In trying to identify potential contributions of the mathematical sciences to the
development of a science of HESs as CASs, we isolate essential features of the latter
that must be captured in a mathematical approach to the subject. Each of these should
be included, in one way or another, in any credible mathematical representation, keeping
in mind that successful model development is an exercise in compromise. Specific
implementations in the form of examples will be detailed in the following section.
The mutual, bidirectional interaction between the composing human and
environmental elements is an essential feature of HESs. As alluded to above, the way
humans bidirectionally modify the environment has been neglected in many
representations of HESs, especially the feedback of evolving environments on human
behavior.
In both the temporal and spatial domains, many different time scales are at play. It may
be that a simplification is equally justifiable and necessary in order to gain insight into the
modeled system. Multiscale modeling is a mature subject in classical mathematical
physics, and uses techniques as such as matched asymptotic expansions and singular
perturbations to address the problem of differing temporal scales. Such methods are
receiving much attention even outside of the environmental sciences, as witnessed by a
full SIAM journal (“Multiscale Modeling & Simulation”) devoted to this topic; the extent to
which these techniques are being developed to address environmental problems is less
clear, but some illustrative examples are presented in the following section.
The modeling process itself should be seen as dynamical and hierarchical. There
is hardly a definitive, comprehensive and “final” representation of a given HES. There is
thus a need for consideration of a hierarchy of representations, addressing progressively
more refined incorporation of the details of the description of the HES under study. In
this respect, diversity pays off: there is considerable insight to be gained by a wide
variety of model building approaches. The role of stochasticity, for example, can be
incorporated in many different ways, and most are complementary and provide deeper
understanding of the robustness of the different hypotheses. Either uncertainty in the
initial states, which are never known to arbitrary precision, or sensitivity to variations in
the numerous parameters constituting the model have to be analyzed, both qualitatively
(e.g., classical results on “continuous dependence on initial data”) and quantitatively
(e.g., more recent “sensitive dependence on initial conditions”, the essence of chaos).
Robustness, and a related property—adaptability—are essential features of
HESs. Resilience is a remarkable trait of the human components of an HES, and it is not
entirely clear how this concept can be expressed unambiguously, mathematically, let
alone incorporated in an HES model. Indeed, it could be one of the many emergent
properties of the system—unpredictable properties of an assembled system that appear
(emerge) when constituent components are assembled and put in dynamical interaction,
but are not clear when the components are considered in isolation: the whole is more
than the sum of its parts. An example of emergence is collective behavior, global
coordination among the agents of a complex system, which may or may not be a direct,
predictable consequence of explicit individual properties. Especially when human
components are incorporated, it is a challenge to ascertain at which level of the model
building these should be included, and it is particularly difficult to mediate between local
and global behaviors.
It is premature to advance a comprehensive mathematical theory of coupled
human-environment systems as complex adaptive systems. Instead, we sketch a
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