The Three Species Problem.

In class, the predator$-$prey problem of paramecium and

yeast was explored. Now, let's consider the interaction

between three groups of imaginary creatures $x,Y,$ and $Z$

competing for the same food. The populations of these

three species are governed by the following Lotka$-$

Volterra equations:

$\frac{dx}{dt}=0\mathrm{.}75x(1-\frac{x}{20})-1\mathrm{.}5xy-0\mathrm{.}5xz$

$\frac{dy}{dt}=y(1-\frac{y}{25})-0\mathrm{.}75xy-1\mathrm{.}25yz$

$\frac{dz}{dt}=1\mathrm{.}5z(1-\frac{z}{30})-xz-yz$

where $x,y,$ and $z$ are the population per unit area of

species $x,Y,$ and $Z.$ Note that there are additional

quadratic terms, differing signs, and carrying capacities

unique to each species when compared with the

predator$-$prey example presented in lab.

Part $($a$)$

Before you start programming, write down the

discretized governing equations for this problem using

the forward Euler method. Use subscripts $k$ and $k+1$ to

denote the know values at timestep $k$ and the values to