I would like for you to attempt to derive “long-ass equation” starting on page 214. I want your attempt, not the correct answer. Take the resource equation, \[ \frac{\mathrm{d}R}{\mathrm{d}t} = rR\left(\left(\frac{k - R}{k}\right) - a_1N_1 - a_2N_2\right), \] and extend it to two resources by (1) writing two equations, one as \(R_1\) and one as \(R_2\). Next, (2) find the equilibria. Assume that the resource dynamics are much faster than comsumption and (3) insert those values into the consumer equations: \[ \frac{\mathrm{d}N_1}{\mathrm{d}t} = b_{11}N_1R_1 + b_{12}N_1R_2 - mN_1 \\ \frac{\mathrm{d}N_2}{\mathrm{d}t} = b_{22}N_2R_1 + b_{21}N_1R_2 - mN_2. \] Last, (4) arrange the equations such that they resemble the distributed logisitc-based Lotka-Volterra equation. You can do this by hand oppsed to \(\LaTeX\) if you desire, and wherever you save your .Rmd file, save a .jog/.png of your equations, and use this html command that you learned in problem set 1.
You saw two different versions of equations describing the dynamics of resources. The first from the book was: \[ \frac{\mathrm{d}R}{\mathrm{d}t} = rR\left(\left(\frac{k - R}{k}\right) - a_1N_1 - a_2N_2\right). \] The second was from your quiz, which was \[\frac{\mathrm{d}R}{\mathrm{d}t} = rR\left(\frac{K - R - a_1N_1 - a_2N_2}{K}\right). \] For each of these equations, find equilibria. Next, take a few minutes to compare the book and quiz diffential equations and their equilibria: which of the two do you prefer? Why?