The population $P$ of a gut bacteria species at time $t$ is described by the differential equation

$\frac{dP}{dt}=kP(r\frac{t^4+3}{t^4+3t^2+1}-P)$

with initial condition $P(0)={10}^4.$

$($a$)$ Write a MATLAB function BactIVP that takes the values $k,r,N$ and tau as inputs, and outputs the following:

$($i$)$ a vector $t$ containing $N$ points from $0$ to tau.

$($ii$)$ a vector $P$ of numerical solution values that are computed using the RungeKutta fourth$-$order method and correspond to the entries in the vector $t.$

$($b$)$ Write MATLAB commands that use BactIVP to produce overlaid plots of the numerical solution to the given initial$-$value problem when $k={10}^{-5},r=2,=100$ and

$($i$)N=10$

$($ii$)N=15$

$($iii$)N=25$

$($iv$)N=50.$

The plot lines corresponding to $N=10,15,25$ and $50$ should be coloured green, red, black and blue respectively.