RESUMEN

The dynamics of hydraulic fracture, described by a system of nonlinear integrodifferential equations, is studied through the development and application of a multiparameter singular perturbation analysis. We present a new single expansion framework which describes the interaction between several physical processes, namely viscosity, toughness, and leak-off. The problem has nonlocal and nonlinear effects which give a complex solution structure involving transitions on small scales near the tip of the fracture. Detailed solutions obtained in the crack tip region vary with the dominant physical processes. The parameters quantifying these processes can be identified from critical scaling relationships, which are then used to construct a smooth solution for the fracture depending on all three processes. Our work focuses on plane strain hydraulic fractures on long time scales, and this methodology shows promise for related models with additional time scales, fluid lag, or different geometries, such as radial (penny-shaped) fractures and the classical Perkins–Kern–Nordgren (PKN) model.

RESUMEN

A model of the chemostat involving two species of microorganisms competing for two perfectly complementary, growth-limiting nutrients is considered. The model incorporates distributed time delay in the form of integral differential equations in order to describe the time involved in converting nutrient to biomass. The delays are included in the nutrient and species concentrations simultaneously. A general class of monotone increasing functions is used to describe nutrient uptake. Sufficient conditions based on biologically meaningful parameters in the model are given that predict competitive exclusion for certain parameter ranges and coexistence for others. We prove that the global asymptotic attractivity of steady states of the model is similar to that of the corresponding model without time delays. However, our results indicate that when the inherent delays are in fact large, ignoring them may result in incorrect predictions.