Catherine A. Roberts

Modeling complex human–environment interactions: the Grand Canyon river trip simulator

Abstract Understanding the impacts of human recreation on natural resources is of critical importance in constructing effective management strategies. The Grand Canyon River Trip Simulator is a computer program that models complex, dynamic human–environment interactions in the river corridor of the Grand Canyon National Park. The system consists of a database and simulator engine. The database contains 487 trip diaries that report all stops for activities and camping along the 447 km Colorado River corridor within the purview of the National Park Service. The computer simulation employs statistics and artificial intelligence in creating an individual-based modeling system. This simulation system successfully models the recreational rafting behavior and captures the decision making of rafting parties as they responsively seek to optimize their experience. The model allows the Park managers to assess the likely impact of various alternative management scenarios for rafting trips on the Colorado River. The Grand Canyon River Trip Simulator advances our abilities to model complex systems in the context of human–environment interactions. It may serve as a suitable template for modeling a suite of other complex adaptive systems including ecosystems.

The one-dimensional heat equation with a nonlocal initial condition☆

Abstract A boundary value problem for the one-dimensional heat equation is considered under the constraint of a nonlocal initial condition. The problem is investigated by conversion to a Fredholm integral equation. The solution of the integral equation is derived as an eigenfunction expansion that is valid for situations in which either uniqueness or nonuniqueness applies. Circumstances under which a resonance effect occurs are discussed and illustrated with an example.

Analysis of explosion for nonlinear Volterra equations

This survey paper presents some analytical tools for determining the existence of a blow-up solution for a wide class of nonlinear Volterra integral equations. The related phenomenon of quenching has also been investigated using similar techniques. In both cases, lower and upper bound estimates for the critical time, as well as asymptotic solution behavior in the key limit are established. This paper surveys recent results in the analysis of such problems and proposes problems for future study.

Thermal Blow-up in a Subdiffusive Medium

The problem of thermal blow-up in a subdiffusive medium is examined within the framework of a fractional heat equation with a nonlinear source term. This model establishes that a thermal blow-up always occurs when a finite strip of subdiffusive material is exposed to the effects of a localized, high-energy source. This behavior is distinctly different from the classical diffusion case in which a blow-up can be avoided by locating the site of the energy source sufficiently close to one of the cold ends of the strip. The asymptotic growth of the solution near blow-up is determined for a nonlinear source whose output increases with temperature in either an algebraic or exponential manner. The blow-up growth rate is found to depend upon the anomalous diffusion parameter that defines the subdiffusive medium. This suggests that such media might be characterized by their response to a reaction-diffusion process.

Dimensional Influence on Blow-up in a Superdiffusive Medium

Thermal blow-up in a superdiffusive medium with a localized energy source is examined for a spatial domain of infinite extent in one, two, and three dimensions. An analysis of a nonlinear model of this problem reveals that the occurrence of a blow-up depends upon the spatial dimension of the superdiffusive medium. In one dimension, a blow-up always occurs. In two and three dimensions, a blow-up can be avoided if the superdiffusive properties are sufficiently enhanced. The asymptotic growth of the temperature near blow-up is determined for energy sources whose output increases in either an algebraic or an exponential manner. The results of the analysis have potential application to porous materials in which a pore filling agent is introduced to enhance the thermal transport properties.