how the sampling procedure affects the properties of the sequence of node degrees in a
graph representing a physical, biological, or social process. A random graph has a
binomial or Poisson distribution of node degrees, a scale-free graph has a power law
distribution of node degrees, and a “small world” graph has an exponential distribution of
node degrees (Albert and Barabási 2002). Different models of graph evolution lead to
these different degree distributions in dynamically evolving networks and can be used to
understand the evolutionary processes underlying changing migration patterns.
Moreover, the differences among such models of graphs and their formation provide
information about the processes that may have produced them. The degree parameters
are also important characteristics of the graph’s structure. For example, they can lead to
an analysis of the vulnerability or stability of a network under the removal of any
particular node (under various measures of vulnerability and stability).
Scientists are predicting human mass migration as a consequence of climate
change: millions of people fleeing from rising sea levels and drought, leading to serious
consequences for both migrants and receiving societies. A mathematical approach to
such mass migration, which modeled the connection between climate change and
human migration, was developed by Perch-Nielsen (2004). Among the mathematical
sciences challenges involving the study of mass human migrations are the development
of models describing analogies to and evolution of patterns of animal migrations, and the
understanding of the complex adaptive systems involved in mass human migrations.
Research Challenge for the Mathematical Sciences: Develop models of the
interplay between climate and migration and the disruption/synchronization of the
processes that allow for seamless integration of multiple mechanisms relevant to
migration; model the spatial and temporal spread of animal and plant populations under
rapidly changing environmental conditions caused by human processes and the impact
of modified human systems on changing migration patterns; understand the evolution of
networks that interconnect migratory routes so as to understand forces threatening the
stability of migration and the resulting impact on human systems.
Example 6: Health of Lakes and Oceans
The quality of water in our lakes, rivers, streams, and oceans is critical to
sustaining life on our planet. The natural processes underlying healthy bodies of water,
large and small, are closely related to processes underlying human activities.
Human and natural processes are often effectively modeled by linked economic-
ecological models, often capable of exhibiting multiple (coexisting) attractors. A simple
example is given by the eutrophication of a lake through phosphorous run-off from
agricultural land, which has been extensively studied by ecologists and economists (see
e.g. Bennett et al. 1999). The driver of change in this setting is often that fertilizer runs
off farmland around the lake and into the lake, particularly when it rains. Phosphorus in
the fertilizer dissolves in water and also is retained by sediment on the lake bottom. At
low concentrations of phosphorus a lake is clear and productive, with many sources of
economic value. At high concentrations it is biologically almost dead and of little or no
economic value. The basic dynamics are that phosphorus leaves the lake through
outflow in the stream that exits the lake, at a rate that is proportional to the concentration
in the water; it flows in off the neighboring cropland, and may also move from the
sediment at the bottom of the lake into solution.