Environmental Engineering Reference
In-Depth Information
Chapter 6
Ordinary Differential Equations
Broder Breckling, Fred Jopp, and Hauke Reuter
Abstract Differential equations represent a centrally important ecological model-
ling approach. Originally developed to describe quantitative changes of one or more
variables in physics, the approach was imported to model ecological processes, in
particular population dynamic phenomena. The chapter describes the conceptual
background of ordinary differential equations and introduces the different types of
dynamic phenomena which can be modelled using ordinary differential equations.
These are in particular different forms of increase and decline, stable and unstable
equilibria, limit cycles and chaos. Example equations are given and explained. The
Lotka-Volterra model for predator-prey interaction is introduced along with basic
concepts (e.g. direction field, zero growth isoclines, trajectory and phase space)
which help to understand dynamic processes. Knowing basic characteristics, it is
possible for a modeller to construct equation systems with specific properties. This
is exemplified for multiple stability and hysteresis (a sudden shift of the models
state when certain stability conditions come to a limit). Only very few non-linear
ecological models can be solved analytically. Most of the relevant models require
numeric approximation using a simulation tool.
6.1 Background and Purpose of the Chapter
Differential equations play a highly relevant role in the history of modern ecology.
The introduction of differential equation-based modelling was an important
achievement in the paradigm shift from of a previously more qualitatively oriented
science to a leading role of quantitative approaches. The concept of differential
equations originated in classical mechanics. It was developed to describe the motion
of mass points, acceleration, and other time dependent processes. Early last century,
differential equations were adapted by a few ecologists, who focused on quantitative
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