Field experiments were conducted from 2019 to 2020, at the Chongzhou Experimental Farm of Sichuan Agricultural University, Sichuan Province, China. The climate of the experimental area is humid and subtropical. The annual mean air temperature is 15.9 °C, of which the mean air temperature of the hottest month (July) is 25 °C, and the coldest month (January) is 5.4 °C ( Supplementary Figure 1 ). The annual mean sunshine hours and the annual mean rainfall are 1161.5 h and 1012.4 mm, respectively. The soil at the experimental site is classified as clayey soil, with total nitrogen 1.6 g kg^-1, total phosphorus 1.3 g kg^-1, total potassium 15.2 g kg^-1, available nitrogen 299.5 mg kg⁻¹, available phosphorus 1.3 g kg^-1, available potassium 169.4 mg kg⁻¹, organic matter content of 24.3 g kg⁻¹ and the pH of the top 0-20 cm soil layer is 7.1.

In this experiment, Zhenghong-505 (semi-compact), a maize cultivar and Chuandou-16 (shade-sensitive), a soybean cultivar, which are mainly planted in southwest China, were used as materials. The experiment was a two-factor randomized block design with three replications. The planting patterns (e.g. maize-soybean strip intercropping system and skip strip monocropping system) were as the main factor, and three soybean planting densities (e.g. PD1 = 8.5 plants m^-2, PD2 = 10 plants m^-2 and PD3 = 12.5 plants m^-2, thereafter abbreviated as PD1, PD2 and PD3 respectively) were as the secondary factor. The classic wide and narrow rows planting with 2 m of each strip width and 6 m of each strip length was adopted by the maize-soybean strip intercropping system. The area of each individual plot was 36 m² (6.0 m×6.0 m) for both strip intercropping and skip strip monocropping. Each individual intercropping plot consisted of 3 maize-soybean strips, and skip strip monocropping plot included three maize or soybean strips. As shown in Supplementary Figure 2 , in the maize-soybean strip intercropping system, two rows of maize were planted in narrow row spacing of 40 cm, with planting space of 20 cm (i.e. 5 plants m^-2), and two rows of soybean were planted in wide row spacing of 160 cm, with planting space of 8 cm, 10 cm and 12 cm (i.e. PD3, PD2 and PD1 respectively) ( Supplementary Figures 2A, C ). In the sole soybean system with skip strips, the planting method was the same as that of intercropped soybean without maize strip, which served as the control ( Supplementary Figures 2B, D ). Both maize and soybean were sown on 2 April in 2019 and 2020, and harvested on 26 July and 2 August in 2019 and 2020, respectively.

All maize strips were treated with compound fertilizer (N: P: K = 15: 15: 15) at 80 g m^-2 as the base fertilizer before sowing. The urea (N ≥ 46%) was applied as fertilizer with 7.8 g m^-2 at the jointing of maize and the second dose of urea (N ≥ 46%) was applied as fertilizer with 13.2 g m^-2 at heading stages of maize (Cheng et al., 2020). Whereas there was no fertilizer application in all soybean strips due to the fertile soil in the experimental area. In addition, weeding, pest control and irrigation are used in the daily management of maize and soybean.

The lodging resistance index (LRI), dry matter growth of soybean (i.e. stems, leaves, branches and pods) and accumulation of sucrose, cellulose and lignin were measured 5 times every 14 days from 35 to 91 days after sowing in 2019 and 2020, respectively. At the mature stage of soybean and maize, soybean seeds and maize cobs were collected respectively for grain yield measurement in 2019 and 2020.

To determine the effect of the light environment of soybean in the maize soybean strip intercropping system, we measured the plant height of soybean and maize, as well as the breadth of maize crown, using a ruler with a measuring range of 0 to 3.0 m and an accuracy of 0.1 cm during the period from 35 to 91 days after sowing. These measurements were taken every 14 days, for a total of 5 times.

The stems, leaves, branches, and pods of 9 soybean plants taken from each treatment, were packed separately into paper bags and dried at 105 °C for 30 min in an electric oven (Sunne, SN-101-3QB, Shanghai, China), and dried at 80 °C until achieving a constant weight, and finally weighed using a balance (Sartorius BSA224S-CW, Beijing, China). After measuring the dry matter weight of each organ, these stems from the 3rd to 5th sections of the soybean were ground with a powder prototype (Hongbang Technology QE-50, Henan, China) to determine contents of lignin, cellulose and sucrose according to previously published methods (Leng et al., 2016; Liu et al., 2016; Wen et al., 2020). All dry leaves from 5 soybean plants taken from each treatment were ground and the sucrose content was measured using the method described by (Leng et al., 2016).

