1. The Lotka-Volterra competition equations (6a and 6b in your text) are described: \[ \frac{\mathrm{d}N_1}{\mathrm{d}t} = r_1N_1\left(\frac{K_1 - N_1 - \alpha _{12}N_2}{K_1}\right) \\ \frac{\mathrm{d}N_2}{\mathrm{d}t} = r_2N_2\left(\frac{K_2 - N_2 - \alpha _{21}N_1}{K_2}\right). \] These are the same equations that you studied in Ecology (BI/ES271). For these equations, (a) first, show your work and algebraically find the nullclines. Next, (b) plot the four nullclines all (not using phaseR), for the 4 different outcomes of competition; i.e., remake figure 8.4. Use segments() or lines(), but don’t use nullclines().

  2. Recreate figure 8.4. using phaseR’s functions flowField() and nullclines(). 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 \(N_2\).