Effect of Drip Irrigation, N, K, and Zn Coupling on $Pn$ of Densely Cultivated Apple on Dwarf Rootstock in Xinjiang, China

Abstract

This study aimed to determine the effect of irrigation amount ($W$), nitrogen ($N$), potassium ($K$), and zinc ($Zn$) on the net photosynthetic rate ($Pn$) of closely planted apple trees on dwarf rootstocks in arid areas of Xinjiang. Taking the “Royal Gala” apple as the experimental material, a mathematical model for $Pn$ was established using the principle of four-factor five-level quadratic regression with a general rotation combination design. The results show that: (1) The regression equations reached significant levels (F = 37.06 > F_0.01(11.11) = 4.54). (2) The effect of $W$, $N$, $K$, $Zn$ on $Pn$ is significant with relative importance $W$ > $N$ > $Zn$ > $K$. (3) The results of single factor analysis showed that with an increase in W, N, K, and Zn, $Pn$ exhibits an n-shaped parabolic response. (4) The positive coupling between $W$ and $N$ is significant, and the positive coupling between $W$ and $Zn$ is also significant. (5) Analysis of the interaction between sets of three factors revealed that $W$, $N,$ and $Zn$ could be combined to best effect, with the maximum value reaching 12.77 $\mathsf{\mu }\mathrm{mol}·{\mathrm{m}}^{-2}·{\mathrm{s}}^{-1}$. Compared with $W\times K\times Zn$ and $W\times N\times K$, the combination of $W\times N\times Zn$ reduces $W$ by 9.2% and 6.3%, respectively, which indicates its suitability for use in the dry and water deficient planting environment in Xinjiang. (6) Within the 95% confidence level, when $W$ is 258–294.75 mm, $N$ is 33.44–39.51 $\mathrm{kg}/{\mathrm{hm}}^2$, $K$ is 53.82–69.39 $\mathrm{kg}/{\mathrm{hm}}^2,$ and $Zn$ is 6.46–7.84 $\mathrm{kg}/{\mathrm{hm}}^2$, the net photosynthetic rate reaches 11 $\mathsf{\mu }\mathrm{mol}·{\mathrm{m}}^{-2}·{\mathrm{s}}^{-1}$.

1. Introduction

Photosynthesis is the basic link between energy absorption, fixation, material transformation, and distribution in terrestrial ecosystems, an important biochemical process of material circulation and energy exchange on the surface, and an important factor affecting crop productivity, providing essential nutrients for crop growth [1,2,3,4]. Under natural conditions, in addition to the plant’s own physiological characteristics, the main factors affecting any change in $Pn$ (net photosynthetic rate) are the amount of irrigation ($W$) and fertilization ($F_t$), but other environmental factors also play a decisive role, which is more obvious in agricultural crops [5,6,7]. Therefore, research into the effect of $W$ and $F_t$ on $Pn$ has always been a hot research issue globally [8].

As an important crop, apple is widely planted, particularly because of its adaptability to the environment and its high nutritional value [9]. According to recent statistics, China has become the largest apple producer in the world, with the planting area and output accounting for about 50% of the global total, and the export volume of apples also ranks among the top in the world [10,11]. Xinjiang is located in an arid area. In this area there is sufficient sunlight and a large temperature difference between day and night, resulting in conditions that are extremely effective for the accumulation of fruit sugar; it has, therefore, become an important high-quality apple production base in China [12]. As a new type of planting mode, dwarfing rootstock combined with close planting has the advantages of high mechanization potential, early bearing, high quality, and high land utilization rate, and has become an important approach in apple cultivation [13]. However, the development of the apple industry in Xinjiang is seriously restricted by limited rainfall, extensive use of irrigation and fertilizer, and serious salinization of the soil [14,15]. Therefore, it is very important to study how the interaction between irrigation and fertilization affects the physiological characteristics of apples during growth and development, in order to build an effective irrigation and fertilization management model, ensuring the efficient production and sustainable development of dwarf, closely planted apples in arid areas of Xinjiang [16,17,18].

Irrigation and fertilization are important factors affecting crop growth, photosynthetic characteristics, and production efficiency. Controlling the relationship between $W$ and $F_t$ can help to promote crop growth, improve $Pn$, and enhance crop yield and water and fertilizer use efficiency [19,20,21]. Of these factors, $W$ is particularly important with respect to photosynthesis [22]. It has been found that a good soil water environment can promote the opening of leaf stomata, increase the absorption of ${CO}_2$ by leaves, facilitate the transport of photosynthetic products and reduce inhibition of photosynthesis due to an accumulation of photosynthetic products in leaves [23,24]. Therefore, appropriately increasing $W$ will promote the $Pn$ of plant leaves [23,25,26]. Soil water deficit will induce plant roots to produce abscisic acid, which will send inhibition signals to reduce plant growth, with signals being transmitted to the crown through the xylem under the effect of water transmission, there will be the closure of the crown stomata, leading to a reduction in $Pn$ [27,28,29,30]. Too much $W$ will lead to poor soil aeration, and decreased root activity, and will indirectly affect photosynthesis [31]. In addition, the nitrogen concentration in most plant leaves is closely related to carbon fixation via photosynthesis [32,33]. Research shows that increasing $N$ can improve the photosynthetic performance of leaves and increase the $Pn$, thereby promoting the yield and harvest of crops [34]. However excessive nitrogen will reduce the activity of key enzymes in photosynthesis, which is not conducive to the improvement of photosynthetic performance [35,36,37].

