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  1. The text describes how to arrive at the phenomenological competition equations by building on a mechanistic framework with explicit resources. They present a 3-state system of equations for one resource, \(R\), and two species that consume it, \(N_1\) and \(N_2\): \[ \begin{align} \frac{\mathrm{d}R}{\mathrm{d}t} &= rR\left(\frac{K - R - a_1N_1 - a_2N_2}{K}\right)\\ \frac{\mathrm{d}N_1}{\mathrm{d}t} &= b_1N_1R - mN_1\\ \frac{\mathrm{d}N_2}{\mathrm{d}t} &= b_2N_2R - mN_2 \end{align} \] They then make the assumption that the resources have fast dynamics relative to the consumers: what do they mean by that?




  1. Find the two equilibria for \(\mathrm{d}R/\mathrm{d}t\). Well, \(R^* = 0\) is one, so find the other ;)







  2. Replace the \(R\) in the equations with the \(R^*\) : \[ \begin{align} \frac{\mathrm{d}N_1}{\mathrm{d}t} &= b_1N_1R - mN_1\\ \frac{\mathrm{d}N_2}{\mathrm{d}t} &= b_2N_2R - mN_2 \end{align} \] Then, arrange each equation so you have three terms: a first-order term of \(N_i\), a second-order term of \(N_i\), and an interaction term with \(N_iN_j\), similar to how we can logistic equation (i.e., \(\mathrm{d}N/\mathrm{d}t = rN - r/KN^2\)), but with the interaction term at the end.
  1. Please identify each equilibrium on the following phase planes for competition (equations below). Additionally, for each equilibrium, label if it’s stable, \(s\) or unstable, \(u\). \[ \begin{align} \frac{\mathrm{d}N_1}{\mathrm{d}t} &= r_1N_1\left(\frac{K_1 - N_1 - \alpha _{12}N_2}{K_1}\right)\\ \frac{\mathrm{d}N_2}{\mathrm{d}t} &= r_2N_2\left(\frac{K_2 - N_2 - \alpha _{21}N_1}{K_2}\right) \end{align} \]

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