Abstract: Integrated temperature, as a measure of heat, has been widely used in the prediction of crop development period, yield, diseases and insect pests. However, more and more studies showed that the integrated temperature is unstable, and the stability of different integrated temperature models is different. Therefore, it is great significant to analyze and understand the stability of different integrated temperature models for the application of integrated temperature in agricultural meteorological work. In this paper, four linear hypotheses about response of growth and development rate to temperature and five integrated temperature models were made. The first linear hypothesis is that the growth and development rate of crops increases linearly with the increase of temperature when the average daily temperature (T) is higher than the lower limit temperature (Tb). It is the hypothesis of the active integrated temperature model (Aa) and the first effective integrated temperature model (Ae1). The second hypothesis is that when the T is between the lower limit temperature (Tb) and the upper limit temperature (Tu) of crops, the growth and development rate of crops increases linearly with the increase of temperature and reaches the maximum (1.0); when the T exceeds Tu, the growth and development rate of crops remains constant with the increase of temperature. It is the hypothesis of the second effective integrated temperature model (Ae2). The third hypothesis is that when the T is between Tb and Tu, the growth and development rate of crops increases linearly with the increase of temperature and reaches the maximum (1.0); when the T exceeds Tu, the growth and development of crops stagnate. It is the hypothesis of the third effective integrated temperature model (Ae3). The fourth hypothesis is that when the T is between Tb and the optimum temperature (T0), the growth and development rate of crops increases linearly with the increase of temperature and reaches the maximum (1.0); when the T is between T0 and Tu, the growth and development rate of crops decreases linearly with the increase of temperature and decreases to 0.0; when the T exceeds Tu, the growth and development of crops stagnate. It is the hypothesis of the fourth effective integrated temperature model (Ae4). Based on these hypotheses, long time series of crop development period observation data and surface meteorological observation data of two winter wheat stations, three spring maize stations and three summer maize stations in Shanxi Province were selected to calculate the active integrated temperature and four effective integrated temperature. Using the coefficient of variation, the average simulation deviation of the crop growth period and the simulation accuracy of the crop growth period as indicators, the stability of the five integrated temperature models were evaluated. The result showed that the coefficient of variation (CV) of Aa model during different growth stages for three representative crops in each station were between 0.062 and 0.143; the CV of Ae1, Ae2 and Ae3 models were between 0.073 and 0.201; the CV of Ae4 model were between 0.072 and 0.179. That is, when using the CV as an indicator to test the stability of each model, the stability of Aa model was highest, that of the Ae4 model was the second and that of the Ae1, Ae2 and Ae3 models were the weakest. The average simulation deviation (SD) of Aa model for different growth periods of three representative crops in each station were between 1.3 and 5.8 days; the SD of Ae1, Ae2 and Ae3 models were between 1.5 and 6.6 days; the SD of Ae4 model were between 2.2 and 6.7 days. The simulation accuracy (SA) of Aa model for different growth periods of three representative crops in each station were between 39.5% and 92.3%; the SA of Ae1, Ae2 and Ae3 models were between 28.6% and 87.2%; the SA of Ae4 model were between 26.5% and 84.6%. That is, when using the SD and SA as the indicators, the Aa model had the best simulation effect for different growth periods of the crops and had the highest stability. The simulation accuracy and stability of Ae1, Ae2 and Ae3 models had no significant difference and were higher than those of Ae4 model. For spring maize and summer maize, the stability of each integrated temperature model was consistent from emergence to tasseling and from tasseling to maturity, and the stability of each integrated temperature model from emergence to tasseling was higher than that from tasseling to maturity. While the stability of each integrated temperature model for winter wheat from jointing to heading and from heading to maturity varied from region to region. Therefore, in practical applications, it is also necessary to select the appropriate base point temperature according to the crop planting region, variety relationship and growth period, and to select a more stable integrated temperature model based on the comprehensive analysis of the stability of multiple integrated temperature models.

Key words: Integrated temperature, Linear hypotheses, Stability, Coefficient of variation, Simulation accuracy of growth period