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

  1. Write a for loop as syntactically accurate as you can if you wanted to index the ith value of the vector vec <- c(1, 3, 5, 7, 9) multiply it by 6, then save it in the empty vector empty_vec <- rep(x = NA, times = length(vec)).








  2. For the following bifurcation diagram, please identify regions where \(\lambda\) values lead to:
  1. Chaos
  2. Two-point oscillations/cycles
  3. Damped oscillations/cycles
  4. Extinction
  5. Four-point oscillations/cycles

    Note that one of the above are indecipherable from this figure 🤓. (So very cute)

  1. For the following Lotka-Volterra Predator-Prey model, match the coefficients with the definition: \[\begin{aligned} \frac{\mathrm{d}\Omega}{\mathrm{d}t} &= \alpha \diamondsuit \Omega - \spadesuit \Omega\\ \frac{\mathrm{d}\alpha}{\mathrm{d}t} &= \clubsuit \alpha - \Omega \alpha \heartsuit \end{aligned}\]
Coefficient Meaning
\(\clubsuit\) Death rate of predators
\(\heartsuit\) Capture/attack rate of predators removing prey
\(\spadesuit\) Conversion rate of attacked prey to predators
\(\diamondsuit\) Intrinsic rate of growth of prey

  1. For the equation above, please find the equilibria for each each equation, separately. Note that now that we have 2 variables, \(\Omega\) and \(\alpha\), when you find equilibria for each, that we want the values of either variable that make the rate of change equal to 0.









  2. For the equilibria you found for \(d\Omega /dt = 0\), plot them on the left plot. For the equilibria you found for \(d\alpha /dt = 0\), plot them on the right plot.