Differential Equations: Problem Set 12

Problems 5.3.2, 5.3.4 on page 202.
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.
Consider the following system:

$\begin{aligned} N^{'} & = 0.4 N (1 - N) - 0.2 N P_{1} - 0.2 N P_{2} \\ P_{1}^{'} & = 0.5 N P_{1} - 0.2 P_{1} \\ P_{2}^{'} & = 0.4 N P_{2} - 0.05 P_{1} P_{2} - 0.1 P_{2} \end{aligned}$
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.