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In this project, we consider logistic models of population growth that have
been modified to account for the fact that the carrying capacity of the
population may change with time.

Recall that the usual logistic differential equation is:

$P^{\prime}=kP(1-\frac{P}{L})$ where $L$ represents the carrying capacity. \
We will now consider $L$ to be a function of time; that is our first attempt
to find a suitable differential equation is:

(1) \ \ \ \ $P^{\prime}=kP(1-\frac{P}{L(t)})$

\ where $L(t)$ is now a function which gives the variational carrying
capacity.$\,\,$ \ We assume that $0<L(t)$ for all $t>0$. \ We can also
assume that $0<P(0)<L(0)$.

\bigskip

\begin{center}
{\Large Calculus Review Questions}

{\Large \bigskip }
\end{center}

Question One. \ Why is it reasonable to assume that $P(0)<L(0)$?

\bigskip

Question Two. \ If $L$ is assumed to be a smooth positive function, given an
initial condition $P(0)=P_{0}$, are we guaranteed to have a unique solution
over our region (in the $t,P$ plane) of interest (which is what region?)? \
Why? \ Note: \ there are both existence and uniqueness considerations here.
\ 

\bigskip

Question Three. \ If we assume that $L(t)$ is an increasing function, what
can we say about the relative growth rate $\frac{P^{\prime }}{P}$?

\bigskip

Question Four. \ \ If \ $L(t_{1})=P(t_{1})$ what can you say about $%
P^{\prime }(t_{1})$? \ If $P^{\prime }(t_{1})=0$, what can you say about $%
L(t_{1})$ and $P(t_{1})$?

\bigskip

Question Five. \ Here is a challenge: \ if $L$ is a strictly increasing
smooth function, conclude that $P$($t)<L(t)$ for all $t>0$ (hint: \ what can
you say about $L^{\prime }(t)$)? \ 

\bigskip

Question Six. \ Here is a challenge: \ If $L$ is a smooth, strictly
increasing function, conclude that $P$($t)<L(t)$ for all $t>0$ (hint: \ what
can you say about $L^{\prime }(t)$)? \ 

\bigskip

Question Seven: \ Calculate $P^{\prime \prime \prime }$ and evaluate at $%
t_{1}$ and leave your answer in terms of $L,L^{\prime }$ and $L^{\prime
\prime }$.

\bigskip

Question Eight: \ Tell how we would get $P^{(n)}$. \ We can obtain $P^{(n)}$
merely by starting with an expression for $P^{(n-1)}$ and taking the
derivative. \ That is, we differentiate both sides of (1) to obtain $%
P^{\prime \prime }$, then both sides of ``$P^{\prime \prime }=$'' to obtain $%
P^{\prime \prime \prime }$, and so on.

\bigskip

\bigskip

\begin{center}
{\Large Computer Project}
\end{center}

\bigskip

\bigskip

Consider the following candidates for $L$:

$L_1(t)=L_0+w_1\,t\,\,\,\,\,(w_1>0)$

$L_2(t)=L_0+w_2\,ln(1+t)\,\,\,\,(w_2>0)$

$L_3(t)=L_0+w_3t^2\,\,\,\,(w_3>0)\,\,$

Use a computer ODE solver to plot direction fields (of the ODE sans the
initial condition) and of \ solutions to (1) using the appropriate carrying
capacity function $L_i$ and given the following table of initial conditions
and model parameters:

\begin{tabular}[t]{|l|l|l|l|l|l|}
\hline
$P(0)$ & $L(0)$ & $k$ & $w_1$ & $w_2$ & $w_3$ \\ \hline
4 & 200 & .03 & 1.4 & 15 & .002 \\ \hline
10 & 200 & .03 & 1.4 & 15 & .002 \\ \hline
50 & 200 & .03 & 1.4 & 15 & .002 \\ \hline
4 & 100 & .05 & 1.4 & 15 & .002 \\ \hline
10 & 100 & .05 & 1.4 & 15 & .002 \\ \hline
4 & 200 & .03 & 2.0 & 45 & .004 \\ \hline
\end{tabular}

Example: \ for the first row, you would plot the direction fields and
solutions to three ODE's:

$P^{\prime }=.03$ $P(1-\frac{P}{200+1.4t})$, $P^{\prime }=.03P(1-\frac{P}{%
200+15\ln (1+t)})$, \ $P^{\prime }=.03P(1-\frac{P}{200+.002\text{ }t^{2}})$,
each of which has initial condition $P(0)=4$.

Note: \ you will be plotting the graphs of \ the solutions to 6*3 = 18
different ODE's. \ Don't worry, you won't have to turn in 18 different
graphs.

Write a few paragraphs describing the effects that \ $P(0)$, $L(0),$ $%
k,\,L_i $ and $w_i$ have on the solution. \ Include some carefully chosen
graphs as part of your explanation. Note that the first row of data is taken
from studies on the population growth of the United States (where $P$ is in
millions).

\bigskip

2. \ Now assume that $L(t)\,=\,200\,+\,1.4t\,+40\,sin(\frac{2\pi t}{50})$,
and that $k=.05$.

Describe the carrying capacity function. \ Use the computer to plot a
direction field and to plot 3 solutions where $P(0)=4,\,P(0)=10,\,P(0)\,=%
\,50 $. \ Notice that your solution curves for $P$ have local maxima and
minima. \ Comment on what must be happening at such points.

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