seasonal predator$-$prey system, where the prey growth is dependent on a nutrient

source, can be modeled by the system of dimensionless differential equations

dx

dt $=1$ x $$ Axy

a $+$ x

dy

dt $=$ Axy

a $+$ x $$ y $$ Byz

b $+$ y

dz

dt $=$ Byz

b $+$ y $$ z

where x represents the nutrient source, y represents the prey population, and z represents

the predator population. The parameters govern the rates at which nutrient are consumed,

as well as measures of the organism$$s ability to subsist on low levels of nutrients.

For this problem, let A $=5,$ B $=2,$ a $=8/115,$ and b $=90/460.$ Use x $(0)=0\mathrm{.}1,$ y $(0)=4,$

and z $(0)=1.$

$($a$)$ Solve this system of equations using MATLAB ODE integrators for t in $[0,40].$ Find

the value of time where the system seems to settle into a steady state, with potentially

predictable cycles of behavior.

$($b$)$ Compare the solution components for two different MATLAB ODE algorithms and the

forward Euler method for systems of equations. Describe differences you see, if any, in

the approximations generated by the methods. What size of the time step is needed for

the forward Euler approximations to be comparable to those generated by MATLAB?