(a) In Examples 3 and 4 of Section 2.1 we saw that any solution P(t) of (4) possesses the asymptotic behavior P(t) :a/b as t → ∞ for P₀ > a/b and for 0 > P₀ , a/b; as a consequence the equilibrium solution P = a/b is called an attractor. Use a root-fi­nding application of a CAS (or a graphic calculator) to approximate the equilibrium solution of the immigration model

dP/dt = P(1 - P) + 0.3e^-P.

(b) Use a graphing utility to graph the function F(P) = P(1 - P) + 0.3e^-P. Explain how this graph can be used to determine whether the number found in part (a) is an attractor.

(c) Use a numerical solver to compare the solution curves for the IVPs

dP/dt = P(1 - P), P(0) = P₀

for P₀ = 0.2 and P₀ = 1.2 with the solution curves for the IVPs

dP/dt = P(1 - P) + 0.3e^-P, P(0) = P₀

for P₀ = 0.2 and P₀ = 1.2. Superimpose all curves on the same coordinate axes but, if possible, use a different color for the curves of the second initial-value problem. Over a long period of time, what percentage increase does the immigration model predict in the population compared to the logistic model?