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.