$(4$ pts$)($Lotka$-$Volterra predator$-$prey model$)$ A popular model to model predator$-$prey

population dynamics is given as follows

$x_1^{}=x_1-x_1{x}_2$

$x_2^{}=-x_2+x_1{x}_2$

where $x_1$ and $x_2$ represent the population densities of the prey and predator, respectively,

and $,,,$ are positive parameters.

$($a$)(2pts)$ Identify all the equilibrium points of the system.

$($b$)(2$ pts$)$ Take $====1$ and the initial conditions $(x_1,x_2)(0)=(0\mathrm{.}5,0\mathrm{.}5).$

Then simulate the system $(3)$ in the time interval $0,10$ and plot your solution in

the $(x_1,x_2)$ plane.

Next, keeping $x_2(0)=0\mathrm{.}5$ fixed, choose the other initial condition $x_1(0)$ from the

set $\{1,1\mathrm{.}5,2,2\mathrm{.}5\},$ and repeat the above $($plot on the same figure$).$ You have just

created a phase portrait of the Lotka$-$Volterra dynamics!

$($You may want to use the Matlab function ode$23$ for this exercise $($similar usage as

ode$45))$

$($bonus: $2$ pts $)$ Show that the quantity

$V(x)=x_1-ln(x_1)+x_2-ln(x_2),x_1>0,x_2>0$

remains constant along any trajectory of the dynamical system $(3).$ This is called a

conserved quantity of a system. This is a generalization of the concept of total energy

$($kinetic $+$ potential$)$ of a mechanical system.