using matlab Use the demographics equation derived in class to make a script which finds the population histories for each of $4$ population groups. Migration into group $2$ will be defined by a pulse function.

SETUP:

parameters $=[$ b$2,$d$1,$d$2,$d$3,$d$4,$t$12,$t$23,$t$34]$ where

b$2,$ birth rate in group $2$

di$,$ death rates in group Pi

tij, transition from group Pi to group Pj due to ageing

M$2,$ rectangular migration pulse defined by $[$Mann$2,$tStartM$2,$tEndM$2]$

initial conditions $=[$ Pinit$1,$Pinit$2,$Pinit$3,$Pinit$4]$

time data $=[$tinit$,$tfinal,delt$]$

RESULTS $($follow conventions or an incorrect answer will result$)$:

P$1,$P$2,$P$3,$P$4,$ group time histories as row vectors

M$2,$ M$2$ time history as a row vector $\%$ Parameters

b$2=\mathrm{.}02$;

d$1=\mathrm{.}004$;

d$2=\mathrm{.}0075$;

d$3=\mathrm{.}012$;

d$4=\mathrm{.}019$;

t$12=\mathrm{.}067$;

t$23=\mathrm{.}04$;

t$34=\mathrm{.}04$;

$\%$ Migration effect M$2$

M$2$ann$=80$;

tStartM$2=20$;

tEndM$2=50$;

$\%$ Initial conditions $($in $1000\text{'}$s for convenience$)$

Pinit$1=1200$;

Pinit$2=1000$;

Pinit$3=800$;

Pinit$4=600$;

$\%$ Solution control data

tinit$=0.$;

tfinal$=100.$;

delt$=0\mathrm{.}5$;

$\%\%$ Your code goes here $\%\%$

$\%$ DISPLAY RESULTS SECTION $-----$ THIS SECTION WILL HELP YOU VISUALIZE YOUR RESULTS

figure

subplot$(3,1,1)$

plot$($t$,$P$1($:$),\text{'}-\text{'},$t$,$P$2($:$),\text{'}--\text{'},$'LineWidth',$2)$

title$(\text{'}$Demographics $1$: P$1,$ P$2$ histories, in $1000\text{'}\text{'}$s$\text{'})$

$\%$xlabel$(\text{'}$t$,$ years'$)$

ylabel$(\text{'}$P$1,$ P$2\text{'})$

legend$(\text{'}$P$1\text{'},\text{'}$P$2\text{'})$

grid on subplot$(3,1,2)$

plot$($t$,$P$1($:$),\text{'}-\text{'},$t$,$P$2($:$),\text{'}--\text{'},$t$,$P$3($:$),\text{'}-.\text{'},$t$,$P$4($:$),\text{'}$:$\text{'},$'LineWidth',$2)$

title$(\text{'}$P$1,$ P$2,$ P$3,$ P$4$ histories, in $1000\text{'}\text{'}$s$\text{'})$

$\%$xlabel$(\text{'}$t$,$ years'$)$

ylabel$(\text{'}$P$1,$ P$2,$ P$3,$ P$4\text{'})$

legend$(\text{'}$P$1\text{'},\text{'}$P$2\text{'},\text{'}$P$3\text{'},\text{'}$P$4\text{'})$

grid on

subplot$(3,1,3)$

plot$($t$,$M$2,\text{'}-\text{'},$'LineWidth',$2)$

title$(\text{'}$Migration effect M$2,$ in $1000\text{'}\text{'}$s$\text{'})$

xlabel$(\text{'}$t$,$ years'$)$

ylabel$(\text{'}$M$2\text{'})$

grid on