MAI, CAI and rotation
Forest growth explained: current, periodic and mean annual increment, why CAI crosses MAI at its peak, types of rotation, and a worked yield table example.
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When should a forest be harvested? Fell too early, and the trees are small and the site's growth potential is wasted. Fell too late, and the stand has slowed down, carrying old timber that adds little each year while the land could be growing a new crop. Foresters answer this with a few measures of growth, called increments, and with the concept of rotation: the planned age at which an even-aged crop is felled.
The key relationship, which exams test every year, is between current annual increment (CAI) and mean annual increment (MAI). It applies to volume, basal area, diameter or height, and to single trees or whole stands.
Kinds of increment
Volume of a stand, V, increases with age, t, following an S-shaped growth curve. Increments describe the rate:
In practice, CAI is rarely measured year by year. PAI over a 5- or 10-year period, from repeated measurements or a yield table, is used as its estimate.
Gross increment includes trees that died or were thinned; net increment covers only the surviving stand. Total MAI includes thinnings removed earlier.
Why CAI crosses MAI at MAI's peak
- Early on, the stand is small, and CAI rises rapidly as trees build crowns and occupy the site. MAI rises too, but more slowly, because it averages in the slow early years.
- CAI peaks first, at the point of inflection of the growth curve: often 10–25 years for fast species and later for slow ones. It then declines as trees age, competition increases and respiration rises.
- MAI keeps rising as long as CAI is above it, because each new year adds more than the current average. When CAI falls below MAI, the average starts to fall.
- MAI is therefore at its maximum exactly when CAI = MAI. This is a mathematical property, the same as how a student's average mark changes when a new mark is above or below it.
On a graph, the CAI curve rises above MAI, peaks, and then cuts through the MAI curve at MAI's highest point.
Rotation
The rotation is the planned number of years between establishing a crop and its final felling. Different objectives give different rotations:
| Rotation | Basis | Typical length |
|---|---|---|
| Physical | The age a species can live or remain sound on the site | Longest |
| Silvicultural | The age at which the stand stays healthy and regenerates reliably | Long |
| Technical | The age at which trees reach the size needed for a product, such as poles, pulpwood or sawlogs | Depends on the product |
| Maximum volume production | The age of maximum MAI (CAI = MAI) | Medium |
| Financial | The age that maximises net present value or land expectation value | Usually shortest |
| Maximum income | The age that maximises average annual net revenue, without discounting | Medium |
The rotation of maximum volume production gives the most wood per hectare per year, averaged over successive rotations. It is the classic yardstick for public forests.
The financial rotation discounts future returns at an interest rate. It is usually shorter than the volume rotation, because money tied up in standing timber has an opportunity cost. Faustmann's formula (1849) gives the land expectation value, and the rotation that maximises it is the economically optimal one when the land can be used for successive crops.
A worked yield table
Finding the rotation of maximum volume
A plantation yield table gives total volume (m3 per ha) at ages 10, 20, 30, 40 and 50 years: 30, 110, 200, 260 and 300.
- MAI: 30 ⁄ 10 = 3.0; 110 ⁄ 20 = 5.5; 200 ⁄ 30 = 6.7; 260 ⁄ 40 = 6.5; 300 ⁄ 50 = 6.0.
- PAI per year over each decade: 10–20 = 8.0; 20–30 = 9.0; 30–40 = 6.0; 40–50 = 4.0.
MAI is highest at 30 years. PAI in the 30–40 period (6.0) has already fallen below the MAI at 30 (6.7), which confirms that MAI culminates around 30 years. The rotation of maximum volume production is about 30 years. At a 6% interest rate, the financial rotation would come out earlier.
Factors affecting rotation and growth
- Species: eucalyptus and poplar plantations peak in MAI at about 6–10 years, while teak, sal and conifers take several decades. Indian teak plantations have traditionally been planned on rotations of 50–80 years, and shorter rotations are being tested for improved planting stock.
- Site quality: better sites grow faster and reach MAI culmination earlier, at a higher peak.
- Thinning: including thinning yields raises total MAI and can change the shape of the curves.
- Markets: the product and its price, such as pulpwood, poles or veneer logs, affect the technical and financial rotations.
- Carbon goals: longer rotations store more carbon in the forest, but shorter ones produce more harvested wood. Rotation decisions now often involve carbon accounting.
Key points
- CAI is growth in one year; PAI is average yearly growth over a period; MAI is total volume divided by age.
- CAI peaks before MAI. MAI is at its maximum where CAI = MAI.
- The rotation of maximum volume production is the age of maximum MAI.
- Rotations can be physical, silvicultural, technical, volume, financial or income-based. Financial rotations are usually the shortest.
- Faustmann's formula gives the land expectation value for financial rotations.
- Better sites reach MAI culmination earlier and higher.
What the exams ask
- Define CAI, PAI and MAI, and explain the relationship between CAI and MAI.
- Draw the CAI and MAI curves and mark the rotation of maximum volume production.
- Calculate MAI and PAI from a yield table and identify the rotation.
- List types of rotation and their bases.
- Explain why financial rotations are shorter.
Described in general terms: the kind of examination, not a past paper. Laisha never reproduces exam questions.
Questions people ask
Why not harvest when CAI is at its peak?
At the CAI peak, the stand is growing fastest. Felling it then would waste the high growth still to come, and MAI is still rising. Wood production over successive rotations is greatest when felling at the MAI peak, when the long-run average is highest.
Does a tree stop growing after its MAI peaks?
No. A tree, or a stand, continues to add volume for many years after MAI culminates. It just adds less each year than its lifetime average. Old trees still grow, and large old trees can add significant carbon each year.
What is a yield table?
A yield table shows how a normal, fully stocked even-aged stand develops with age on sites of different quality. It lists stems per hectare, mean diameter, top height, basal area and volume, often including thinnings. The Forest Research Institute in Dehradun has published yield tables for teak, sal, deodar, chir pine and eucalyptus.
Sources
- Avery, T. E. and Burkhart, H. E. Forest Measurements, 5th edition. McGraw-Hill, 2002.
- Davis, L. S., Johnson, K. N., Bettinger, P. and Howard, T. E. Forest Management: To Sustain Ecological, Economic, and Social Values, 4th edition. McGraw-Hill, 2001.
- Chaturvedi, A. N. and Khanna, L. S. Forest Mensuration and Biometry. International Book Distributors, Dehradun, revised edition 2011.
- Faustmann, M. "Calculation of the value which forest land and immature stands possess for forestry" (1849). English translation in Journal of Forest Economics 1, 7–44 (1995).
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