SimuLab 18: Percolating Forests

1. Start the application called Blaze.

2. Choose a tree density, here called p, with a value of 0.500 by using the mouse to click on the numbers. This means that the probability is one-half that a given square will have a tree on it; the same as the probability that a coin flip will be heads.

3. Click on Go and see what happens. A picture of the forest appears on your screen, with trees shown in green. A fire starts burning on the left side of the forest and spreads toward the right. As each tree burns it turns red, then the ashes left behind turn blue.

4. There is a helicopter in this game that we will use later on to try to stop the fire from spreading. For now, though, just watch the forest burn. Don't move the mouse or click it until the fire burns out.

5. How far does the fire spread across the forest? Does it percolate-that is, does it reach the opposite side of the grid?

6. Grow and burn more forests using a tree probability of 0.500. Do any of these forests have fires that percolate? Carry out at least 10 trials, and note in your lab book what percent of your trials produce fires that percolate.

7. Repeat the experiment using tree probability p = 0.600. Burn 10 or more forests. What fraction of these p = 0.600 forests have a fire that percolates?

8. Repeat the experiment using tree probability p = 0.700. Burn 10 or more forests. What fraction of these p = 0.700 forests have a fire that percolates?

9. Stop and think. Do you notice a significant change in the number of fires that percolate between probabilities p = 0.5 and 0.7? Guess: Will the number of fires that percolate be radically different for p = 0.3 than for p = 0.5? Will the number of fires that percolate be radically different for p = 0.9 than for p = 0.7? Try a few at these lower and higher values of p to check out your guesses.

The tree probability p at which approximately half of the fires percolate is called the critical probability. Below the critical probability, the fire is unlikely to percolate, spread across the forest. Above the critical probably the fire almost always percolates.

 Q7.18: What do you think is the critical probability for the forest modeled in Blaze?

 Q7.19: What's so special about the critical probability? If you were a forest ranger, you might want to keep the forest density below the critical probability, so that if a fire starts the whole forest won't burn down.

 Q7.20: Can you describe the relationship between the different test tubes of Jell-O at various concentrations and the forest generated by the Blaze simulation?

Now let's go back to the Blaze program and test your skills as a forest ranger!

1. Start the Blaze program again. Set the tree probability at p = 0.600.

2. Your job is to maneuver your helicopter and drop water to stop the spread of the fire. The helicopter moves toward the position where you place the cursor and drops water when you click the mouse button. Any tree that you dump water on does not burn. The idea is to wet trees ahead of the fire and to stop its spread. Are there certain places-so-called tree bridges-where you can wet just one or two trees and cut off the fire easily?

3. When the fire is finished burning, your score flashes on the screen. The score depends on the number of trees you saved, and the quantity of water used; your score decreases the more water you use to stop the fire. If your score is 0, you were not able to stop the fire from spreading. Try again!

 Q7.21: What happens if you play with a tree probability of p = 0.5, or p = 0.7? Which of these cases leads to a higher score?

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