$[$Solve $($a$),($b$),($c$)$ by hand$-$writing, $($d$),($e$)$ in computer$]$

Rabbits and wolves live in the same area. The only food source for wolves is rabbits. Rabbits die either because of wolf attacks or because of limited food resources. When a wolf attacks a rabbit, it is successful in $1\%$ of cases per year. The coefficient for the competition for available food between the rabbits is $0\mathrm{.}08/$year$.$ Rabbits have a net birth rate of $6$ rabbits per one rabbit each year. Since the number of rabbits in the field is also limited$,$ wolves compete to catch rabbits with a coefficient of $0\mathrm{.}03/$ year. The wolves that are not successful in catching rabbits will eventually die.

$($a$)$ Write the ODE$-$based models of changes in the number of rabbits and wolves in this area over time.

$($b$)$ Identify equilibrium point$($s$).$

$($c$)$ Check if the equilibrium point$($s$)$ are stable or not by using the eigenvalue$-$based approach.

$($d$)$ It is given that initially there are $4$ rabbits and $2$ wolves in the area. Use ODE solver in Python to identify the number of rabbits and wolves three years later. Generate also a plot of the numbers versus time.

$($e$)$ Given that there are initially $4$ rabbits and $2$ wolves in the area, use ODE solver in Python to identify how many years it takes to reach steady state. Generate also a plot of the numbers versus time.