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A model for the spruce budworm population u(t) is governed by (d u)/(d t)=r u(1-(u)/(q))-(u^2)/(1 u^2) where r and q are positive dimensionless parameters. The nonzero steady states are thus given by the intersection of the two curves U(u)=r(1-(u)/(q)), V(u)=(u)/(1 u^2) Show, using the conditions for a double root, that the curve in r, q space which divides it into regions where there are I or 3 positive steady states is given parametrically by r=(2 a^3)/((1 a^2)^2), q=(2 a^3)/(a^2-1) Show t
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