Question

Use a numerical solver to compare the solution curves for the IVPs$\frac{\mathbf{d}\mathbf{P}}{\mathbf{d}\mathbf{t}}\mathbf{=}\mathit{P}\mathbf{(}\mathbf{1}\mathbf{-}\mathit{P}\mathbf{)}\mathbf{,}$${\mathbf{\text{P(0) =P}}}_{\mathbf{0}}$, for ${\mathit{P}}_{\mathbf{0}}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{2}$and ${\mathit{P}}_{\mathbf{0}}\mathbf{=}\mathbf{1}\mathbf{.}\mathbf{2}$with the solution curves for the IVPs $\frac{\mathbf{d}\mathbf{P}}{\mathbf{d}\mathbf{t}}\mathbf{=}\mathit{P}\mathbf{(}\mathbf{1}\mathbf{-}\mathit{P}\mathbf{)}\mathbf{+}\mathbf{0}\mathbf{.}\mathbf{3}{\mathit{e}}^{\mathbf{-}\mathbf{P}}$, $\mathit{P}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}{\mathit{P}}_{\mathbf{0}}$and ${\mathit{P}}_{\mathbf{0}}\mathbf{=}\mathbf{1}\mathbf{.}\mathbf{2}$. 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 increases does the immigration model predict in the population compared to the logistic model?

Short answer

The percentage increases do the immigration model predict in the population compared to the logistic model is$9\%$

Step-by-step solution

Define population model

A population model is a type of mathematical model that is applied to the study of population dynamics.

$\frac{\mathbf{d}\mathbf{P}}{\mathbf{d}\mathbf{t}}\mathbf{=}\mathit{P}\mathbf{(}\mathbf{1}\mathbf{-}\mathit{P}\mathbf{)}\mathbf{+}\mathbf{0}\mathbf{.}\mathbf{3}{\mathit{e}}^{\mathbf{-}\mathbf{P}}$

Find percentage increases of the immigration model

We can plot these solutions in Maple.

Solutions of the first equation are in yellow and red, yellow for $P_0=0.2$,

red for$P_0=1.2$.

Solutions of the second equation are in blue and green, blue for$P_0=0.2$and green forlocalid="1664168308114" $P_0=1.2$.

Over a long period of time, the solution of the first equation will tend to 1 and the solutions of the second equation will tend tolocalid="1664168320176" $1.092155213$(as we showed in (a)).

This is approximately localid="1664168330674" $9\%$ increase.

localid="1664168342146" $$\begin{gathered} with\left(plots\right): \\ p:=dsolve\left(\right\{D\left(P\right)\left(t\right)=P\left(t\right)·(1-P(t\left)\right),P\left(0\right)=0.2\},P(t),type=numeric): \\ q:=dsolve\left(\right\{D\left(P\right)\left(t\right)=P\left(t\right)·(1-P(t\left)\right),P\left(0\right)=1.2\},P(t),type=numeric): \\ r:=dsolve\left(\right\{D\left(P\right)\left(t\right)=P\left(t\right)·(1-P(t\left)\right)+0.3\exp (-P(t\left)\right),P\left(0\right)=0.2\},P(t),type=numeric): \\ {P}_p:=odeplot(p,[t,P\left(t\right)],0...10,colo=yellow): \\ {P}_q:=odeplot(q,[t,P\left(t\right)],0...10,colo=red): \\ {P}_r:=odeplot(r,[t,P\left(t\right)],0...10,colo=blue): \\ {P}_s:=odeplot(s,[t,P\left(t\right)],0...10,colo=green): \\ display(P_p,P_q,P_r,P_s); \end{gathered}$$

By using a graphical tool, the graph is

The percentage increases do the immigration model is$9\%$.