$1.$ Does the system have a steady state? Explain your answer.

$2.$ Change parameter values, in particular those representing the strengths of activation and$/$or inhibition; report the impact of each change.

$3.$ Add an independent input variable and also a variable that receives material from X$6.$ What are the effects of these additions?

$4.$ Convert the independent input variable into a dependent variable. How does the system respond to this conversion?

$5.$ The question we want to answer now is: Which combination of values for g and h give us the highest value of X$6?$

To answer the question, set up PLAS code that allows you to execute $1,000$ Monte Carlo simulations with the system. In particular, replace g $=-1$ and h $=2$ with random numbers between $-2$ and $0$ and between $0$ and $4,$ respectively. Run the system several times and look at the responses.

Study the results $($including g and h$)$ in the $$Message Board.$$ Close this board after every simulation, unless you want to accumulate results.

Copy the results of one simulation into Excel and custom$-$sort them by X$6$ to see which combinations of g and h tend to lead to low or high output values. Do the $$best$$ values for g and h represent the absolutely best solution within the set ranges for the parameter values or could there be better solutions? Summarize your findings in a report.

X$1\text{'}=$ a$1$ X$0$ X$4^$g $-$ b$1$ X$1-$ c$1$ X$1^2$ X$4^$h

X$2\text{'}=$ b$1$ X$1-$ b$2$ X$2^\mathrm{.}4$

X$3\text{'}=$ b$2$ X$2^\mathrm{.}4-$ b$3$ X$3^\mathrm{.}7$

X$4\text{'}=$ b$3$ X$3^\mathrm{.}7-$ b$4$ X$4^\mathrm{.}8$

X$5\text{'}=$ c$1$ X$1^2$ X$4^$h $-$ c$2$ X$5$

X$6\text{'}=$ c$2$ X$5-$ c$3$ X$6$

X$1=2$

X$2=4$

X$3=6$

X$4=\mathrm{.}5$

X$5=\mathrm{.}1$

X$6=\mathrm{.}1$

X$0=1$

a$1=5$

b$1=\mathrm{.}2$

b$2=2$

b$3=2$

b$4=2$

c$1=3$

c$2=0\mathrm{.}5$

c$3=0\mathrm{.}5$

g $=-4$

h $=2$

t$0=0$

tf $=10$

hr $=\mathrm{.}1$