Although potassium does not directly participate in the synthesis of important organic substances in plants, it is frequently an activator of enzymes, and thus indirectly participates in important metabolic activity in plants and has a significant effect on $Pn$ [38,39,40,41]. Topdressing with potassium at the fruit expansion stage can improve $Pn$ quality and yield [42]. Potassium stress will limit transmission, transformation, and other non-stomatal factors associated with light energy, reduce photosynthetic capacity and cause a decline in biomass accumulation [43,44]. Zinc is an essential trace element for crops and has important nutritional and physiological functions playing a role in stabilizing, regulating, or catalyzing various enzymes [45,46,47]. Increasing $Zn$ can enhance leaf photosynthesis and facilitate the transportation of photosynthetic products [48,49]. During the growth of fruit trees, the loss of $Zn$ will cause $IAA$ synthesis in the plant to cease, and eventually lead to the development of young leaves and stems of the plant being blocked; this is usually referred to as “small leaf disease” and “clump leaf disease” and ultimately affects the photosynthetic ability of leaves [50,51].

In general, the existing research on irrigation and fertilizer regulation has mostly focused on the effects of $W$ and $F_t$ as individual factors or on different fertilizer ratios on plant growth, Research into the combined effects of $W$, $N$, $K$, $Zn$, on photosynthesis is relatively rare [18,52,53]. Especially in arid areas, it is very important to study the synergistic effect of irrigation and fertilization. In this study, the apple variety “Royal Gala” on dwarf rootstock closely planted in the arid region of Xinjiang was used as the experimental material, in order to determine the effect of irrigation amount ($W$), nitrogen ($N$), potassium ($K$), and zinc ($Zn$) on net photosynthetic rate ($Pn$) of apple trees planted closely with dwarf stocks in arid areas of Xinjiang. The field experiment was carried out using a four-factor five-level quadratic regression general rotation combination design [54,55,56]. The effects of $W$, $N$, $K,$ and $Zn$ on $Pn$ were analyzed in order to provide reference data and effective suggestions for extending drip irrigation and the use of closely planted apples on dwarf rootstock in arid areas of Xinjiang.

2. Materials and Methods

2.1. Overview of the Study Area

The study site is located in the Apple Garden of the 10th Regiment of Alar City, the First Division of the Xinjiang Production and Construction Corps (40°39′14″ N, 81°16′21″ E), which belongs to a typical inland extremely arid climate zone. The annual precipitation is about 150 $\mathrm{mm}$, the average annual temperature is about 11 $℃$, the annual evaporation is about 2100 $\mathrm{mm}$, the annual sunshine hours amount to about 2900, the frost-free period is more than 200 $\mathrm{days}$, and the groundwater depth is more than 3 $\mathrm{m}$. The tested soils were sandy with a field holding capacity of 13.7% from 0 to 120 $\mathrm{cm}$ soil depth, an average capacity of 1.52 $\mathrm{g}/{\mathrm{cm}}^3$, organic matter content of 11.05 $\mathrm{g}/\mathrm{kg}$, available phosphorus and available boron contents of 3.20 $\mathrm{mg}/\mathrm{kg}$ and 0.60 $\mathrm{mg}/\mathrm{kg}$, rapid potassium content of 33 $\mathrm{mg}/\mathrm{kg}$, alkaline nitrogen and total nitrogen contents of 10 $\mathrm{mg}/\mathrm{kg}$ and 176 $\mathrm{mg}/\mathrm{kg}$, ammonium nitrogen and nitrate nitrogen contents of 2.01 $\mathrm{mg}/\mathrm{kg}$ and 1.00 $\mathrm{mg}/\mathrm{kg}$, respectively, $PH$ of 8.71, and $EC$ value of 154.60 $\mathsf{\mu }\mathrm{s}/\mathrm{cm}$.

2.2. Experimental Materials

The experiment was conducted from 10 April to 1 September 2022, using “Royal Gala”, an early maturing apple variety. The M195 rootstock was the trees grafted and the apple trees used in the experiment were six years old. The total growth period (TGP) was 150 days. The anthesis fruit setting stage (AFS) was from 20 April to 1 May, the young fruit development stage (YFS) from 2 May to 1 June, the fruit expansion stage (FES) from 2 June to 1 August, and the fruit ripeness stage (FRS) from 2 August to 20 August. The planting density 2850 $\mathrm{plants}/{\mathrm{hm}}^2$, the plant spacing 1 m, and the row spacing 3.5 $\mathrm{m}$. The drip pipe was set 60 $\mathrm{cm}$ from the ground on each side of the fruit trees; the drop head flow was 4 $\mathrm{L}/\mathrm{h}$, the drop head spacing was 50 $\mathrm{cm}$, and the tanks were pressure differential fertilization tanks, which were individually configured for each test plot. The distribution of precipitation and daily average temperature is shown in Figure 1. The total precipitation was 195.3 $\mathrm{mm}$, mainly concentrated in July and August.

2.3. Experimental Design

$W$, $N$, $K$, $Zn$ were treated as independent variables, and net photosynthetic rate ($Pn$) was used as the dependent variable. A total of 23 treatments were set up using a four-factor five-level quadratic regression general rotation combination design (with 50% implementation) [54,55,56]. Each treatment was repeated three times with an experimental plot area of 35 m² ($10 \mathrm{m}\times 3.5 \mathrm{m}$) and 10 apple trees were planted in each plot, 690 apple trees in total. The irrigation method is drip irrigation. Irrigation was controlled by a water meter (precision 0.001 m³) during the test. Since 10 April, the water has been irrigated every 4 days, 30 times. Each test plot was equipped with its own fertilizer tank; fertilizer was dissolved before application. During the experiment, nitrogen was supplied in the form of urea (46%), potassium as potassium sulfate ($K_2{SO}_4$ 52%) and zinc as zinc sulfate heptahydrate ($Zn{SO}_4$, 56%) [57,58]. During the total growth period of the apple, a total of 15 times fertilization. Fertilize every 8 days. The level coding for the four factors is presented in

Table 1. Factor level code table for the four factors.

