please use python

When a predator is introduced to an ecosystem some species are seen to rapidly disappear,while others remain. In this problem we will look at modelling this situation. To this end we will consider a predator species $F$ interacting with twe prey species R and S$,$ as dencribed by the foltowing model:

$\frac{dF}{dt}=$aFS$+bFR-cF$

$\frac{dR}{dt}=dR-bFR$

$\frac{dS}{dt}=eS-$aFS

where $a,b,c,d$ and $e$ are all greater than zero.

$($a$)$ Identify the dependent variables, independent variables and parameters in this system.

$($b$)$ Provide the Python code for a function derivs which takes in values for the independent and dependent variables and the parameters, and returns the derivatives $d\frac{F}{d}t,d\frac{R}{d}t$ and $d\frac{S}{d}t.$

$($c$)$ Integrate the system of equations from $t=0$ to $t=200,$ given the following initial conditions $F(0)=0\mathrm{.}1,R(0)=1,S(0)=1,$ with $a=0\mathrm{.}5,b=0\mathrm{.}1,c=0\mathrm{.}2,$

$d=0\mathrm{.}2$ and $e=0\mathrm{.}1.$ Sketch the resulting output for $F(t),R(t)$ and $S(t)$ versus

$t$ and identify which line corresponds to which function. Provide an indication of

the scale on the $y$ axis and identify numerically the maximum value of $F$ on your

sketch.

$($d$)$ Let's now explore the dependence of the dynamics on the parameters. We need some metric to characterise the system. Let's use the maximum population observed in each species in the period $t=150$ to $t=200.$ Sketch the dependence

of these maxima versus $e.$ Vary $e$ in the range $0\mathrm{.}01,1.$ Interpret your results. Is

it possible for all three species to coexist? Keep all other parameters and initial

conditions the same as in part $($c$).$