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Acta Agronomica Sinica ›› 2019, Vol. 45 ›› Issue (1): 144-152.doi: 10.3724/SP.J.1006.2019.81014

• RESEARCH NOTES • Previous Articles     Next Articles

Comparison of different mathematical models describing flag leaf senescence process of wheat and characteristics of leaf senescence process

Guo-Feng LYU1,2(),Jin-Ping FAN2,Bo-Qiao ZHANG2,De-Rong GAO2,Hui WANG2,Ye-Yu LIU2,Su-Lan WU2,Kai CHENG2,Xiu-E WANG1,*()   

  1. 1 Nanjing Agricultural University, Nanjing 210095, Jiangsu, China
    2 Institute of Agricultural Sciences for Lixiahe Region of Jiangsu Province / Yangzhou Sub-center of National Wheat Improvement Center, Yangzhou 225007, Jiangsu, China
  • Received:2018-02-03 Accepted:2018-10-08 Online:2018-11-07 Published:2018-11-07
  • Contact: Xiu-E WANG E-mail:lgf@wheat.org.cn;xiuew@njau.edu.cn
  • Supported by:
    This study was supported by the Key Research and Development Programs of Jiangsu Province(BE2017340);the National Key Research and Development Programs of China(2017YFD0100800)

Abstract:

To clarity leaf senescence patterns and characteristics of wheat, we used 91 and 105 varieties or advanced lines in 2011 and 2012 respectively, to fit the leaf senescence process by Logistic, Gompertz, and Richards models. The varieties tested were classified into stay green (SG), moderately stay green (MSG), moderately non-stay green (MNSG) and non-stay green (NSG) types. Development of the flag leaf senescence was divided into initial senescence stage, rapid senescence stage and terminal senescence stage in all variety types. The senescence rates of the three stages showed a slow-fast-slow changing pattern. The decrease of green leaf area (% GLA) after anthesis occurred mainly in the middle and late senescence stages. Logistic, Gompertz and Richards models were able to simulate the leaf senescence process of all variety types and the fitness of Richards and Gompertz models was better than that of Logistic model. The fitting goodness of Gompertz model ranked as NSG type > MNSG type > MSG type > SG type. There was a significant difference in time of maximum rate of senescence, green leaf area duration and average rate of senescence derived from Gompertz equation in all varieties. The average rate of senescence showed a trend of NSG type > MNSG type > MSG type > SG type, while the remaining two parameters showed a trend of SG type > MSG type > MNSG type > NSG type. Our results indicate that Gompertz model is more suitable than Logistic model to describe the leaf senescence process of wheat and the characteristic parameters for leaf senescence process can be used to evaluate stay green difference between wheat varieties.

Key words: wheat, leaf senescence model, curve fitting, stay green

Table 1

Characteristic parameters of equations fitted by Gompertz, Logistic, and Richards models"

模型
Model
数学式
Equation
最大衰老速度时间
TMRS
最大衰老速度时的%
GLA ymax
最大衰老速度
MRS
绿色叶面积持续期
GLAD
平均衰老速度
ARS
Gompertz $y=a{{\text{e}}^{-{{\text{e}}^{-b(x-c)}}}}$ $c$ $\frac{a}{\text{e}}$ $\frac{ab}{e}$ $c-\frac{\ln (-\ln 0.01)}{b}$ $\frac{-99a}{\text{GLAD}}$
Logistic $y=\frac{a}{1+{{\text{e}}^{-b(x-c)}}}$ $c$ $\frac{a}{2}$ $\frac{ab}{4}$ $c-\frac{\ln 99}{b}$ $\frac{-99a}{\text{GLAD}}$
Richards $y=\frac{a}{{{[1+{{\text{e}}^{-}}^{b(x-c)}]}^{\frac{1}{d}}}}$ $c-\frac{\lg d}{b}$ $\frac{a}{{{(d+1)}^{\frac{1}{d}}}}$ $\frac{ab}{{{(d+1)}^{(\frac{1}{d}+1)}}}$ $c-\frac{\ln ({{100}^{d}}-1)}{b}$ $\frac{-99a}{\text{GLAD}}$

Table 2

Decline process of %GLA of four types of variety tested"