$\mathbf{z}$  $\left({\mathit{x}}_{\mathit{t}}\right)$	${\mathit{z}}_1$  $\left(\mathit{W}\right)$	${\mathit{z}}_2$  $\left(\mathit{N}\right)$	${\mathit{z}}_3$  $\left(\mathit{K}\right)$	${\mathit{z}}_4$  $\left(\mathit{Z}\mathit{n}\right)$
$z_{2j}$($r$)	800.0	150.0	300.0	22.5
$z_{0j}+{∆}_j$(1)	700.0	135.0	270.0	15.8
$z_{0j}$(0)	550.0	112.5	225.0	11.3
$z_{0j}-{∆}_j$(−1)	400.0	90.0	180.0	6.7
$z_{1j}$($-r$)	300.0	75.0	150.0	0
${∆}_j=\frac{z_{2j}-z_{ij}}{2r}$	148.6	22.3	44.6	6.7
$x_j=\left(z_j-z_{0j}\right)/{∆}_j$	$x_1=\frac{z_1-550}{148.6}$	$x_2=\frac{z_2-112.5}{22.5}$	$x_3=\frac{z_3-225}{44.6}$	$x_4=\frac{z_4}{6.7}$

[56]. Therein, $z_t$ represents the coded formula, and $x_1,x_2,x_3,x_4$ represent the coded levels of $W$, $N$, $K$, and $Zn,$ respectively [56]. Five levels were assigned to $W$, $N$, $K$, $Zn$, and the coded levels were $\pm r,\pm 1,0$. The specific irrigation and fertilization regimes were executed in accordance with

Table 2. Irrigation and fertilization regimes.

Factors	Stage	Factor Levels
		−1.682	−1	0	1	1.682
$K$  $(\mathrm{g}/35{\mathrm{cm}}^3)$	AFS	26.25	31.50	39.25	47.25	52.50
	YFS	105.00	126.00	157.00	189.00	210.00
	FES	288.75	346.50	431.75	519.75	577.50
	FRS	105.00	126.00	157.00	189.00	210.00
	TGP	525.00	630.00	785.00	945.00	1050.00
$N$  $(\mathrm{g}/35{\mathrm{cm}}^3)$	AFS	26.25	31.50	39.38	47.25	52.50
	YFS	131.25	157.50	196.88	236.25	262.50
	FES	105.00	126.00	157.40	189.00	210.00
	FRS	0	0	0	0	0
	TGP	262.50	315.00	393.75	472.50	525.00
$W$  ${(\mathrm{m}}^3/35{\mathrm{cm}}^3)$	AFS	0.53	0.70	0.96	1.23	1.40
	YFS	1.58	2.10	2.89	3.68	4.20
	FES	6.30	8.40	11.55	14.70	16.80
	FRS	2.10	2.80	3.85	4.90	5.60
	TGP	10.50	14.00	19.25	24.50	28.00
$Zn$  $(\mathrm{g}/35{\mathrm{cm}}^3)$	AFS	0	7.10	17.50	27.90	35.00
	YFS	0	7.10	17.50	27.90	35.00
	FES	0	7.10	17.50	27.90	35.00
	FRS	0	0	0	0	0
	TGP	0	21.30	52.45	83.70	105.00

Note: TGP represents the total growth period of the apple tree, AFS represents the anthesis fruit setting stage of the apple tree, YFS represents the young fruit development stage of the apple tree, FES represents the fruit expansion stage of the apple tree, FRS represents the fruit ripeness stage of the apple tree.

.

2.4. Data Acquisition and Analysis

(1)

Sample collection

At the FRS, we randomly selected three apple trees in each test plot and selected three leaves from the middle of each tree, east, south, west, and north for the observation of the net photosynthetic rate, and measured the net photosynthetic rate three times for each leaf. The $Pn$ was determined using a $Li-6800$ portable photosynthesis analysis system (USA) between 9:00 and 11:00 a.m. During the assay, the photometric flux setting was 1800 $\mathsf{\mu }\mathrm{mol}·{\mathrm{m}}^{-2}·{\mathrm{s}}^{-1}$, and the blade temperature was set to 28 $℃$.

(2)

Construction of regression models

Based on the quadratic regression general rotation combination design, the regression statistics were generated; with $p$ variables, the general formula of the quadratic orthogonal regression model is:

$$\widehat{y}=b_0+\sum _{j=1}^p{b}_j{X}_j+\sum _{j=1}^p{b}_{jj}{X}_j^2+\sum _{i<j}^p{b}_{ij}{X}_i{X}_j$$

(1)

where $\widehat{y}$ is the response variable; $b_0$ is a constant term; $b_j$ is the coefficient for $X_1$; $b_{jj}$ is the quadratic term coefficient; $b_{ij}$ is the interaction term coefficient; $p$ is the number of dependent variables; $1\le j\le p$, $1\le i\le p$ and $i<p$.

The corresponding regression coefficients were calculated:

$$b_0=A\sum _{i=1}^{N_a}{y}_i+E\sum _{j=1}^p\left(\sum _{i=1}^{N_a}{x}_{ij}^2{y}_i\right)$$

(2)

$$b_j=e^{-1}\sum _{i=1}^{N_a}{x}_{ij}{y}_i=\frac{B_j}{D_j}$$

(3)

$$b_{hj}=m_c^{-1}\sum _{\begin{array}{c}i=1\\ h>j\end{array}}^{N_a}\left(x_{ih}{x}_{ij}\right)y_i=\frac{B_{hj}}{D_{hj}}$$

(4)

$$b_{jj}=\left(F_a-G\right)\sum _{i=1}^{N_a}{x}_{ij}^2{y}_i+G\sum _{j=1}^p\sum _{i=1}^{N_a}{x}_{ij}^2{y}_i+E\sum _{i=1}^{N_a}{y}_i$$

(5)

where $A,E,N_a,F_a,G,m_c,p$ all represent the necessary parameters of quadratic regression general rotation combination design, $x\mathrm{a}\mathrm{n}\mathrm{d}y$, respectively, represent coding and the corresponding $Pn$.

