"Most of the places in these parts that burned out last summer are dry in the summer and wet in the winter. Our idea is to store some of the winter water in cisterns to be held on hilltops. If a hilltop holds enough water to put out one fire on that hill, that should help a lot to douse the wildfires next summer."

"Mr. Laney, I'm a mathematician, sometimes a windsurfer, not a firefighter, and certainly not a hunter," Ecco confessed. "The last time I shot an arrow was when I was put in front of an archery target at the age of 9."

"How did you do, uncle?" 12-year-old Liane asked.

"Let's just say that I didn't hit the bull's eye," Ecco suppressed a grin. He then turned to Laney, "How could I possibly help you?"

"We don't need an archer, Doc," Laney replied. "We think we need a mathematician. We can figure out how big a cistern each hilltop needs in order to put out one fire, maybe two. Before we contact the forestry service, we want to be able to offer them some good ideas about where to put 'em. One of my buddies is an engineer and he said the following might help you."

Laney laid out a sheet of paper. It showed the hills and mountains of a section of forest of the northwest. On the bottom right were the notes.

"These assumptions are simplified, but should give us enough to go by when we approach the forestry service," it began. "We divide the forest into a rectangular grid of cells, each cell corresponding to a square kilometer. The rectangular grid is 50 east-west and 100 north-south, for a total of 5000 cells. During the 100-day fire season, lightning will hit about 1 out of 1000 cells each day.

"Assuming we have good satellite support, we will be able to detect fires soon after they start. Assume our aerial firefighters can put out fires in four cells per day (whether just started or caused by spreading fire).

"If a fire starts because of lightning or is burning in a cell at the start of day d, it will spread to four other cells in the direction of the wind on day d unless it's obstructed. A fire that starts on day d in a cell will burn out in the cell at the end of d+1. So, in an area with no obstructions, a fire that begins at location (i,j) on day d will spread to (i+1,j), (i+2,j), (i+3,j), and (i+4,j) on day d, if the wind is blowing eastward. It will then spread to the square whose corner points are (i,j), (i+4,j), (i+4,j+4), and (i,j+4) by the end of day d+1 if the wind is blowing northward on that day. By the end of d+1, the fire will burn out in (i,j), (i+1,j), (i+2,j), (i+3,j), and (i+4,j). If, on day d+2, the wind is going southward, then the fire will burn in the rectangle, with the corners (i+1, j-4), (i+4, j-4), (i+1, j+4), and (i+4, j+4). You can see the progression by looking at Figures 1 2, 3 and 4, though the first day's wind goes toward the north in those figures.

"Once a square is burnt out, it won't burn again that season. If a fire is put out in a cell, the cell cannot burn again that season. Fires won't burn beyond the forest edge, because that area is either desert or river. There are enough firefighting teams to put out fires in four cells per day. These can be either cells where a fire is just starting or cells where fires are still burning. While a team is fighting a fire in a cell, the fire does not spread from that cell to other cells. Any cell with a cistern can be protected by special cistern management teams, so the cistern will prevent that cell from burning during the whole season without any regular fire team being involved. (That may require a lot of water, but we figure that enough water for one or two fires will probably be sufficient.)

"Every day, lightning strikes five cells, though some may be protected or already burnt out. Wind directions vary regularly: east, north, west, south, east, north, west, south, and so on.

You have 500 cisterns that you may place anywhere. So the only random element is where the lightning strikes. "The question is where to place the cisterns and how to allocate your firefighters (the ones unconcerned with the cisterns). That is, should they go for new fires or old ones? If new ones, which ones? The goal is to minimize the number of cells that burn out over the 100-day fire season. The elk need the cover."

Reader: Ecco and Liane were able to place the cisterns and assign firefighting teams in such a way that, over 30 simulations, only about 850 cells would be lost (out of 5000) with a standard deviation of 100.

"Trouble is," said Laney, "I don't think those engineers can store enough water so that if a cistern puts out a fire one day it will have enough water for the next fire. What happens then?"

Reader: Which arrangement would you use then and with what result? Ecco figured to lose about 1000 cells. Can you do better?

"We also want to show the forestry service that the cisterns help," Laney said. "How many cells would burn out if there were no cisterns, just firefighting teams?"

Reader: In Ecco's best strategy having no cisterns, the number was nearly 1500 cells. Can you do better?

"That's a big help, Doc," Laney said. "I reckon we're ready now."

Last Month's Solution

Figure 5 is a possible solution to the acyclic circuit. In text form, the edges are: AG, GB, AB, BE, BH, GE, HD, GD, BC, and HF.

Figure 6 is a solution to the cyclic circuit. In text form, that's: JK, KE, JL, JH, HK, EH, LG, LA, LB, IC, KC, HC, GF, HD, and LI.

Reader Notes

The question was to see which topology of connections among habitats led to the most rapid decrease in biodiversity (DDJ, October 2000). The most interesting response came from Mark Murphy and Andrew Palfreyman who correctly noted the advantage of chains over other connections and who further questioned what happens when the 10 pairs are divided into two connected components of five. The result was the loss of biodiversity within each group in about 1/3 the time as for all 10. As Murphy points out: "It appears that splitting a population has a dramatic effect on biodiversity." Dramatic and pernicious.

DDJ