Modify the code below so that creates Figure $1$ exactly as it appears.

Note that in this case $K=10$ and $r=0\mathrm{.}5.$

The solution to the logistic equation $P^{\text{'}}=rP(1-\frac{P}{K})$ can be written as

$P(t)=\frac{K{P}_0{e}^{rt}}{K+P_0(e^{rt}-1)}$

State $($e$.$g$.,$ type$)$ the Logistic differential equation describing the dynamics of a Population $($P$)$ that

varies in time $(t)$ with a growth rate $(r)$ and carrying capacity $($K$).$ State its classification.

The parts in the code that need to be changed are highlighted in yellow. Perform the following task and report them in accordance to the $$What to Submit$$ section:

$1.$ Modify the code below so that creates Figure $1$ exactly as it appears.

$2.$ Note that in this case K$=10$ and r$=0\mathrm{.}5.$

$3.$ The solution to the logistic equation P$=$ rP $(1-$P$/$K$)$

$4.$ State $($e$.$g$.,$ type$)$ the Logistic differential equation describing the dynamics of a Population $($P$)$ that

varies in time $($t$)$ with a growth rate $($r$)$ and carrying capacity $($K$).$ State its classification.

$5.$ The parts in the code that need to be changed are highlighted

$\%$ Code for Logistic Slope Field with Solutions.

r$=0\mathrm{.}5$; $\%$growth rate

K$=10$; $\%$Carrying capacity

t$=0$:$0\mathrm{.}2$:$20$; $\%$ vector for time

$\%$ Define the values of x and y that we will use to create the slope field

$\%$ We use tS and pS to identify the values of time and population

$\%$ to be used for slope field.

$[$tS$,$ pS$]=$ meshgrid$(\_\_\_\_\_)$;

$\%$ Define the values of the slope $($p$\text{'})$

s $=$ r$*$P$*(1-$P$/$K$)$;

$\%$ The following line is needed to find the size of each vector $(1,$s$)$ so

$\%$ that we can scale all of the arrows to have the same magnitude$=1$

L $=$ sqrt$(1+$ s$.^2)$;

figure$(2)$

hold on $\%$this command tell MATLAB to draw everything on the current figure

quiver$($tS$,$pS$,1./$L$,$s$./$L$,0\mathrm{.}5),$ axis tight

xlabel$(\text{'}$time$\text{'})$;

ylabel$(\text{'}$population$\text{'})$;

title$(\text{'}$Direction field for dp$/$dt $=$ r$*$p$*(1-$p$/$K$)\text{'})$;

$\%$ The nex part adds the solution curves to the slope field

$\%$ there are $3$ meaningful cases which we will draw in red, green, magenta

p$0=\_\_\_\_\_\_\_\_$;

p$=\_\_\_\_\_\_\_\_$;

plot$($t$,$p$,\text{'}$g$\text{'})$

p$0=$K;

p$=\_\_\_\_\_\_\_\_$;

plot$($t$,$p$,\text{'}$r$\text{'})$

p$0=5$;

p$=\_\_\_\_\_\_\_\_$;

plot$($t$,$p$,\text{'}$m$\text{'})$

hold off $\%$this command tell MATLAB to stop drawing on figure