小麦旗叶衰老过程不同数学模型拟合比较及衰老特征分析

doi:10.3724/SP.J.1006.2019.81014

Abstract

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

Guo-Feng LYU,Jin-Ping FAN,Bo-Qiao ZHANG,De-Rong GAO,Hui WANG,Ye-Yu LIU,Su-Lan WU,Kai CHENG,Xiu-E WANG. Comparison of different mathematical models describing flag leaf senescence process of wheat and characteristics of leaf senescence process[J].Acta Agronomica Sinica, 2019, 45(1): 144-152.

Figures/Tables 7

Table 1

Table 2

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
类型 Type	品种数 No. of varieties	模型 Model	方程 Equation	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
类型 Type		品种数 No. of varieties		模型 Model		方程 Equation		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 3

Table 4

Table 5

Fig. 1

Table 6

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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 y_max	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	$y=\frac{a}{{{[1+{{\text{e}}^{-}}^{b(x-c)}]}^{\frac{1}{d}}}}$	$c-\frac{\lg d}{b}$	$\frac{a}{{{(d+1)}^{\frac{1}{d}}}}$	$\frac{ab}{{{(d+1)}^{(\frac{1}{d}+1)}}}$	$c-\frac{\ln ({{100}^{d}}-1)}{b}$	$\frac{-99a}{\text{GLAD}}$

类型 Type	品种数 Variety number		DAA10		DAA15		DAA20		DAA25		DAA30
类型 Type	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

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

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
类型 Type	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

类型 Type	TMRS		MRS		GLAD		ARS
类型 Type	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

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

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