstrate the applicability of the method on a variety of problems.

The sterile insect release method (SIRM) for pest control is modeled by a reaction-diffusion equation. Conditions on the strip size of the favorable environment and the initial population to avoid an outbreak state are obtained by using a combination of analytical and numerical techniques.