Differential Equations: Problem Set 12

  1. Problems 5.3.2, 5.3.4 on page 202.

  2. In the last part of problem 5.3.4, you produced a phase plane by hand and then used pplane for comparison. It looks like the equilibrium is a center and that the trajectories are closed periodic orbits.

    1. But Hartman-Grobman is inconclusive in this case. Why?
    2. Prove that the trajectories are, in fact, closed periodic orbits.

  3. Consider the following system:

    \[ \begin{align*} N' &= 0.4N(1-N)-0.2NP_1-0.2NP_2 \\ P_1' &= 0.5NP_1-0.2P_1 \\ P_2' &= 0.4NP_2-0.05P_1P_2-0.1P_2 \end{align*} \]

    1. Find all the equilibria, \((\bar{N},\bar{P_1},\bar{P_2})\).
    2. In terms of predators and prey, what do these equations model?
    3. Is there any (long-term) scenario where the three species can coexist? Explain.