with a relatively coarse-grained spatial-temporal description of modest scope) is well
beyond the reach of current-day solvers, and improving them to be relevant to such a
setting is a significant mathematical challenge. Furthermore, it remains to be seen to
what extent the crude approximations in this approach - the particular discretization of
both time and space, or the linearization of the objective that completely abstracts away
the spatial dimension - lead to results that would align with more detailed models (which
are even further beyond our computational capabilities).
Alternatively, one might also investigate more sophisticated models that seek to
capture stochasticity inherent in changes linked to climate change, such as the
increased frequency of future droughts. Understanding the alignment between
differently-scaled models is particularly important in this context, since more
sophisticated models, while not solvable in a real-time setting, would be useful in
validating decisions.
Forest insect pests are a major threat to forest health (Liebhold et al., 1995).
Many forest insect pests both respond to the damage of trees by fires, and also can
damage trees and make them more susceptible to fires. Particular pests that illustrate
some of the mathematical issues involved in management range from gypsy moth to
mountain pine beetle and other bark beetles. The goal is either eradication or reduction
of the population level to a low enough level so that there are no damages. The
management of forest insect pests is a complex problem that requires attention to spatial
and temporal heterogeneity both in the trees and in the population dynamics of the
insects in a control problem that must include stochasticity at many levels.
Although there is a long history in mathematics of attention to control problems,
management of forest pests introduces a series of extra complications that are typical of
biological problems that are not well understood as well as requiring attention to different
statistical and estimation questions. The population dynamics is likely to be modeled
best as a nonlinear stochastic integro-difference equation, and control problems for this
sort of model are not well understood. In the case of bark beetles there are additional
important issues related to the interactions between individuals in their movement.
Estimation of populations in a spatial context introduces difficult problems of estimation
that are necessary both to start the control problem and to determine whether
management has been successful. Other approaches could include determining what
simpler versions of the models would still be sufficient for management.
2.3. Agricultural systems
Agricultural systems present numerous opportunities to utilize diverse
mathematical approaches and offer various challenges requiring new mathematics.
These arise in part due to the coupling of agricultural systems to many environmental
components, human behaviors, and the intimate linkages between various sub-
components of agriculture. There is a very long history of quantitative approaches in
cropping systems, animal production systems, and economics including the use of large
systems approaches and more general mathematical formulations (Thornley and
France, 2007). These systems present unique opportunities to project the impacts of
various management and policy actions on food availability for the growing world
population, water resources, waste production, and contributions to greenhouse gas
emissions.