1. 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 consumption and (3) insert those values into the consumer equations: \[ \frac{\mathrm{d}N_1}{\mathrm{d}t} = b_{11}N_1R_1 + b_{12}N_1R_2 - mN_1 \\ \frac{\mathrm{d}N_2}{\mathrm{d}t} = b_{22}N_2R_1 + b_{21}N_2R_2 - mN_2. \] Last, (4) arrange the equations such that they resemble the distributed logistic-based Lotka-Volterra equation. You can do this by hand opposed to \(\LaTeX\) if you desire, and wherever you save your .Rmd file, save a .jpg/.png of your equations, and use this html command that you learned in problem set 1.

(1) Resource equations

\[ \frac{\mathrm{d}R_1}{\mathrm{d}t} = rR_1\left(\left(\frac{k_1 - R_1}{k_1}\right) - a_1N_1 - a_2N_2\right)\\ \frac{\mathrm{d}R_2}{\mathrm{d}t} = rR_2\left(\left(\frac{k_2 - R}{k_2}\right) - a_1N_1 - a_2N_2\right) \]

(2) Resource equation equilibira

\[ R_1^* = k_1\left(1 - a_1N_1 - a_2N_2\right) \\ R_2^* = k_2\left(1 - a_1N_1 - a_2N_2\right) \]

(3) Resource equilibira in consumer equations

\[ \frac{\mathrm{d}N_1}{\mathrm{d}t} = b_{11}N_1k_1\left(1 - a_1N_1 - a_2N_2\right) + b_{12}N_1k_2\left(1 - a_1N_1 - a_2N_2\right) - mN_1 \\ \frac{\mathrm{d}N_2}{\mathrm{d}t} = b_{22}N_2k_2\left(1 - a_1N_1 - a_2N_2\right) + b_{21}N_2k_1\left(1 - a_1N_1 - a_2N_2\right) - mN_2 \]

(4) Arrange consumer equations

Distribute: \[ \frac{\mathrm{d}N_1}{\mathrm{d}t} = b_{11}N_1k_1 - b_{11}N_1k_1a_1N_1 - b_{11}N_1k_1a_2N_2 + b_{12}N_1k_2 - b_{12}N_1k_2a_1N_1 - b_{12}N_1k_2a_2N_2 - mN_1 \\ \frac{\mathrm{d}N_2}{\mathrm{d}t} = b_{22}N_2k_2 - b_{22}N_2k_2a_1N_1 - b_{22}N_2k_2a_2N_2 + b_{21}N_2k_1 - b_{21}N_2k_1a_1N_1 - b_{21}N_2k_1a_2N_2 - mN_2 \] Arrange as first- and second-orders of \(N_i\), then \(N_j\): \[ \frac{\mathrm{d}N_1}{\mathrm{d}t} = \left(b_{11}k_1 + b_{12}k_2 - m\right)N_1 - \left(b_{11}k_1a_1 + b_{12}k_2a_1\right)N_1^2 - \left(b_{11}k_1a_2 + b_{12}k_2a_2\right)N_1N_2 \\ \frac{\mathrm{d}N_2}{\mathrm{d}t} = \left(b_{22}k_2 + b_{21}k_1 - m\right)N_2 - \left(b_{22}k_2a_2 + b_{21}k_1a_2\right)N_2^2 - \left(b_{22}k_2a_1 + b_{21}k_1a_1\right)N_2N_1 \]

  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 differential equations and their equilibria: which of the two do you prefer? Why?

The first, from the book: \[ \frac{\mathrm{d}R}{\mathrm{d}t} = rR\left(\left(\frac{k - R}{k}\right) - a_1N_1 - a_2N_2\right) \\ 0 = rR\left(\left(\frac{k - R}{k}\right) - a_1N_1 - a_2N_2\right) \\ 0 = r\left(\left(\frac{k - R}{k}\right) - a_1N_1 - a_2N_2\right) \\ 0 = r - \frac{r}{k}R - ra_1N_1 - ra_2N_2 \\ \frac{r}{k}R = r - ra_1N_1 - ra_2N_2 \\ \frac{1}{k}R = 1 - a_1N_1 - a_2N_2 \\ R = k\left(1 - a_1N_1 - a_2N_2\right) \] The second, from your quiz: \[ \frac{\mathrm{d}R}{\mathrm{d}t} = rR\left(\frac{K - R - a_1N_1 - a_2N_2}{K}\right) \\ 0 = rR\left(\frac{K - R - a_1N_1 - a_2N_2}{K}\right) \\ 0 = r\left(\frac{K - R - a_1N_1 - a_2N_2}{K}\right) \\ 0 = r\left(\frac{K - a_1N_1 - a_2N_2}{K}\right) - \frac{r}{K}R\\ \frac{r}{K}R = r\left(\frac{K - a_1N_1 - a_2N_2}{K}\right) \\ \frac{1}{K}R = \frac{K - a_1N_1 - a_2N_2}{K} \\ R = K - a_1N_1 - a_2N_2 \\ \]