642:527ASSIGNMENT 7FALL 2009Turn in starred problems Tuesday 10/27/2009.Section 7.4: 2 (a), (b)*, (f)*See instructions below.Section 7.4: 7Section 7.5: 4*Problem 7.A*Two interacting populationsx(t),y(t) are described by the equationsx′= (1-x)x ,y′= (3-y-x)y .(a) Find all the critical points of this system. You do not need to classify these.(b) Sketch the first quadrantx≥0,y≥0 of the phase plane, indicating, by arrows or other-wise, regions wherexandyare increasing,xis increasing andydecreasing, etc., and where thetrajectories are horizontal and vertical.(c) For each initial condition below, find (from your sketch or otherwise) limt→∞bracketleftbiggx(t)y(t)bracketrightbigg:(i)x(0) = 0,y(0) = 1;(ii)x(0) = 3,y(0) = 3;(iii)x(0) = 0,y(0) = 0.In 7.4 (b), (f), please do the following:•Find all singular points;•Obtain thelinearized systemnear each singular point and classify the origin of that system (asa saddle point, unstable or stable node, unstable or stable focus, or center). If the origin is asaddle point or a node, obtain the special straight-line trajectories. Sketch the phase plane of
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Technological singularity, The Singularity Is Near, phase plane, special straight-line trajectories