MCED Chapter 06
Title
Ordinary Differential Equations
Authors
Broder Breckling, Fred Jopp, and Hauke Reuter
Abstract (from the book)
Differential equations represent a centrally important ecological modelling approach. Originally developed to describe quantitative changes of one or more variables in physics, the approach was imported also 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 a 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). Almost no relevant nonlinear ecological model cannot be solved analytically but requires numeric approximation using a simulation tool.
Software used to produce Chapter 6 simulations
For the simulation of the differential equations presented in this chapter we used SIMILE. The programme is available under http://www.simulistics.com/. It provides a graphic user interface, and tabular as well as graphical output. Adequate for most ecological applications is the Runge Kutta 4th order integration routine (as an alternative to Euler integration, which is not recommendable for equations even or the complexity of the Lotka Volterra system, which is an elementary starting point for population dynamic studies. An advantage of the software is its accessibility. Plots of variables over time are easy, a bit more difficult to find in the menu is the option for x/yplots (one variable over another).
A drawback is the lacking option to display trajectories in a 3dcoordinate system.
Additional material
Lecture material developed (among others) for the course “Systems Analysis” in the Master of Science Programme “International Studies in Aquatic Tropical Ecology” at the University of Bremen during the years 1999 – 2011. It partly extends the content covered in MCED Chapter 6 Differential Equations.
 Part 1: Basic aspects of Dynamic Systems (PDF)
 Part 2: Difference Equations (Discrete systems) (PDF)
 Part 3: Differential Equations (Continuous Systems) (PDF)
 Part 4: Ecological Stability (PDF)
 Part 5: Differential equations with interesting properties (PDF)
Further Reading
(from the book and additional recommendations)

Many textbooks exist on ordinary differential equations, often with a very specific focus. A list of books relating to the ecological context can be found at http:// homepage.ruhrunibochum.de/michael.knorrenschild/embooks.html (Knorrenschild M (2010) List of textbooks on ecological modelling). From our perspective we would select the following books and webpages that expand on the contents provided in this chapter:

EdelsteinKeshet L (2004) Mathematical models in biology, 2nd edn. SIAM, 586 p
 Jeffries C (1989) A workbook in mathematical modeling for students of ecology. Springer, Heidelberg Kot M (2001) Elements of mathematical ecology. Cambridge University Press, Cambridge,
http://www.cambridge.org/us/catalogue/catalogue.asp?isbn=9780521001502 
Sharov A (n.d) Quantitative population ecology. OnLine Course.
http://home.comcast.net/~sharov/PopEcol/popecol.html 
William SC, Gurney WSC, Nisbet RM (1989) Ecological dynamics. Oxford University Press, Oxford, New York. http://www.stams.strath.ac.uk/ecodyn/

Wiki book on differential equations.
http://en.wikibooks.org/wiki/Ordinary_Differential_Equations 
Yodzis P (1989) Introduction to theoretical ecology. Harper & Row, New York