(3)

Data processing and mapping

Data were processed using Microsoft Excel 2020 (Microsoft, Redmond, WA, USA); frequency analysis and regression analysis were conducted in SPSS 25.0 (IBM SPSS, Chicago, IL, USA), and one-way effect plots as well as two- and three-factor interac-tion plots were produced by Origin 2021 pro (Northampton, MA, USA) and Matlab 2021 (Natick, MA, USA).

3. Results and Analysis

3.1. Analysis of Pn in Apple Leaves

It can be seen from

Table 3. Net photosynthetic rate ($Pn$ ) change for different treatment.

No.	Factor Code				Implementation				$\mathit{P}\mathit{n}$  $(\mathsf{\mu }\mathbf{mol}·{\mathbf{m}}^{-2}·{\mathbf{s}}^{-1})$
	${\mathit{x}}_1$	${\mathit{x}}_2$	${\mathit{x}}_3$	${\mathit{x}}_4$	$\mathbf{W}$ (mm)	$\mathbf{N}$ (kg/hm2)	$\mathbf{K}$ (kg/hm2)	$\mathbf{Z}\mathbf{n}$ (kg/hm2)
T1	1	1	1	1	700	135.0	270	15.825	12.960 a
T2	1	1	−1	−1	700	135.0	180	6.675	11.226 ef
T3	1	−1	1	−1	700	90.0	270	6.675	10.428 hi
T4	1	−1	−1	1	700	90.0	180	15.825	10.797 gh
T5	−1	1	1	−1	400	135.0	270	6.675	10.227 ij
T6	−1	1	−1	1	400	135.0	180	15.825	10.106 ij
T7	−1	−1	1	1	400	90.0	270	15.825	10.062 ij
T8	−1	−1	−1	−1	400	90.0	180	6.675	9.517 k
T9	−1.682	0	0	0	300	112.5	225	11.250	9.900 jk
T10	1.682	0	0	0	800	112.5	225	11.250	11.692 d
T11	0	−1.682	0	0	550	75.0	225	11.250	10.119 ij
T12	0	1.682	0	0	550	150.0	225	11.250	11.168 efg
T13	0	0	−1.682	0	550	112.5	150	11.250	11.025 ef
T14	0	0	1.682	0	550	112.5	300	11.250	11.431 de
T15	0	0	0	−1.682	550	112.5	225	0	10.897 fg
T16	0	0	0	1.682	550	112.5	225	22.500	11.659 d
T17	0	0	0	0	550	112.5	225	11.250	12.733 ab
T18	0	0	0	0	550	112.5	225	750.000	12.580 bc
T19	0	0	0	0	550	112.5	225	750.000	12.643 abc
T20	0	0	0	0	550	112.5	225	750.000	12.752 ab
T21	0	0	0	0	550	112.5	225	750.000	12.677 ab
T22	0	0	0	0	550	112.5	225	750.000	12.265 c
T23	0	0	0	0	550	112.5	225	750.000	12.575 bc

Note: $Pn$ is the net photosynthetic rate. T1–T23, respectively, represent 23 treatments set up in the experiment. $Pn$ is expressed in average values, and different letters indicate significant difference at $p<0.05$.

that there are significant differences between T1, T2, T3 T4. The results show that $Pn$ is affected only by the $W$. Significant differences were found between T9, T10, and T17, and the $Pn$ response is ranked T17 > T10 > T9. It appears that fixing $N$, $K$, and $Zn$, and changing $W$ significantly affects $Pn$. As $W$ increases, $Pn$ first increases and then decreases. These results indicate that the effect of water fertilizer coupling on $Pn$ can be altered by changing $W$ and $F_t$ individually. Based on the differences between treatments (T15, T16, T17), it can be seen that an appropriate increase in $Zn$ can increase the $Pn$, when $W$, $N$, $K$ are fixed; however, when the application is excessive, it will inhibit $Pn$. Fixing $N$ (T3, T4, T7, T8), P (T1, T3, T5, T7), and Zn (T2, T3, T5, T8), respectively, resulted in the $Pn$ differing significantly, indicating that each factor has a significant effect on $Pn$. It follows that an increase in irrigation water and fertilization alone does not necessarily contribute to fruit tree growth, which in turn favors yield increase, and that fruit tree growth can be better enhanced and fruit yield improved only by an appropriate water-to-fertilizer ratio.

3.2. Regression Analysis

With $Pn$ as the response variable ($y_{Pn}$), and $W$ ($x_1$), $N$ ($x_2$), $K$ ($x_3$), $Zn$ ($x_4$) as independent variables, regression models were constructed according to general rotation combination design and Formula (1) [56]. Coefficients values ($b_0,b_j,b_{jj},b_{hj}$) for the regression model were obtained according to Formulas (2)–(5), respectively [56].

$$\begin{gathered} y_{pn}=12.5517+0.6238x_1+0.4011x_2+0.1988x_3+0.2787x_4-0.5666x_1^2 \\ -0.62x_2^2-0.4134x_3^2-0.3955x_4^2+0.2759x_1{x}_2+0.0872x_1{x}_3+0.2097x_1{x}_4 \end{gathered}$$

(6)

To determine the validity and reliability of the regression models, the regression coefficients and partial regression coefficients were examined using the F-test. The results are shown in

Table 4. Analysis of variance for the regression relationships.

