Modelling Complex Ecological Dynamics pp 67-91 | Cite as

# Ordinary Differential Equations

## Abstract

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 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.

## Keywords

Phase Space Equilibrium Point Pool Size Prey Population Multiple Equilibrium## Further Readings

- 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.ruhr-uni-bochum.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:
- Edelstein-Keshet L (2004) Mathematical models in biology, 2nd edn. SIAM, 586 pGoogle Scholar
- Jeffries C (1989) A workbook in mathematical modeling for students of ecology. Springer, HeidelbergGoogle Scholar
- 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. On-Line 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/Differential_Equations
- Yodzis P (1989) Introduction to theoretical ecology. Harper & Row, New YorkGoogle Scholar