Simple nonlinear＇ model for the relationship between maize yield and cumulative water amount

Both the additive and multiplicative models of crop yield and water supply are polynomial equations, and the number of parameters increases linearly when the growing period is specified. However, interactions among multiple parameters occasionally lead to unreasonable estimations of certain parameters, which were water sensitivity coefficients but with negative value. Additionally, evapotranspiration must be measured as a model input. To facilitate the application of these models and overcome the aforementioned shortcomings, a simple model with only three parameters was derived in this paper based on certain general quantitative relations of crop yield （Y） and water supply （W）. The new model, Y/Y-W＊/（W＊＋w＊）, fits an S or a saturated curve of crop yield with the cumulative amount of water. Three parameters are related to biological factors： the yield potential （Y＊）, the water requirement to achieve half of the yield potential （half-yield water requirement, wh）, and the water sensitivity coefficient （k）. The model was validated with data from 24 maize lines obtained in the present study and 17 maize hybrids published by other authors. The results showed that the model was well fit to the data, and the normal root of the mean square error （NRMSE） values were 2.8 to 17.8% （average 7.2%） for the 24 maize lines and 2.7 to 12.7% （average 7.4%） for the 17 maize varieties. According to the present model, the maize water-sensitive stages in descending order were pollen shedding and silking, tasselling, jointing, initial grain filling, germination, middle grain filling, late grain filling, and end of grain filling. This sequence was consistent with actual observations in the maize field. The present model may be easily used to analyse the water use efficiency and drought tolerance of maize at specific stages.