Please place your name(s) in the upper-right corner.

  1. For the Rosenzweig-MacArthur predator-prey model, please find the nullclines for each of the two equations. Next, plot them. Last, using graphical reasoning (e.g., shifting intercepts, slopes, allowed curvature), create any graphs that might have different numbers of positive (both species densities are > 0) equilibira.

\[\begin{aligned} \frac{\mathrm{d}V}{\mathrm{d}t} &= rV\left(\frac{K-V}{K}\right) - \frac{cV}{g + V}P\\ \frac{\mathrm{d}P}{\mathrm{d}t} &= \frac{bV}{g + V}P - mP \end{aligned}\]




























  1. What is a bifurcation diagram?




  1. For the logistic map below (a type of bifurcation diagram), please label where on the figure we see:
    1. Chaotic dynamics
    2. Asymptotic stability
    3. Four-point cycles
    4. Two-point cycles

  1. Please draw a hypothetical 2-dimensional bifurcation diagram. Let’s imagine that we have a two species interaction with parameters, \(a\) and \(b\), respectively representing growth and attack rate. At high levels of growth rates and attack rates the populations will cycle. At low growth rates populations go extinct. All other rates of growth and attack lead to asymptotic stability.