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

  1. For the density-dependent predator-prey model, please find the isoclines for each of the two equations (i.e., set each to 0 and solve for the variables).

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














  1. Next, plot all 4 isoclines you found in 1 on the single blank plot, below. Draw the predator and prey isoclines as either different colors or different line types.

  1. What is a bifurcation diagram?


  2. 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.

















  3. Please identify each of these figures as either type I, II, or III functional responses.