Agent-Based Models and Optimal Control in Biology: A Discrete Approach - Mathematical Concepts and Methods in Modern Biology - page 153

￿ Cost function: a function that assigns a value to each possible solution. In general,
the cost function will depend upon a combination of state variables and other
quantifiable aspects of the model dynamics.
￿ Optimal solution: in a general optimization problem the optimal solution is a
solution that achieves the stated optimization goal in the best way. In an optimal
control problem, it is the solution that is associated with the minimum value of
the cost function.
Use the modified rabbit and grass model file RabbitsGrass.nlogo at
http://admg.vbi.vt.edu/software/rabbitsgrass-netlogo-files/ to answer the questions in
this section. Once you have opened the file, you will find a description of the interface
settings under the Info tab.
Exercise 5.5. On the left-hand side of the model, you will notice sliders for
the following values: initial-rabbits, initial-grass, birth-
threshold, grass-grow-rate, grass-energy, move-cost,
world-size, poison-strength . Are these model parameters or state vari-
ables? Explain.

Exercise 5.7. Suppose the control objective is to minimize the number of rabbits
while also minimizing the number of days on which poison is used. What is the
difference between a solution and the optimal solution?

Exercise 5.6. List three state variables and one control variable for this model.

Exercise 5.8. Click restore-defaults ,set poison-strength to 0.5, and
then click setup .Now,click poison repeatedly until you have run through the
entire simulation time. Note the cost. Now click degrade-poison so that it is in
the On position. Click setup and then repeatedly click poison again. What do
you notice about the cost this time? Is it higher or lower? How can you explain the
difference?

Exercise 5.9. Click restore-defaults ,set poison-strength to 0.5, and
click setup . Using the buttons poison and don't poison as you wish, run
through the simulation several times, taking note of the cost for each solution. What
is the lowest cost you are able to achieve? If you now reduce poison-cost to 5
and increase rabbit-cost to 15, you will notice that the cost when using the same
schedule is now different. Is it lower or higher? Why do you think this is? Try to find
a schedule that reduces the cost to less than 10,000. How about less than 5000? Turn
degrade-poison? to the On position and try to achieve similarly low costs. Do
you notice any patterns in the solutions in each case?

Exercise 5.10. Run the same schedule twice (for 1 run each time). You should
notice that the costs are different each time. Why does this happen? Explain why this
is an important issue, and suggest a method for dealing with this issue.

Exercise 5.11. Click restore-defaults , then setup . Change num-runs
to 100, then click go . Note the average cost over these 100 runs. Now change the
number of runs to 25, then 3 (note that unchecking the view updates box toward