Source	Df	SS	MS	F Value	p Value
$x_1$	1	5.314	5.314	82.62	0.0001 **
$x_2$	1	2.197	2.197	34.16	0.0001 **
$x_3$	1	0.540	0.540	8.39	0.015 *
$x_4$	1	1.061	1.061	16.49	0.002 **
$x_1^2$	1	5.103	5.103	79.33	0.0001 **
$x_2^2$	1	6.109	6.109	94.97	0.0001 **
$x_3^2$	1	2.717	2.717	42.23	0.0001 **
$x_4^2$	1	2.486	2.486	38.65	0.0001 **
$x_1{x}_2$	1	0.609	0.609	9.47	0.011 *
$x_1{x}_3$	1	0.061	0.061	0.95	0.352
$x_1{x}_4$	1	0.352	0.352	5.47	0.039 *
Regression	11 ($f_r$)	26.225 ($S_r$)	2.384	$F_r$ = 37.06	0.0001 **
Residual error	11 ($f_R$)	0.708 ($S_R$)	0.064
Lack-of-fit	5 ($f_{If}$)	0.546 ($S_{If}$)	0.109	$F_{If}$ = 4.04	0.059
Error	6 ($f_e$)	0.162 ($S_e$)	0.027
Sum	22	26.225

Notes: * indicates significant at the 0.05 level, ** indicates significant at the 0.01 level; SS indicates stdev square, MS is mean square, Df is degree of freedom.

. Based on these results, a lack-of-fit test was conducted.

$$F_{If}=\frac{S_r/f_{If}}{S_e/f_e}=4.04<F_{0.05}\left(\mathrm{5,6}\right)=4.36$$

(7)

The lack-of-fit of the regression model was not significant, indicating that the four factors selected in this experiment are meaningful for investigating the change in $Pn$ of apple and suitable for estimation. For the regression equations, the test was as follows:

$$F_r=\frac{S_r/f_r}{S_R/f_R}=37.06>F_{0.01}\left(\mathrm{11,11}\right)=4.54$$

(8)

This indicates that the relationship between four factors ($W$, $N$, $K$, $Zn$) and $Pn$ was significant ($P<0.01$), and the regression model could well reflect the correlation between independent variables and dependent variables. In addition, Table 4 also shows that partial regression coefficients in the model reached significant or extremely significant levels for all but the $x_1{x}_3$($W$·$K$) interaction term. Further indicating that the single factor variations in $W$, $N$, $K$, and $Zn$ were strongly related to the variation in $Pn$ and there was a strong correlation between the quadratic term and $Pn$; $W\times N$ coupling and $W\times Zn$ coupling had significant effects on $Pn$.

(1)

Main effect analysis

Regression models were subjected to principal factor analysis. There was no correlation between the coefficients of the linear term in the model and the linear term coefficients and the interaction term coefficients were also uncorrelated. Therefore, the effect of the linear term for each factor on $Pn$ was determined by comparing the magnitude of the absolute value of the regression coefficients. The results showed that $W$ and $N$ had the greatest important effect on $Pn$, followed by $K$ and $Zn$, respectively, and that all factors had a positive effect on $Pn$. It can be concluded from the linear term coefficient that $Pn$ increased with the increasing application until the maximum tested application was reached.

The quadratic coefficients of $W$, $N$, $K$, and $Zn$ were all negative. With increasing $W$, $Pn$ showed first an increasing and then decreasing trend, appearing as an n-shaped parabolic curve. We believe that the optimal solution for $W$, $N$, $K$, and $Zn$ is at the highest point of the parabolic curve when $Pn$ reaches its peak value. Because of the correlation between the quadratic term coefficients in the orthogonal trial designs, the absolute value of the coefficients cannot be used directly to compare the magnitude of the quadratic term effect, and therefore, further analysis and validation are required. The coefficients of the $W\times N$ coupling, $W\times K$ coupling, and $W\times Zn$ coupling were all positive, therefore, increasing their combined effect is important to increase net photosynthesis in fruit trees, showing a positive interaction, indicating that the factors can work synergistically to increase $Pn$.

(2)

Single factor effects

The “dimension reduction method” was used to simplify the regression model, coding one factor within the range of investigated values, with the remaining factors all set to zero; this approach eliminates the influence of other factors on the analysis of the target factor. A single-factor model was obtained after dimension reduction elimination:

$$\mathrm{y}=12.5517+0.6238x_1-0.5666x_1^2$$

(9)

$$\mathrm{y}=12.5517+0.401x_2-0.62x_2^2$$

(10)

$$\mathrm{y}=12.5517+0.1988x_3-0.4134x_3^2$$

(11)

$$\mathrm{y}=12.5517+0.2787{\mathrm{x}}_4-0.3955{\mathrm{x}}_4^2$$

(12)

