# What kind of bifuraction/fixed point is this?

Homework Statement

Consider the ODE system:

dx/dt = x * (1 - x - d*y)
dy/dt = g * y * (1 - y)

where d and g are parameters.

I'm trying to find and classify the bifurcations of this system. The parameter g doesn't appear to influence any bifurcation, only the parameter d does. I have found there is a bifurcation when d = 1, but I'm having trouble classifying it.

The attempt at a solution

If you graph the nullclines of the system there are always fixed points at (0,0) (a source) and (1,0) (a saddle). Additionally there are two more fixed points at (0,1) and (1 - d, 1).

When d < 1, (0,1) is a saddle and (1-d, 1) is a sink.

As d increases, the sink slowly moves to the left toward the saddle.

When d = 1, the two fixed points coalesce into one. I cannot determine what type of fixed point this is. The associated eigenvalues are 0 and g. Some trajectories go into the fixed point, and others seem to be repelled. Intuition says this is a stable line of equilibria, but I'm not sure.

When d > 1, there are again two fixed points. The former sink is now a saddle and the former saddle is now a sink, i.e. they've switched stability. As d increases, the saddle continues to move along the same line the sink did when d < 1.

The fixed point has a bifurcation at d = 1 but it seems like it's a saddle-node (two FP become one) AND a pitchfork (stability switch). Is that possible? If you can also let me know what kind of fixed point (1,0) when d = 1, I would really appreciate it!

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