The R Book

Phase Planes

Suppose that we have two competing species (named 1 and 2) and we are interested in modelling the dynamics of the numbers of each species (N₁ and N₂). We want to draw a phase plane showing the behaviour of the system close to equilibrium. Setting the derivates to zero and dividing both sides by N_rN_i we get

which is called the isocline for species 1. It is linear in N₁ and N₂ and we want to draw it on a phase plane with N₂ on the y axis and N₁ on the x axis. The intercept on the y axis shows the abundance of N₂ when N₁ = 0: this is 1/α₁₂. Likewise, when N₂ = 0 we can see that N₁ = 1/α₁₁ (the value of its single-species equilibrium). Similarly,

describes the isocline for species 2. The intercept on the y axis is 1/α₂₂ and the value of N₁ when N₂ = 0 is 1/α₂₁. Now we draw a phase plane with both isoclines, and label the ends of the lines appropriately. We might as well scale the axes from 0 to 1 but we want to suppress the default tick marks:

plot(c(0,1),c(0,1),ylab="",xlab="",xaxt="n",yaxt="n",type="n")
abline(0.8,-1.5)
abline(0.6,-0.8,lty=2)

The solid line shows the isocline for species 1 and the dotted line shows species 2.

Now for the labels. We use at to locate the tick marks – first the x axis (axis = 1),

axis(1, at = 0.805, lab = expression(1/alpha[21]))
axis(1, ...