A single-factor plot of $Pn$ effects was produced from the single-factor model described above (Figure 2). From calculations, we know that the corresponding $Pn$ is 9.9 $\mathsf{\mu }\mathrm{mol}·{\mathrm{m}}^{-2}·{\mathrm{s}}^{-1}$, 10.123 $\mathsf{\mu }\mathrm{mol}·{\mathrm{m}}^{-2}·{\mathrm{s}}^{-1}$, 11.047 $\mathsf{\mu }\mathrm{mol}·{\mathrm{m}}^{-2}·{\mathrm{s}}^{-1}$, and 10.964 $\mathsf{\mu }\mathrm{mol}·{\mathrm{m}}^{-2}·{\mathrm{s}}^{-1}$, respectively, when the single-factor minimum coded level for $W$, $N$, $K$, $Zn$ is −1.682 within the experimental design. When the coded level increased to 0, the corresponding $Pn$ values all increased to 12.552 $\mathsf{\mu }\mathrm{mol}·{\mathrm{m}}^{-2}·{\mathrm{s}}^{-1}$. The corresponding $Pn$ reached its maximum value within the range of the experimental design when the coded values of $W$, $N$, $K$, and $Zn$ increased to 0.799, 0.493, 0.323, and 0.561, the corresponding applications were 669.775 $\mathrm{mm}$, 123.586 $\mathrm{kg}/{\mathrm{hm}}^2$, 236.298 $\mathrm{kg}/{\mathrm{hm}}^2$ and 13.815 $\mathrm{kg}/{\mathrm{hm}}^2$, respectively, and the $Pn$ values were 12.689 $\mathsf{\mu }\mathrm{mol}·{\mathrm{m}}^{-2}·{\mathrm{s}}^{-1}$, 12.599 $\mathsf{\mu }\mathrm{mol}·{\mathrm{m}}^{-2}·{\mathrm{s}}^{-1}$, 12.573 $\mathsf{\mu }\mathrm{mol}·{\mathrm{m}}^{-2}·{\mathrm{s}}^{-1}$ and 12.584 $\mathsf{\mu }\mathrm{mol}·{\mathrm{m}}^{-2}·{\mathrm{s}}^{-1}$. When the coded levels were increased to 1.682, the corresponding $Pn$ for each factor decreased to 11.998 $\mathsf{\mu }\mathrm{mol}·{\mathrm{m}}^{-2}·{\mathrm{s}}^{-1}$, 11.471 $\mathsf{\mu }\mathrm{mol}·{\mathrm{m}}^{-2}·{\mathrm{s}}^{-1}$, 11.715 $\mathsf{\mu }\mathrm{mol}·{\mathrm{m}}^{-2}·{\mathrm{s}}^{-1}$, and 11.902 $\mathsf{\mu }\mathrm{mol}·{\mathrm{m}}^{-2}·{\mathrm{s}}^{-1}$, respectively. It follows that the maximum $Pn$ that can be achieved under a single-factor influence and the coded levels of that single factor are not the same, but the trend of change in $Pn$ is similar under each in-factor influence, showing a gradual increase in $Pn$ with increasing levels for $W$, $N$, $K$, and $Zn$ before the maximum levels are reached, after which the $Pn$ again exhibits a gradually decreasing trend. This indicates that excessive irrigation and fertilization do not promote $Pn$ but rather have an inhibitory effect.

The sensitivity of individual single factors to $Pn$ can be judged by examining the curve of the parabola: the tighter the curve, the more sensitive the $Pn$ is to the factor, and the greater the extent to which it is affected. Figure 2 reveals that the single factor effect plots in relation to $Pn$ are parabolic and n-shaped, and the sensitivity ranking of the factors to the $Pn$ is: $W$ > $N$ > $Zn$ > $K$.

(3)

Two-factor interaction effect analysis

Fixing the coding value of two factors to zero, we can model the interaction between the other two factors with respect to the $Pn$:

$$\mathrm{y}=12.5517+0.6238x_1+0.4011x_2-0.5666x_1^2-0.62x_2^2+0.2759x_1{x}_2$$

(13)

$$\mathrm{y}=12.5517+0.6238x_1+0.1988x_3-0.5666x_1^2-0.4134x_3^2+0.0872x_1{x}_3$$

(14)

$$\mathrm{y}=12.5517+0.6238x_1+0.2787x_4-0.5666x_1^2-0.3955x_4^2+0.2097x_1{x}_4$$

(15)

Figure 3 represents the above models. Figure 3a shows that the effects of $W\times N$ coupling can be represented as a domed surface. That is, when other factors are set to zero, $Pn$ changes are represented by a parabola as $W$ and $N$ increase, and the interaction coefficient is 0.2759. This suggests that the interaction between $W$ and $N$ promotes $Pn$. When the coded levels for $W$ and $N$ were 0.68 and 0.48, respectively, $Pn$ reached its maximum of 12.85 $\mathsf{\mu }\mathrm{mol}·{\mathrm{m}}^{-2}·{\mathrm{s}}^{-1}$. The actual levels for $W$ and $N$ were 652 $\mathrm{mm}$ and 123.3 $\mathrm{kg}/{\mathrm{hm}}^2$, respectively. With a continuing increase in $W$ and $N$ application, $Pn$ began to decline.

Figure 3b,c show similar effect trends for the $W\times Zn$ coupling and $W\times K$ coupling. With respect to the degree of effect $W>K,Zn$. When the $W$ was at its middle or upper level, the $Pn$ remained at a high level. When the $W$ was at a lower level, increasing the Zn or K had little effect on $Pn$. The $W\times Zn$ coupling produced a maximum $Pn$ of 12.82 $\mathsf{\mu }\mathrm{mol}·{\mathrm{m}}^{-2}·{\mathrm{s}}^{-1}$ with coded levels of 0.64 and 0.52, respectively. The corresponding $W$ and $Zn$ actual levels are 646 mm and 13.63 $\mathrm{kg}/{\mathrm{hm}}^2$, respectively. For $W$ and $K$, $Pn$ reached a maximum value of 12.76 $\mathsf{\mu }\mathrm{mol}·{\mathrm{m}}^{-2}·{\mathrm{s}}^{-1}$ when the coded levels were 0.56 and 0.28, respectively. The corresponding $W$ and $K$ actual levels are 634 $\mathrm{mm}$ and 237.6 $\mathrm{kg}/{\mathrm{hm}}^2$, respectively.

