Systems and complexitySystems thinking and modelling

Equilibria and Stability Analysis: The Phase Plane [Systems thinking & modelling series]

This is part 54 of a series of articles featuring the book Beyond Connecting the Dots, Modeling for Meaningful Results.

When looking at model results we have been focused on time series plots and we have mainly been interested in the trajectory of the variables and stocks over time. For the mathematical analysis of differential equations, however, the primary graphical tool is not this time series plot; instead it is what is known as a phase plane plot.

Phase planes are almost like scatterplots. They show one of the state variables plotted against another of the state variables. A scatterplot could be used to show the path for these two variables over the course of a simulation. In the predator-prey model the results of a scatterplot of the wolf and moose population will be an ellipsoid. The two populations these two variables over the course of a simulation. In the predator-prey model the results of a scatterplot of the wolf and moose population will be an ellipsoid. The two populations will cycle continuously. A phase plane plot is similar to this, but rather than just showing one of these cycles for a given simulation run, the phase plane shows the trajectories for all combinations of moose and wolf population sizes.

Predator-prey phase plane plot. The trajectory for a single set of initial conditions is highlighted in red.
Figure 1. Predator-prey phase plane plot. The trajectory for a single set of initial conditions is highlighted in red.

Figure 1 illustrates a phase plane plot for the predator-prey system. The trajectory for one set of parameter and state variable values is highlighted in red and, as expected, we see a continual oscillation. We can also see the trajectories for all the other combinations of state variables. We see that the system will always oscillate and the size of this oscillation depends on the initial conditions for the state variables. This illustration provides us with a good deal of information in a single graphic; the phase plane plot is a great way to summarize the behavior of a system with two state variables.

Let’s quickly explore the phase plane plots for a simpler system. Take a system consisting of two state variables, both of which grow (or decay) exponentially1. These state variables will be assumed to be independent from each other, so the value of one does not affect the value of the other:

Clearly, there is an equilibrium point for this model at X = 0 and  Y = 0. There are four general types of behavior around this equilibrium:

  1. When α > 0 and β > 0.
  2. When α < 0 and β > 0.
  3. When α > 0 and β < 0.
  4. When α < 0 and β < 0.

The phase planes for each of the four cases are shown in Figure 2.

Phase planes for a simple two state variable exponential growth model
Figure 2. Phase planes for a simple two state variable exponential growth model.

From these plots we can visually determine how the stability of the equilibrium point at X = 0, Y = 0 changes as we change α and β. When α < 0 and β < 0, we have a stable equilibrium. In all other cases we have an unstable equilibrium.

Exercise 8-7
Sketch out the phase plane for the differential equation model:

Exercise 8-8
Sketch out the phase plane for the differential equation model:

Next edition: Equilibria and Stability Analysis: Stability Analysis.

Article sources: Beyond Connecting the Dots, Insight Maker. Reproduced by permission.

Header image source: Beyond Connecting the Dots.

Notes:

  1. A helpful reminder: if you are starting to get lost in some of this differential equation jargon, a “state variable” is just a stock. Return to the table at the beginning of The Mathematics of Modeling section to see how these terms relate to the system dynamics modeling terminology we have already learned.

Scott Fortmann-Roe and Gene Bellinger

Scott Fortmann-Roe, creator of Insight Maker, and Gene Bellinger, creator of SystemsWiki, have written the innovative interactive book "Beyond Connecting the Dots" to demystify systems thinking and modelling.

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