mathematical models of ocean food-limited fishery. A challenge is to incorporate the
effects of global change and ocean acidification into such models. The effect of climate
change on fish migration is clearly amenable to mathematical models. For example, one
can readily state a system of partial differential equations that includes a transport term,
diffusion, and nonlinear interactions among the species. For example, for fish seeking
cooler environment, the transport term may be proportional to the gradient of the
temperature and the acidity of the ocean water. The diffusion matrix is such that it tends
to discourage too many fish converging on one spot. There are many variants in such
differential equation models and challenging problems arise from analyzing them. Animal
migration on land including predator-prey relationship and plant migration or invasion
can be modeled by a similar system, where one incorporates land cover types such as
water and different kinds of vegetation.
Birds migrate long distances as seasons change. With changing climate, these
migration patterns are changing. Not all species react in the same way or adapt as
quickly to changing environmental conditions. Thus, for example, there are cases of
birds migrating earlier than before, but arriving at their destination before their traditional
food sources are available (see, e.g. Miller-Rushing et al. 2008). A key to ecosystem
health is the delicate balance among interacting cyclic processes, and climate change
can disrupt long-developed synchronicities in timing among these processes. This type
of problem – and prediction of its impact on natural populations – leads to serious
mathematical challenges. It connects closely to human well-being since birds play an
important role in insect control, which in turn affects the growth of crops and other
agricultural products for human consumption.
Understanding animal and plant migrations requires us to understand
interactions among biological entities from ecological and evolutionary perspectives in a
dynamic and disturbed global environment (Agrawal et al. 2007). Graph theory provides
a flexible conceptual model that can clarify the relationship between structures and
processes in such applied problems, including the mechanisms of configuration effects
and compositional differences. Graph concepts apply to many ecological and
evolutionary phenomena, including interspecific associations, spatial structure, dispersal
in landscapes, and relationships within meta-populations and meta-communities. Spatial
graph properties can be used for description and comparison of migration patterns as
well as to test specific hypotheses about migration. The analysis of animal movement
can focus either on the attractive or avoidance effects of each patch of land, or on the
directionality and volume of movement between patches (Croft et al. 2008). Spatially
explicit graph analyses of these two aspects can be examined separately or together
using gravity models, spatial graphs, or other new models not yet developed. Numerous
questions about migration can be addressed this way, for example: What is the
relationship between changes in spatial habitat structure and gene flow? How do these
changes affect species survival? How does the pattern of migration routes affect the
spread of disease? The mathematical sciences have much to contribute in answering
such questions.
Sampling processes create serious limitations for the interpretation of metrics
that describe the property of a graph when there is no assurance that there is a
complete census of the objects depicted in the graph. The comparison of metrics is a
significant problem to be addressed in the mathematical sciences. The distribution of
degrees of nodes in a graph presents a relevant example, and we need to understand
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