(4)

Three-factor interaction effect analysis

By setting the value of one factor in the fixed model to zero, the combined effect of the other three factors on $Pn$ can be examined. The model was, thus, used to derive the relationships shown in Figure 4. Figure 4a. reveals the combined effects of $W$, $K$, and $Zn$ on the $Pn$: when coded levels reach 0.496, 0.187, and 0.221, respectively, the corresponding actual levels are 624.33 $\mathrm{mm}$, 233.42 $\mathrm{kg}/{\mathrm{hm}}^2$, and 12.06 $\mathrm{kg}/{\mathrm{hm}}^2$, and the $Pn$ reaches 12.75 $\mathsf{\mu }\mathrm{mol}·{\mathrm{m}}^{-2}·{\mathrm{s}}^{-1}$. Figure 4b. reveals that when the $Pn$ reaches its highest value, 12.77 $\mathsf{\mu }\mathrm{mol}·{\mathrm{m}}^{-2}·{\mathrm{s}}^{-1}$, under the combined effects of $W$, $K$, and $Zn$, the corresponding coded levels are 0.454, 0.222 and 0.231, respectively, and the actual levels are 618.06 $\mathrm{mm}$, 117.5 $\mathrm{kg}/{\mathrm{hm}}^2$ and 12.31 $\mathrm{kg}/{\mathrm{hm}}^2$. Compared with Figure 4a, the maximum $Pn$ exhibits little difference. The $Zn$ is increased by 4.5% and the $W$ is decreased by 9.2%. Figure 4c shows the combined effect of $W$, $N$, and $K$ on the $Pn$. The maximum $Pn$ is 12.76 $\mathsf{\mu }\mathrm{mol}·{\mathrm{m}}^{-2}·{\mathrm{s}}^{-1}$. When the $Pn$ reaches its maximum, the coded levels for each factor are 0.483, 0.216, and 0.189 respectively; the corresponding actual levels are 622.51 $\mathrm{mm}$, 117.36 $\mathrm{kg}/{\mathrm{hm}}^2$, and 233.51 $\mathrm{kg}/{\mathrm{hm}}^2$. By comparing Figure 4c to Figure 4b, it can be seen that when the $Pn$ reached its maximum, $W$ increased by 6.2%. Contrasting Figure 4a–c, it can be found that the maximum $Pn$ values for the $W\times Zn\times K$ coupling, $W\times K\times N$ coupling, $W\times Zn\times N$ coupling do not differ much, but in the case of $W\times Zn\times N$ coupling, the $W$ required for $Pn$ to reach a maximum is minimal, which is more suitable for the arid conditions in Xinjiang. This demonstrates that the combined effect of $W\times Zn\times N$ coupling can be manipulated to deliver the best results. This may be related to the content of soil nitrogen and zinc in the experimental plot, or it is possible that under the experimental conditions, $Pn$ is more sensitive to the changes in $N$ and $Zn$ in the soil. It may also be that the synergy between soil water and nitrogen and zinc is strong, so the promotion of $Pn$ is more obvious than other water and fertilizer combinations.

(5)

Optimal combination scheme

The regression model was based on five levels (−1.682, −1, 0, 1, 1.682) and we used simulation optimization and frequency analysis, resulting in 129 irrigation fertilizer couplings with $Pn$ > 11 $\mathsf{\mu }\mathrm{mol}·{\mathrm{m}}^{-2}·{\mathrm{s}}^{-1}$, accounting for 20.6% of the total test protocol. It can be seen from

Table 5. Optimization scheme and frequency of target output.

Code	${\mathit{x}}_1$		${\mathit{x}}_2$		${\mathit{x}}_3$		${\mathit{x}}_4$
	t	f	t	f	t	f	t	f
1.682	36	0.279	25	0.194	25	0.194	18	0.140
1	51	0.395	46	0.357	36	0.279	41	0.318
0	39	0.302	49	0.380	39	0.302	41	0.318
−1	3	0.023	9	0.070	25	0.194	18	0.140
−1.682	0	0	0	0	4	0.031	0	0
total	129	1	129	1	129	1	129	1
Weighted average	0.843		0.621		0.369		0.565
95% Confidence interval	0.720–0.965		0.486–0.756		0.196–0.542		0.411–0.720
Application rate	658.00–694.75 mm		123.44–129.51 kg/hm2		233.82–249.39 kg/hm2		13.13–14.54 kg/hm2

Notes: t indicates times, f indicates frequency.

that when the value range for each factor value reflects the 95% confidence interval for $W$-coded levels (0.72–0.965), $N$-coded levels (0.486–0.756), $K$-coded levels (0.196–0.542) and $Zn$-coded levels (0.411–0.72), the corresponding actual level of $W$, $N$, $K$, and $Zn$ are 658–694.75 $\mathrm{mm}$, 123.44–129.51 $\mathrm{kg}/{\mathrm{hm}}^2$, 233.82–249.39 $\mathrm{kg}/{\mathrm{hm}}^2$ and 13.13–14.54 $\mathrm{kg}/{\mathrm{hm}}^2$, respectively.

4. Discussion

Photosynthesis is the main driving force affecting dry matter assimilation and organ formation, and it is the basis of plant production [59,60]. Irrigation and fertilizer are important factors affecting crop photosynthesis. Appropriate irrigation and fertilizer management can change the environmental conditions for crop growth and improve crop $Pn$ [61,62].

This study has shown that soil moisture has a significant positive effect on the $Pn$; that is, with increased $W$, the $Pn$ first showed an increasing trend. This is consistent with the research results of Liao et al. [63] and Zhen et al. [64]. An increase in soil moisture promotes the synthesis of hormones and related enzymes in plants, enhances the material transport capacity, and accelerates the transport rate of photosynthetic products, whilst, at the same time, the stomata open, the transpiration rate increases, and the $Pn$ increases [65,66]. However, when the $W$ increases beyond a certain threshold, the $Pn$ decreases. This may be because too much soil moisture leads to poor soil ventilation and decreased root activity, thus affecting water transmission in plants, and indirectly inhibiting photosynthesis [67].