At the mature stage of soybean, 15 soybean plants with uniform growth from each treatment were collected and dried in an air-drying room. These dried plants were used to measure the number of full-pods, non-full-pods and the number of branches per plant, as well as the seed yield per unit area. Similarly, at the mature stage of maize, 15 uniform complete maize cobs from each plot were collected and dried in an air-drying room, which was used to only measure the seed yield per unit area.

The size-dependent changes in soybean dry matter allocation models were divided into stems, leaves, branches and pods by using the method of allochronic analysis (Eqn. 10) (Zhang et al., 2020). The daily allocation was computed as the ratio of the daily growth rate of each soybean organ (e.g. stems, leaves, branches and pods) to the sum of daily growth rates of each soybean organ.

Where AI_i is the dry matter allocation index for an organ i in soybean, d w i d t represents the daily growth rate for an organ i in soybean, i stands for stems, leaves, branches and pods, respectively.

All the data were sequentially collected and sorted out through Excel 2019 software. The statistical software Origin Pro 2020b was applied for two-way analysis of variance (ANOVA), weighted nonlinear fitting and graphing. The significant differences among treatments were separated according to the LSD at p ≤ 0.05 and p ≤ 0.01. The hypothesis of nonlinear regression model was statistically checked by Chi-square-test and F-test to ensure the correctness of the conclusion based on beta and logical growth curves. The R² in models represented the fitting degree of the curves. The root mean square error (RMSE) (Eqn.13) was used to evaluate the effectiveness of the curve models. The determinate coefficient (R²) was used to characterize the correlation intensity of LR, LRI and PD, as well as the correlation intensity of LRI with Stem/Leaf, lignin and cellulose.

Where RMSE is the root mean square error, N is the total number of observations, S_i is the fitted values, Q_i is the actual values.

Plant height of maize and soybean, along with maize-crown breadth, were measured to indirectly assess the impact of maize on the light environment dynamics of soybean in the strip intercropping system ( Figure 1A ). From day 35 after sowing, the height and crown breadth of maize plants exhibited rapid growth, which continued until day 63. During this period, the canopy width of maize surpassed 60 cm, which was the spacing between maize rows and soybean rows, while the height of maize plants reached 194 cm ( Figure 1B ). These findings indicate that the shading effect of maize on soybean gradually intensified from day 35 after sowing and reached a stable level by day 63.

Both planting densities and patterns had significant effects on the grain yield, grain plumpness and branches of soybean, while soybean planting density had no effect on maize yield in the maize-soybean strip intercropping system ( Table 1 ). As planting densities increased, the number of full-pods and branches in soybean decreased significantly, whereas the number of non-full-pods increased ( Table 1 ). Furthermore, intercropped soybean exhibited significantly lower numbers of full-pods and branches compared to monocropped soybean. Additionally, soybean grain yield in strip intercropping was 55%, 55%, and 59% lower than that in skip strip monocropping at PD1, PD2, and PD3, respectively, over the course of two years ( Table 1 ). Thus, the reduced yield of soybean in strip intercropping may be attributed to a lower number of branches and full-pods compared to skip strip monoculture.

According to Figure 2 , the LRI (Lodging resistance index) of soybean decreased with the increase of planting density, irrespective of planting patterns. After 49 days of sowing, the LRI of intercropped soybean is lower than that of soybean in monoculture. Furthermore, intercropped soybean reached its maximum LRI at 63 days and then stabilized, followed by a declining trend. In contrast, the LRI of soybean in monoculture continued to exhibit a growth trend until harvest.

The dynamics of soybean organ growth (e.g. stems, leaves, branches, and pods) in strip intercropping and skip strip monocropping were characterized using beta growth curve models due to plant organ shedding and sucrose transfer and transformation in leaves associated with aging ( Figure 3 ). The dry matter growth and daily growth rates of each soybean organ exhibited an initial increase followed by a subsequent decrease during the growth and development of soybean ( Figures 3 , 4 ). Furthermore, both the dry matter growth and daily growth rates of each soybean organ were significantly lower in strip intercropping compared to skip strip monocropping, and exhibited a significant decrease with increasing planting densities ( Figures 3 , 4 ). Additionally, the peak time for both dry matter accumulation and daily growth rates of each soybean organ occurred earlier in strip intercropping compared to the corresponding skip strip monocropping ( Figures 3 , 4 ). Furthermore, during the middle and later stages of growth and development, soybeans in strip intercropping exhibited a more pronounced decrease in the growth rate of leaf dry matter compared to other organs, especially at high planting density ( Figure 4 ). This phenomenon can be attributed to shading caused by maize, which accelerates soybean leaf senescence.

