1. The Lotka-Volterra competition equations (6a and 6b in your text) are described: \[ \frac{\mathrm{d}N_1}{\mathrm{d}t} = r_1N_1\left(\frac{K_1 - N_1 - \alpha _{12}N_2}{K_1}\right) \\ \frac{\mathrm{d}N_2}{\mathrm{d}t} = r_2N_2\left(\frac{K_2 - N_2 - \alpha _{21}N_1}{K_2}\right). \] This is similar to what you’ve seen with mutualism and predator-prey, but with both of the interaction terms having negative signs. For these equations, first algebraically find the nullclines. Next, plot them all (not using phaseR), for the 4 different outcomes of competition; i.e., remake figure 8.4.

  2. Recreate figure 8.4. using phaseR. For each of the outcomes, plot two trajectories using trajctory() (see help for details): with one set of initial conditions with a larger \(N_1\) and a smaller \(N_2\) and one set with a smaller \(N_1\) and larger \(N_2\).