Plant and Soil

, Volume 298, Issue 1–2, pp 81–98 | Cite as

A dynamic model for the combined effects of N, P and K fertilizers on yield and mineral composition; description and experimental test

  • Kefeng Zhang
  • Duncan J. Greenwood
  • Philip J. White
  • Ian G. Burns
Regular Article


This paper describes an integrated model for calculating the interactive effects of N, P and K fertilizers on crop response by combining routines from separate N, P and K models which used readily available inputs. The new model uses the principle of the ‘Law of the Minimum’ to calculate actual daily increments in plant weight and uptake of each nutrient based on the nutrient least able to meet the plant requirements, although account is also taken of soil factors such as the dependence of soil solution K on the level of mineral N. The validity of the model was tested against the results of 4 field experiments with different combinations of crop species, times of harvest, and levels of N, P and K fertilizers. The integrated model gave good overall predictions of the plant dry weight (excluding fibrous roots) and %N of the dry weight. However, predictions of its %P and %K in the dry weight were less satisfactory, especially in the luxury range. Simulation studies with low levels of nutrients showed that, while most interactive effects on final yield conformed to the Law of the Minimum type of response, the inter-dependence of K and nitrate concentrations in the soil solution resulted in responses to K at different levels of N that were better represented by the Mitscherlich equation or the Multiple Limitation Hypothesis. Thus the adaptability of the model allowed it to reproduce crop responses predicted by three quite different non-mechanistic static equations previously used in the literature to summarise nutrient interaction data. This suggests that the model has the potential to provide a mechanistic basis for interpreting factorial NPK fertilizer trials. Opportunities for improving the model were provided by the experimental findings that %P was strongly correlated with %N throughout the entire range of treatments; that K fertilizers failed to increase %K when %N was low; that fertilizer N increased plant %K when the level of K fertilizer was substantial but not otherwise; and that fertilizer-P depressed plant %K. The model validation also showed that there is a need to improve the parameterization for major crops.


Crop response Fertilizer Interaction Model Law of the minimum Multiple limitation hypothesis 



coefficient defining the dependence of N crit on W


depth of soil containing 90% of roots

D1D2D3 and D4

different statistical methods for assessing discrepancies between measured and simulated parameters

K1 and K2

coefficients in a growth rate equation




weight of plant K as a percentage of W


critical %K


a term for the depressive effect in growth associated with N fertilizer


root length when plant weight is W


Law of the Minimum


Multiple Limitation Hypothesis


Mitscherlich Growth Hypothesis


nitrogen response model




fertilizer-N equivalent supplied by Rhizobium


weight of plant N as a percentage of W


critical %N


weight of plant P as a percentage of W


critical %P




phosphate response model


%P at which growth ceases


potassium response model


the daily increments in W calculated from N_ABLE, PHOSMOD and POTAS respectively


the daily increment in W after taking account of R N R P and R K


dry weight of the entire plant excluding fibrous roots



The authors would like to thank the UK Department for Environment, Food and Rural Affairs who financed this work through project HH3507SFV.


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Copyright information

© Springer Science+Business Media B.V. 2007

Authors and Affiliations

  • Kefeng Zhang
    • 1
  • Duncan J. Greenwood
    • 1
  • Philip J. White
    • 1
    • 2
  • Ian G. Burns
    • 1
  1. 1.Warwick HRIThe University of WarwickWellesbourneUK
  2. 2.Scottish Crop Research InstituteInvergowrieUK

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