21 Phase portraits
A Graphical analysis
In analysing a model we would often like to visualise the dynamics of a system. Most commonly we might do this by plotting the time-courses of our variables, showing how the density of our populations change over time. If we have two populations – as we regularly do in this course – we’d therefore be plotting two curves together, which can make such plots a bit messy, certainly if we want to look at different initial conditions. An alternative is to sketch a phase portrait. In this plot we leave time implicit, and instead plot how the two densities change together.
Suppose we have two populations with densities [latex]X[/latex] and [latex]Y[/latex] for which we have ordinary differential equations describing their dynamics. We can imagine a plot that takes [latex]X[/latex] and [latex]Y[/latex] as the two axes. We could then mark on the two densities at their initial values, [latex]X(0)[/latex] and [latex]Y(0)[/latex] as a single point on the plot. We might then go forward in time a little and mark on a second point. We can then continue moving forward in time and adding on points, then ‘join the dots’ to form a trajectory, showing how the two densities change over time.
This approach assumes we know the actual densities at various time-points, but as we have seen, for most non-linear models we do not have that luxury without using numerical solvers in a programming package. However, we can get an idea of the qualitative behaviour by sketching certain details onto a phase portrait.
Algorithm
The basic algorithm for construcing a phase portrait is as follows:
- Draw axes of the two variables.
We can only really sketch the phase portrait for two-dimensional systems. Usually we’d have the variable for our first ODE on the horizontal axis and the second on the vertical axis, but sometimes it makes sense to go the other way around (particularly if the nullclines are awkward for one of the two ODEs – see step 3). - Determine how much of the phase plane is biologically feasible.
An advantage of mathematical biology models is that we rarely need to worry about negative densities, so we can usually just draw the upper-right quadrant of the plane. Sometimes we can reduce it even more – see the SIR models in chapters 4 and 5 for examples - Calculate nullclines and draw them on your plot.
Nullclines are curves on your plot along which one of the ODEs is equal to zero. In turn, set each ODE to zero and (assuming you placed [latex]X[/latex] on your horizontal axis) re-arrange it into the form [latex]Y=f(X)[/latex]. This will hopefully give you a curve you can sketch, though may occasionally look unpleasant. Because the ODE is zero along this line, trajectories must cross these nullclines either vertically or horizontally depending on which ODE it came from – again assuming [latex]X[/latex] is on the horizontal axis, the nullcline that gives[latex]dX/dt=0[/latex] must be crossed vertically, since the densities should not b