A planet supporting life

Published on December 13th, 2012 | by Emma

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# Using mathematics to help manage forests

By Associate Professor Michael McCarthy, University of Melbourne

You may have heard lately that Victoria’s faunal emblem, Leadbeater’s possum, is facing extinction. Its decline is almost inevitable because it relies on large trees for nesting, which have declined due to a combination of timber harvesting and fires.

It would be foolhardy to assume we can control fires in these forests reliably. High fuel loads from accumulation of dead leaves and bark make them one of the most flammable forests in the world. Almost every mountain ash tree in the Wallaby Creek catchment near Kinglake was killed by the Black Saturday fire.

Simple mathematics, like probability theory, can provide useful predictions for difficult environmental problems. While the average fire interval can be estimated, we cannot be sure when or where fires will occur. However, we can represent the occurrence of fire as a randomevent, and determine the chance of forest stand (group of trees) surviving a fire. It is a little like determining the occurrence of fire by rolling a die (pl. dice) each year. If a one turns up, then a fire occurs; otherwise the forest escapes fire for another year. In the case of mountain ash forest, we would be using a 100-sided die rather than a regular six-sided die. The chance of the forest becoming old growth (say, 200+ years) relies on not getting a single 1 in 200 consecutive rolls.

We don’t need to physically roll dice to determine this probability. A little bit of mathematics does the trick. The probability of escaping a fire each year is 0.99, so the chance of a tree reaching one year is 0.99, and the chance of reaching two years is 0.99 x 0.99 (the chance it survives its first year and its second year).

We can continue similarly to determine the chance of the forest reaching 200 years. This is obtained by multiplying 0.99 by itself 200 times, which is 0.99 to the power of 200 = 0.13. In the face of unplanned fires, we would only expect about 10 to 15 per cent of mountain ash forest that is set aside from timber harvesting to be old growth. Given the vast majority of mountain ash forest has been exposed to timber harvesting in the last century, it should come as no surprise that only about one per cent of this forest is now old growth.

This same calculation can help predict timber production in mountain ash forests. Let us assume that the nominal age at which mountain ash trees are harvested is 80 years. The probability a forest stand will reach this age without a fire occurring is about 45 per cent (0.99 to the power of 80). Therefore, we should expect about half of the forest estate to burn prior to it reaching its merchantable age.

This calculation has serious implications when predicting the sustainable timber yield in mountain ash forests. These yields, and the associated harvesting rates, are typically calculated under the
assumption that fires will not occur in the future.

## 2 comments on “Using mathematics to help manage forests”

1. Michael McCarthy on said:

A couple of points:

“Given the vast majority of mountain ash forest has been exposed to timber harvesting in the last century” is not 100% correct. The point I was trying to convey was that the relatively limited areas of old forest were mostly reserved from timber harvesting, yet we shouldn’t be surprised that many of the old forests in those areas have burnt. The regenerating forests in other areas have arisen from both timber harvesting and fire.

Thinking about fires as random events also influences expected water yields from these forests. I have modelled this on my wordpress blog, where I treat the time between fires as a continuous random variable rather than as a discrete random variable as used here. See:

http://mickresearch.wordpress.com/2012/12/21/effects-of-timber-harvesting-on-water-yield-from-mountain-ash-forests/

2. Thanks for the link, Michael!

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