A simple deterministic description of this system is given by an ordinary
differential equation, where the rate of change of the concentration of phosphorous
equals the difference between inflow and outflow of phosphorus. This simple system
exhibits three equilibria where inflow and outflow are equal, two of which are locally
stable and one unstable. Such a system is “normally” in a stable equilibrium at the low
concentration, and is economically and biologically productive. But a sudden heavy rain
can wash in enough fertilizer to shift the concentration of phosphorus to within the basin
of attraction of the right hand equilibrium, leaving it in a far less productive state. So
could a very hot dry spell, by evaporating water from the lake and increasing
concentration above a critical level. This is a very simple example of dynamical
behaviors that can emerge from ecosystems under stress from human economic
activities. These systems can exhibit multiple attractors (not necessarily equilibrium
attractors) with complex basins of attraction of varying sizes. Understanding how these
systems may move stochastically between these attractors is critical. We sometimes
understand the local dynamics of these basins but questions on the outcomes of
stochastic movements tipping complex systems from one attractor to another are mostly
open.
Eutrophication is just one example of a freshwater water quality problem that is
related to the interplay of natural and human systems and is amenable to analysis using
mathematical models. There are many others. For example, novel mixing patterns in
run-off and natural ground water collection caused by change in climate can lead to
novel mixing patterns of otherwise benign contaminants whose combination could
potentially lead to unwanted impacts. Research is required in the fundamental
understanding of these mixing patterns. For another example, water supply is both
required for agricultural processes and affected by them. How can we model the effect
on water supply of changing agricultural practices due to climate change? We also need
early warning of changing availability of good quality water for agricultural use. How can
machine learning and data mining give us early warning of areas of shortage of water
arising from climate change? - see, for example, Dzeroski et al. (2000), Policastro et al.
(2004). Rain is the input for water in hydrological cycles, yet spatial and temporal
estimates of water amounts, at the national or regional level, are poorly understood.
Traditional methods of rain gauging need to be supplemented with remote sensing and
there are many mathematical challenges arising from placement of sensors to finding
patterns from reports from a network of distributed sensors; see, e.g., Schultz (1993).
Speaking more broadly, can we develop mathematical models that will allow us to
predict regional water shortages due to changing climate?
Water in the oceans is critical for the health of the planet in that the life cycle of
many of the world’s species is intimately tied to the oceans and related wetlands, and
also tied to a major source of food of many of our planet’s inhabitants. There is great
concern that increased dissolved organic carbon in the world’s oceans and resulting
ocean acidification, tied to human activities, is a threat to the health of our oceans.
Large-scale computational models are required to better understand both long-term and
short-term carbon cycling in the oceans (Caldeira and Wickett 2005). The field of ocean
science has long emphasized the connection between oceans and climate, and has
used sophisticated numerical analysis methods to model this interconnection. New
challenges require adaptation of these models to understand the connection between
carbon dioxide in the air and dissolved carbon in the water. Getting early warning of
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