Solving for Center Manifold with Parameter

Question

I have a system of ODEs given by

$$\frac{dX}{d\tau}=\beta X\left(1 - \frac{X+Y}{N}\right)$$

$$\frac{dY}{d\tau}= Y\left(1 - \frac{X+Y}{N}\right)$$

where $\beta $ is a parameter.

How should I approach this problem and find the center manifold and its stability?

Answer 1

Drawing a phase diagram would probably have shown you right away that the nonnegative quadrant $Q=(X\geqslant0,Y\geqslant0)$ is stable by the dynamics, that the equilibria in the quadrant $Q$ are the point $(X,Y)=(0,0)$ and the segment $S=(X\geqslant0,Y\geqslant0,X+Y=N)$, and that $(0,0)$ is an unstable node. At every point $(X_0,Y_0)$ on $S$, an eigenvalue is $0$ (in the direction of $S$), the other one is real negative, and the center manifold is stable with equation $XY_0^\beta=Y^\beta X_0$.

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