$[$Lotka$-$Volterra model$]$ This dynamical system models the population of two species interacting as predator and prey.

$x^{}=x-xy$

$y^{}=xy-y$

i Solve this equation using MATLAB ode $45$ solver $($or with any suitable solver in any language$)$ from a nonzero initial condition and plot $x(t)$ and $y(t)$ with respect to $t.$

ii Find out the fixed points $($steady$-$states$)$ of this system.

$[4$ points$]$

$[2$ points$]$

iii Linearize around each fixed points and find the Jacobian matrix $A.$

$[4$ points$]$

Solve without MATLAB