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RE: Foxes hunting Bunnies: Population Modelling with the Predator-Prey Equations
Nice article, as always! I have a question: How is the direction field in Figure 1 derived? Differential equations aren't my strength and I haven't seen this type of graph yet.
If you're asking mathematically, direction fields are simply drawn by choosing values of F and R and working out dF/dR from those chosen values. dF/dR is then simply the slope at the chosen (F, R). So say when you choose F to be 20, at R = 100 the slope dF/dR is quite steep in the negative direction and gradually flattens and becomes positive for increasing values of R.
Graphically, it is drawn in Desmos using this tool: https://www.desmos.com/calculator/p7vd3cdmei
I was asking for exactly that! Thanks. So basically, there is no definite resolution to this, you could technically choose as many points as you liked and calculate the slope at that point, correct?
That's right. There are an infinite number of solutions. The direction field will give you an idea of what a particular solution might look like because it shows you how the slope changes from point to point.
Huh. I wonder if one could increase the resolution to say, the resolution of your monitor and color code the slopes instead of using lines to get a more detailled field. I am not sure if this is at all necessary or helpful, but it seems like a fun project nonetheless.
Sure. Why not? I guess it depends on your goal. For global behaviour of a differential equation, increasing the number of points and resolution may not add so much additional benefit - it may even have the opposite effect of increasing clutter.
The one I have drawn in this post is borderline in my opinion. I could have made the little slope lines a bit bigger and used 1/2 the amount of points, and the slope field (aka direction field) would have told the same story.
However, if you want to zoom in on a behaviour near a critical point, you may want to increase the resolution to accurately depict what's going on.
To expand on the answer a bit. So for 2-dimensional first order ODEs there is the Poincaré–Bendixson theorem. This tells you all the possible types of solutions that can occur. Using this and studying the vector field it is easy to prove which solutions occur.
In the case of equilibria (for n-dimensional ODEs) you have the Stable Unstable and Center Manifold Theorem. These theorems tell you what happens close to an equilibrium.