Environmental Engineering Reference
In-Depth Information
considerations varying over time (Kingsland 1995). The introduction of differential
equations successively refined analysis of ecological relations and helped to assess
the backgrounds and driving forces of quantitative changes in natural systems.
Differential equations describe the change of one or more variables over time. The
change can be influenced by the quantity of the variable itself, by external impact or
by responses to other variables. Differential equations can be used to describe single
elements as well as complex networks of dynamic systems (Bertalanffy 1976).
This chapter introduces the use of differential equations in ecological models. It
does not provide a complete overview of the mathematical theory. Here, those
aspects are selected and explained, that are most important to understand the
contribution of differential equations in ecological theory and its applications. We
facilitate an understanding of which kind of dynamic representations have particu-
lar relevance for describing ecological processes. The chapter addresses the central
terms and topics that are required for model construction and that are useful for
understanding the scope and the limitations of the approach. For this purpose, a
selection was made that reduces the difficulty of mathematical formalism. The
selection of topics builds on lecture experiences at the University of Bremen, and
feedback from a large number of students over the years. Not only are the mechan-
isms of increase and decrease of different variables explained, but also the phe-
nomena of multiple equilibria, hysteresis and deterministic chaos.
6.2 What Are Differential Equations?
Differential equations represent a concept of abstraction with very specific require-
ments. Though the approach is used in a modelling strategy called “general systems
theory” (Bertalanffy 1976) differential equations can be successfully applied only
when specific preconditions hold:
l
They deal with homogeneous quantities. They do not describe the internal
heterogeneities that may lie behind particular variables. The focus of interest,
is on “how much?”, assuming that the internal
quality
of what the variable
represents, is invariant and structurally homogeneous.
l
Differential equations are deterministic and functional. A given state of the
system always determines precisely the subsequent states. Stochastic influences
are excluded.
l
Differential equations are continuous. They describe the succession of states in
infinitely small intervals, i.e. for any point in time of the considered simulation
interval.
These specifications can best be visualized by a pool metaphor: a pool - to be
filled with water (or whatever imaginary liquid), having an inflow and an outflow
(Fig.
6.1
). Together with the initial filling level of the pool, the regulation of inflow
and outflow determines the filling state (i.e. the value of the variable), which we can
also call the compartment size. In this metaphor, the equations describe the operation
