Consider the following Lotka-Volterra model of competition

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with positive k and x, y 0. The variables x and y represent different species.

(a) For k = 3, find and classify the fixed points. Also, draw the fixed points and nullclines in the phase plane. Use arrows to show the direction of the flow along the nullclines.

(b) What does index theory suggest about the possibility of closed orbits in this model?

(c) One of the fixed points xC represents a state of co-existence between species, where x, y > 0. Classify this fixed point in dependence of k.

(d) What type of bifurcation takes place when k = 4?