Law of Maximum

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This article is about the crop growth principle. For the "Law of the Maximum" established during the French Revolution, see general maximum.

The Law of Maximum also known as Law of the Maximum is a principle developed by Arthur Wallace which states that total growth of a crop or a plant is proportional to about 70 growth factors. Growth will not be greater than the aggregate values of the growth factors. Without the correction of the limiting growth factors; nutrients, waters and other inputs are not fully or judicially used resulting in wasted resources.[1][2][3]


The growth factors are arithmetically additive. The factors range from 0 for no growth to 1 for maximum growth. Actual growth is calculated by the total multiplication of each growth factor. For example, if ten factors had a value of 0.5, the actual growth would be:

  • 0.5 x 0.5 x 0.5 x 0.5 x 0.5 x 0.5 x0.5 0.5 x 0.5 x 0.5 or 0.001 of optimum (0.1%).

If each of ten factors had a value of 0.9 the actual growth would be:

  • 0.9 x 0.9 x 0.9 x 0.9 x 0.9 x 0.9 x0.9 0.9 x 0.9 x 0.9 or 0.349 of optimum (34.9%).

Hence the need to achieve maximal value for each factor is critical in order to obtain maximal growth.

Demonstrations of "Law of the Maximum"[edit]

The following demonstrates the Law of the Maximum. For the various crops listed below, one, two or three factors were limiting while all the other factors were 1. When two or three factors were simultaneously limiting, predicted growth of the two or three factors was similar to the actual growth when the two or three factors were limits individually and then multiplied together.

Percent of Optimum growth

(Measured growth)

predicted growth

(calculated growth)

Low phosphorus 83%
Low nitrogen 71%
Both low phosphorus actual result 57% 59% (83% x 71%)
and low nitrogen
Low moisture 66%
Low nitrogen 27%
Both low moisture actual 18% 18% (66% x 27%)
and low nitrogen
Bush beans
Nickel addition 42%
Copper addition 71%
Vanadium addition 39%
All 3 metals actual 10% 11% (42% x 70% x 39%)
Presence of insects 37%
Nutrient deficiencies 78%
Both presence of insects and nutrient deficiencies actual 29% 29% (37% x 78%)
Tomato – poor physical soil conditions
Correction with organic matter 43%
Correction with P.A.M. 82%
No corrections 33% 18% (66% x 27%)
Corrections with both organic matter and P.A.M. 100%

Growth Factors[edit]

A. Adequacy of Nutrients[edit]

  1. Nitrogen
  2. Phosphorus
  3. Potassium
  4. Calcium
  5. Magnesium
  6. Sulfur
  7. Iron
  8. Zinc
  9. Manganese
  10. Copper
  11. Boron
  12. Chlorine
  13. Cobalt
  14. Sodium
  15. Nickel

B. Non-nutrient elements and nutrients excesses that cause toxicities (stresses)[edit]

  1. Nickel
  2. Cadmium
  3. Nickel
  4. Copper
  5. Boron
  6. Sodium chloride
  7. Aluminium
  8. Bicarbonate
  9. others

C. Interactions of the nutrients[edit]

  1. Cation ratios (metals)
  2. Carbon:Nitrogen ratio
  3. recycling available soil
  4. ratios of heavy metals
  5. ratios of all the nutrients
  6. Nitrate vs ammoniacal nitrogen

D. Soil Conditioning requirement and physical processes[edit]

  1. Low pH (soluble Al)
  2. high pH
  3. salinity, EC (electrical conductivity) either too low or too high,
  4. ratios of sand vs. silt vs. clay
  5. presence of rocks
  6. soil organic matter
  7. soil aeration
  8. limestone
  9. soil moisture conditions (frequency of rain or irrigation)
  10. depth to water table
  11. other subsoil conditions
  12. earthworms
  13. cation exchange capacity
  14. soil erosion (dust and water)
  15. redox, soil crusting
  16. structures of soil that give aeration and water penetration
  17. fixation of nutrients by soil
  18. hydrophobic conditions
  19. other aspects of soil quality, slope and topography of lands

E. Additional biology[edit]

  1. Favorable and unfavorable microorganisms
  2. Mycorrhizae
  3. Diseases
  4. Insects
  5. Weeds
  6. other macro organisms
  7. and anaerobic decomposition products

F. Weather factors[edit]

  1. Intensity, amount, frequency of rain events
  2. Wind
  3. Hail
  4. Snow
  5. day and night temperature
  6. times of freezing, humidity
  7. light hours and percent of light saturation (cloud cover)
  8. photoperiods
  9. air pollution

G. Management[edit]

  1. Crop rotations
  2. timing of operations
  3. choice of tillage
  4. use of appropriate cultivars and varieties
  5. irrigation
  6. fertilizer distribution
  7. use of soil conditioners to correct problems

External links[edit]


  1. ^ Wallace, A.; Garn A. Wallace (1993). "Limiting Factors, High Yields, and Law of the Maximum". Horticultural Reviews. 15. doi:10.1002/9780470650547.ch10. 
  2. ^ Arthur Wallace (1994). "Generalized environmentally sound rules for use of fertilizers". Communications in Soil Science and Plant Analysis. 25 (1 & 2): 77–86. doi:10.1080/00103629409369009. 
  3. ^ Arthur Wallace; Richard E. Terry (1998). "Handbook of Soil Conditioners: Substances That Enhance the Physical Properties of Soil": 29–39. ISBN 0-8247-0117-8.