类型
Type
品种数
Variety number
DAA10 DAA15 DAA20 DAA25 DAA30
2011 2012 2011 2012 2011 2012 2011 2012 2011 2012 2011 2012
延绿 SG 10 22 99.2 a 96.6 a 98.7 a 91.7 a 95.6 a 82.3 a 89.0 a 56.5 a 53.0 a 11.4 a
中等延绿 MSG 21 32 98.0 ab 96.3 a 95.7 b 90.0 a 91.5 ab 75.1 b 74.5 b 37.8 b 9.8 b 3.7 b
中等早衰 MNSG 28 40 97.5 b 94.6 b 95.0 b 85.8 b 89.7 b 64.4 c 45.9 c 13.2 c 0.8 c 0.7 c
早衰 NSG 32 11 98.3 ab 92.7 c 95.2 b 81.5 c 85.6 c 26.9 d 11.6 d 0.5 d 0 c 0 c

Table 3

Curve fitting of Logistic, Gompertz, and Richards models for the flag leaf senescence process of varieties tested"

类型
Type
品种数
No. of varieties
模型
Model
方程
Equation
方程系数Coefficients 模型适合性
Fitness of model
a b c d R2 RMSE
2011
延绿
Stay green
10 Logistic $y=a/1+{{\text{e}}^{-b(x-c)}}$ 98.40 -3.67 1.35 0.9536 4.149
Gompertz $y=a{{\text{e}}^{-{{\text{e}}^{-b(x-c)}}}}$ 98.53 -3.15 1.48 0.9577 3.961
Richards $y=a/{{[1+{{\text{e}}^{-b(x-c)}}]}^{\frac{1}{d}}}$ 99.20 -1.97 6.30 1.57×10-4 0.9426 4.662
中等延绿
Moderately stay green
21 Logistic $y=a/1+{{\text{e}}^{-b(x-c)}}$ 95.59 -4.93 0.93 0.9646 6.436
Gompertz $y=a{{\text{e}}^{-{{\text{e}}^{-b(x-c)}}}}$ 96.71 -2.95 1.09 0.9710 5.798
Richards $y=a/{{[1+{{\text{e}}^{-b(x-c)}}]}^{\frac{1}{d}}}$ 97.05 -2.62 7.40 6.85×10-8 0.9713 5.858
中等早衰
Moderately non-stay green
28 Logistic $y=a/1+{{\text{e}}^{-b(x-c)}}$ 95.43 -5.39 0.72 0.9846 4.760
Gompertz $y=a{{\text{e}}^{-{{\text{e}}^{-b(x-c)}}}}$ 96.91 -3.12 0.86 0.9893 3.976
Richards $y=a/{{[1+{{\text{e}}^{-b(x-c)}}]}^{\frac{1}{d}}}$ 97.10 -3.07 5.76 2.98×10-7 0.9887 4.098
早衰
Non-stay green
32 Logistic $y=a/1+{{\text{e}}^{-b(x-c)}}$ 97.39 -7.58 0.33 0.9948 3.167
Gompertz $y=a{{\text{e}}^{-{{\text{e}}^{-b(x-c)}}}}$ 97.47 -5.07 0.50 0.9949 3.124
Richards $y=a/{{[1+{{\text{e}}^{-b(x-c)}}]}^{\frac{1}{d}}}$ 97.36 -3.86 3.54 8.40×10-6 0.9880 4.832
2012
延绿
Stay green
22 Logistic $y=a/1+{{\text{e}}^{-b(x-c)}}$ 94.31 -2.84 0.78 0.9516 7.150
Gompertz $y=a{{\text{e}}^{-{{\text{e}}^{-b(x-c)}}}}$ 96.29 -1.95 0.98 0.9609 6.424
Richards $y=a/{{[1+{{\text{e}}^{-b(x-c)}}]}^{\frac{1}{d}}}$ 96.29 -2.04 6.01 3.62×10-5 0.9596 6.564
中等延绿
Moderately stay green
32 Logistic $y=a/1+{{\text{e}}^{-b(x-c)}}$ 94.81 -2.79 0.52 0.9735 5.808
Gompertz $y=a{{\text{e}}^{-{{\text{e}}^{-b(x-c)}}}}$ 98.08 -1.78 0.73 0.9797 5.075
Richards $y=a/{{[1+{{\text{e}}^{-b(x-c)}}]}^{\frac{1}{d}}}$ 98.07 -1.78 3.52 7.01×10-3 0.9795 5.118
类型
Type
品种数
No. of varieties
模型
Model
方程
Equation
方程系数Coefficients 模型适合性
Fitness of model
a b c d R2 RMSE
中等早衰
Moderately non-stay green
40 Logistic $y=a/1+{{\text{e}}^{-b(x-c)}}$ 93.31 -3.42 0.20 0.9809 5.367
Gompertz $y=a{{\text{e}}^{-{{\text{e}}^{-b(x-c)}}}}$ 96.30 -2.12 0.39 0.9840 4.917
Richards $y=a/{{[1+{{\text{e}}^{-b(x-c)}}]}^{\frac{1}{d}}}$ 95.96 -2.16 4.57 1.17×10-4 0.9836 4.992
早衰
Non-stay green
11 Logistic $y=a/1+{{\text{e}}^{-b(x-c)}}$ 93.46 -4.30 -0.23 0.9916 3.783
Gompertz $y=a{{\text{e}}^{-{{\text{e}}^{-b(x-c)}}}}$ 94.44 -3.17 -0.08 0.9917 3.768
Richards $y=a/{{[1+{{\text{e}}^{-b(x-c)}}]}^{\frac{1}{d}}}$ 94.03 -3.65 0.02 0.50 0.9898 4.216