Under the conditions of this experiment, the effects of the linear term and quadratic term for $N$, $K$, and $Zn$ reached significant levels. Single-factor analysis showed that, with increased applications, $Pn$ first increased and then declined. The effects of the three factors on the $Pn$ were ranked as follows: $N$ > $Zn$ > $K$. This may be because all three factors participate in photosynthesis in plant leaves, but they have different functions. Braun et al. [68] reported that the presence of $K^{+}$ helps to maintain the transmembrane proton gradient of chloroplasts and thylakoids under light, keeping the chloroplast interstitium at the higher PH required for ${CO}_2$ assimilation, promoting photophosphorylation and ${CO}_2$ assimilation, and improving the $Pn$. Makino et al. [37] and Sperling et al. [69] reported that nitrogen is the main element in chloroplasts, present in chloroplasts, proteins, and lamellar membranes and playing an important role in photosynthesis. Zinc is an important component and activator of many enzymes in photosynthesis, and also an essential nutrient for protein, nucleic acid, and sugar metabolism in chloroplasts [70,71,72]. After zinc application, the stomatal resistance of plant leaves decreases, stomatal conductance increases, and the transpiration and $Pn$ of leaves increases [48,73]. Wang et al. [70,74] also found that a lack of zinc will lead to a decrease in chlorophyll content, stomatal conductance, intercellular concentration, and $Pn$, and a reduction in plant photosynthetic performance. These research results support, to some extent, the results of our experiment. However, with the continuous application of nitrogen, potassium, and zinc, the effective content of the three elements in the soil increases, and the plants can also absorb a lot, thus promoting their growth. At the same time, due to the initial content of nitrogen, potassium, and zinc in the soil at the experimental site, the three elements reached the optimum threshold at different application rates, and the activities of photosynthesis-related enzymes were reduced due to non-stomatal factors, which further led to the inhibition of plant photosynthesis with increasing application rates [75].

A large number of studies have shown that the coupling of water and fertilizer indirectly affects the photosynthetic rate of plant leaves by expanding leaf area, increasing leaf transpiration rate, increasing stomatal conductance, increasing intracellular water concentration, and reducing intracellular carbon dioxide concentration [76]. Wang et al. [77] showed that water/nitrogen coupling had a significant positive effect on the $Pn$ of leaves, which is similar to the results of the current study. The reason may be that, with the increase in irrigation and nitrogen fertilizer application, the available nitrogen in the soil increase. Because there is sufficient soil water, the nutrient transport efficiency of the tree is significantly improved. Therefore, the combined effect of the two factors is that the absorption of nitrogen increases, and it is quickly and effectively transported to the leaves, promoting the synthesis of chlorophyll and thus enhancing the $Pn$ [78,79,80]. With regard to the interaction between $W$ and $Zn$, we found that when there was limited $W$, the $Pn$ remained at a low level with increasing $Zn$, and when there was more $W$, the $Pn$ increased somewhat with the appropriate increase in $Zn$. This shows that water plays the dominant role in the interaction between $W$ and $Zn$. Some studies have shown that the effect of applying zinc fertilizer on plant biomass is better when there is sufficient water. Spraying $Zn{SO}_4$ on the leaf surface can improve the leaf water conditions [48,49]. Water and potassium are important factors affecting plant photosynthesis [81].

Some studies have shown that when soil moisture content increases, soil mechanical resistance decreases, which facilitates the flow of nutrients, thus promoting the absorption of nutrients by the root system. Potassium itself promotes photosynthesis. Studies by Nieves-Cordones have shown that under the condition of insufficient potassium fertilizer supply, excess energy in plants can induce the production of more reactive oxygen species, destroy chloroplast structure, accelerate chloroplast decomposition, and then inhibit photosynthesis. These negative effects caused by potassium deficiency will be improved with the increase in potassium application. The results of this experiment showed that the photosynthetic rate of apple leaves increased significantly with the increase in irrigation amount under the condition of constant potassium application. The results showed that the increase in irrigation water promoted the absorption of potassium in apple trees. There was a significant coupling effect between irrigation amount and potassium application amount, and it had a positive effect on Pn increase. So, the combined effect of $W$ and $K$ has a somewhat synergistic effect on the $Pn$ [43,82,83,84].

It should be pointed out that due to spatial and temporal differences in the coupling of irrigation and fertilizer, different regions, soil textures, and soil nutrient contents will lead to different conclusions from such testing. Therefore, in both production and application, we must adapt measures to local conditions and consider the prevailing conditions if we are to gain the best effect.

5. Conclusions

With drip irrigation of closely planted dwarf stock in the arid area of Xinjiang, $W$, $N$, $K$, and $Zn$, had significant effects on the $Pn$ of apple trees, but the influence of each factor on the $Pn$ differed and can be ranked: $W>N>Zn>K$. By examining the interaction between irrigation and each of the fertilizers, we found the following ranking of effects: $W\times N$ > $W\times Zn$ > $W\times K$. Applying zinc can improve the Pn, thus enhancing the storage of nutrients in the tree and promoting growth and development, and thus further improving the yield of apple trees.

The test simulation optimization and frequency analysis showed that with $W$ in the range 258–294.75 $\mathrm{mm}$, $N$ in the range 33.44–39.51 $\mathrm{kg}/{\mathrm{hm}}^2$, $K$ in the range 53.82–69.39 $\mathrm{kg}/{\mathrm{hm}}^2$, and $Zn$ in the range 6.46–7.84 $\mathrm{kg}/{\mathrm{hm}}^2$, within the 95% confidence level, the net photosynthetic rate reaches 11 $\mathsf{\mu }\mathrm{mol}·{\mathrm{m}}^{-2}·{\mathrm{s}}^{-1}$. This is the best irrigation and fertilizer management plan, under the test conditions.

Based on previous research results and our data, we consider that the indicators and methods targeted by this test have certain limitations. More plant physiological growth indicators are required to establish the relationship between irrigation and fertilizer factors and various growth indicators. Finally, the relationship between yield and economic benefits should be established to better serve the development of commercial agriculture and forestry.