Consider a nonlinear system for two populations with equilibrium points (0,0), (0, N), ($\frac{N}{2}, \frac{N}{2}$), and Jacobian
$\begin{aligned} J(x, y) = \begin{bmatrix} -2x + y & x\\ -y & -x + N - 2y \end{bmatrix} \end{aligned}$
a) What is the critical bifurcation value, $N_c$, which changes the number of eq. pts.?
b) Evaluate the Jacobian at each equilibrium point and determine eq. pt. type for N>$N_c$:
(0,0) is a _____
(0,N) is a _____
($\frac{N}{2}, \frac{N}{2}$) is a _____
c) Sketch the geometric behavior of these solutions in the first quadrant of the phase plane (y vs x) for N=2.