Table 4

Curve fitting of Gompertz model for the flag leaf senescence process of varieties tested"

年份
Year
品种数
Number of varieties
R2均值
Mean of R2
R2变幅
Range of R2
RMSE均值
Mean of RMSE
RMSE变幅
Range of RMSE
2011 91 0.997 0.948-1.000 2.062 0.007-10.640
2012 105 0.996 0.964-1.000 2.546 0-8.750

Table 5

Parameters of three flag leaf senescence stages in different stay-green types of varieties"

类型
Type
衰老起始期Initial senescence stage 快速衰老期Rapid senescence stage 衰老结束期Terminal senescence stage
T1 (d) P1 (%) V1 (% d-1) T2 (d) P2 (%) V2 (% d-1) T3 (d) P3 (%) V3 (% d-1)
2011
延绿 SG 28.9 a -31.4 a -1.1 a 5.9 a -60.3 a -11.4 a 1.7 a -6.2 a -4.0 a
中等延绿 MSG 25.8 b -30.8 a -1.2 b 4.5 ab -59.1 a -15.1 a 1.3 ab -6.1 a -5.3 a
中等早衰 MNSG 23.6 c -30.8 a -1.3 c 4.2 b -59.1 a -14.8 a 1.2 b -6.1 a -5.2 a
早衰 NSG 21.9 d -31.0 a -1.4 d 3.4 b -59.6 a -19.5 b 1.0 b -6.2 a -6.9 b
2012
延绿 SG 23.7 a -30.6 a -1.3 a 7.1 ab -58.8 a -9.0 a 2.1 ab -6.0 a -3.1 a
中等延绿 MSG 21.3 b -31.3 a -1.5 b 7.7 a -60.1 a -8.6 a 2.2 a -6.2 a -3.0 a
中等早衰 MNSG 19.7 c -30.5 a -1.6 b 6.3 bc -58.6 a -10.2 a 1.8 bc -6.0 a -3.6 a
早衰 NSG 17.3 d -30.0 a -1.7 c 4.4 c -57.5 a -13.6 b 1.3 c -5.9 a -4.7 b

Fig. 1

Curve fitting flag leaf senescence process by Gompertz model for four types of stay green varieties g1 to g4 represent curves fitted by Gompertz model for SG, MSG, MNSG, and NSG types, and gla_g1 vs t1, gla_g2 vs t2, gla_g3 vs t3, and gla_g4 vs t4 represent %GLA values of the four variety types, respectively."

Table 6

Parameters of flag leaf senescence in four stay-green types of variety"

类型
Type
TMRS MRS GLAD ARS
2011 2012 2011 2012 2011 2012 2011 2012
延绿SG 32.14 a 26.71 a -16.64 a -13.09 a 37.60 a 32.08 a -2.67 a -3.00 a
中等延绿MSG 27.92 b 24.55 b -24.20 b -11.74 a 31.04 b 31.10 a -3.14 b -3.15 a
中等衰老MNSG 25.27 c 22.63 c -22.53 b -13.90 a 28.57 c 27.65 b -3.38 c -3.42 b
早衰NSG 23.34 d 19.94 d -29.44 c -15.29 b 25.90 d 24.67 c -3.78 d -3.85 c
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