April 2010

“Despite the increasing importance of mathematics to the progress of our economy and society, enrollment in mathematics programs has been declining at all levels of the American educational system. Yet the application of mathematics is indispensable in such diverse fields as medicine, computer sciences, space exploration, the skilled trades, business, defense, and government. To help encourage the study and utilization of mathematics, it is appropriate that all Americans be reminded of the importance of this basic branch of science to our daily lives.”

In recent years, ecologists have begun to refine the mathematical study of complex networks of interactions between species within an ecosystem.   In particular, Network Environs Analysis (NEA) has made use of linear algebra to formalize analysis of the relative importance of direct and indirect connections between species.  One of the many indicators used to quantify such description is homogenization, which is a measure of the propensity of a network to distribute material throughout all species along all connections.  We examine a matrix-theoretic formulation of the homogenization indicator and describe how linear algebraic tools can be refined to determine which species and paths may exert strongest influence on the behavior of an ecosystem.

Ecologists and conservation biologists often need to know what roles species perform in ecosystems in order to make informed management decisions and allocate limited resources most effectively. However, quantifying the functional roles of species has proved difficult because species are embedded in complex networks of interactions and a species’ importance cannot be understood in isolation. Here we introduce environ centrality (EC), a metric that quantifies the relative roles of species in generating ecosystem activity and demonstrate its functionality over four empirically-based ecosystem network models. Our results highlight the relative evenness of species roles in ecosystems and demonstrates how indirect relationships promote this evenness.

Mathematical modeling of the spreading of air pollutants in a windy atmosphere has become a crucial field of study over the past several decades. With the advances in computer technology and development of sophisticated programming languages, scientists have become able to evaluate evolutionary partial differential equations using numerical methods with a high degree of accuracy. Computer simulations of air pollution dynamics will help to reveal crucial properties about the forces interacting in gaseous systems, and will also help scientists to create accurate models that predict global weather patterns, climate change, global warming, air pollution spread through cities, and many other critical studies. Two very common numerical methods for solving evolutionary partial differential equations will be presented in this research; finite difference methods and finite volume methods. A thorough investigation into the accuracies and faults between each of these two drastically different methods will be discussed. A detailed discussion on the hydrodynamical equations used to model the spread of air pollution will be approached in an elementary, but intense physical sense. The parameters involved, as well as the source terms that provide the forcing for the emission of pollutants, will be introduced and thoroughly dissected. An analysis of the mass density profiles of the transport of pollutants in the atmosphere of the Earth obtained from this study will be presented at the end. A discussion about the accuracy of the theoretical simulations and its relationship between actual experimental data will ultimately be considered.