Dynamic $($unsteady state$)$ balances require solutions to differential equations. The MATLAB function ode$45$ is a

useful tool that we will use frequently. There is some sample MATLAB code in the Supplementary Resources

Module $($under the MATLAB sub$-$folder$).$ The sample code has lots of explanatory comments to help you figure

out the details.

The problem that is illustrated in the code is that of a Predator$-$Prey problem. Consider the case of a population

of rabbits and foxes on an island. In a simplified model, the number of rabbits is a function of two factors: i$)$ new

births $($proportional to the number of rabbits$)$; ii$)$ deaths $($eaten by foxes$).$ The number of foxes is a function of

two factors: i$)$ new births $($proportional to the number of foxes and the number of rabbits$)$; ii$)$ deaths

$($proportional to the number of foxes$).$

a$)$ Review the MATLAB code and see if you can figure out the governing equations and rate constants for

the various rate laws.

b$)$ Run the code and comment on the plots.

c$)$ Vary some of the rate constants and examine the difference in behavior relative to the baseline case.

d$)$ The model is very simple although the response is complex. Provide at least two $(2)$ ways the model

fidelity could be increased to make it more representative of the actual situation. Note that you do not

need to model this behavior, just come up with some good ways to improve the model