Model (a) In Examples 3 and 4 of Section 2.1 we saw that

Chapter 3, Problem 24E

(choose chapter or problem)

Immigration Model (a) In Examples 3 and 4 of Section 2.1 we saw that any solution P(t) of (4) possesses the asymptotic behavior \(P(t) \rightarrow a / b\) as \(t \rightarrow \infty\) for \(P_{0}>a / b\) ab and for \(0<P_{0}<a / b\); 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

\(\frac{d P}{d t}=P(1-P)+0.3 e^{-P}\).

(b) Use a graphing utility to graph the function \(F(P)=P(1-P)+0.3 e^{-F}\)F(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

\(\frac{d P}{d t}=P(1-P), \quad P(0)=P_{0}\)

for \(P_{0}=0.2 \text { and } P_{0}=1.2\) with the solution curves for the IVPs

\(\frac{d P}{d t}=P(1-P)+0.3 e^{-P}, \quad P(0)=P_{0}\)

for \(P_{0}=0.2 \text { and } P_{0}=1.2\). 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?

Text Transcription:

P(t) : a/b

t \rightarrow \infty

t :  for P_0

0 < P_0 < a/b

FP=P(1 - P) + 0.3 e^P-F F(P)

P_0 = 0.2 and P_0 = 1.2