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1) The first order system

dx/dt = x(1 − 0.5x − 0.5y),
dy/dt= y(−0.25 + 0.5x)

can be interpreted as describeing interaction of two species with population densities x and y. Perform the steps given below:

(i) Determine the critical points.

(ii) For each critical point determine corresponding linear system.

(iii) Determine the eigen values and eigen vectors of linear systems.

(iv) Categorize each critical point as to kind and find out whether it is asymptotically stable, stable or unstable.

(v) Draw the trajectories in neighbourhood of each critical points.

(vi) Sketch the phase portrait.

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