Planting density, planting patterns, and their interaction significantly influenced the w_m , c_m and t_e of soybean organs ( Supplementary Table 1 ). On the one hand, the w_m , c_m and t_e of soybean organs in strip intercropping were lower than those in skip strip monocropping and decreased with increasing planting densities, irrespective of planting patterns ( Supplementary Table 1 ). Strip intercropped soybean in higher planting density (PD3) exhibited a decrease of 28% and 11% in c_m and t_e , respectively, compared to the average of low planting density (PD1) over two years ( Supplementary Table 1 ). On the other hand, compared to skip strip monocropping, the t_e for soybean organs in strip intercropping was observed 10-14 days earlier, suggesting that interspecific and intraspecific shading reduced the duration of soybean green leaves, leading to decreased soybean dry matter growth and accumulation. Additionally, strip intercropping resulted in a significant advancement in t_e for the growth of stems, leaves, and branches, compared to skip strip monocropping, particularly at high planting density ( Supplementary Table 1 ).

Logistic growth curves were effective in accurately fitting the continuous accumulation of lignin and cellulose throughout the entire life cycle of soybean ( Figures 5E–H ). In contrast, the beta growth curves were well-suited for fitting sucrose accumulation in leaves and stems due to its transfer or transformation among organs ( Figures 5A–D ). Sucrose accumulation in leaves and stems exhibited an initial increase followed by a subsequent decrease during the growth and development of soybean, whereas lignin and cellulose accumulation in stems displayed a continuous increasing trend throughout the lifespan of soybean ( Figure 5 ). Over the two years of this study, the accumulation of sucrose, lignin, and cellulose and the daily accumulation rate of soybean in strip intercropping were significantly lower than those of corresponding skip strip monoculture and decreased with the increase of planting density ( Figures 5 , 6 ). Likewise, the w_max , c_max , t_e and t_c of soybean in the strip intercropping were lower than those in corresponding skip strip monocropping and decreased with increasing planting density ( Supplementary Table 2 ). In comparison to the average of low planting density (PD1) over a two-year period, strip intercropped soybean in PD3 exhibited a decrease of 29% in c_max for sucrose (in leaves and stems), lignin, and cellulose (in stems). Furthermore, the t_c of these carbohydrates in intercropped soybean in PD3 decreased by 13%. The peak time for the accumulation of sucrose, lignin, and cellulose in strip intercropped soybean occurred earlier compared to the corresponding skip strip monocropped soybean ( Figure 5 ).

Strip intercropped soybean exhibited a greater decrease in sucrose accumulation in leaves and stems compared to sole soybean with skip strip during the middle and later growth stages, particularly at high planting density ( Figures 5A–D ).

Planting density, planting patterns, and their interaction significantly influenced the w_max , c_max , t_e and t_c of lignin, cellulose and sucrose in soybean ( Supplementary Table 2 ). Firstly, in the strip intercropping, the w_max , c_max and t_e o values of soybean organs were lower than those in skip strip monoculture, and these values decreased with increasing planting densities ( Supplementary Table 2 ). Secondly, the t_e for sucrose and the t_c for lignin and cellulose in the strip intercropping were observably advanced by 16-20 days compared to skip strip monocropping ( Supplementary Table 2 ). This observation suggests that shading stress from interspecific and intraspecific interactions reduced the time for sucrose accumulation, consequently affecting the continuous accumulation of lignin and cellulose in soybean ( Figures 5E–H ).

The assimilate partitioning index (AI) among soybean organs, such as stems, leaves, branches, and pods, can be calculated using the biomass growth rate functions specific to each organ. Our data showed that, during the seedling stage of soybean, a greater proportion of dry matter was allocated to the leaves in terms of time ( Figures 7C, D ). During the middle stage of soybean, a higher proportion of dry matter was allocated to the stems and branches ( Figures 7A, B, E, F ). In the later growth stage, regardless of strip intercropping or skip strip monocropping, a larger portion of dry matter was transferred to the pods to facilitate seed formation ( Figures 7G, H ). When comparing skip strip monocropping with strip intercropping, we observed that the AI among soybean organs in strip intercropping was significantly lower than that of sole soybean with skip strip during the same growth period ( Figure 7 ), which was closely associated with shading from maize. Furthermore, our research revealed that the AI stems showed an increase with increasing planting density during the early stage (0-49 days after soybean sowing), whereas the opposite trend was observed during the later stage ( Figures 7A, B ). Moreover, the AI in leaves ( Figures 7C, D ), branches ( Figures 7E, F ) and pods ( Figures 7G, H ) decreased with the increase in planting density in both strip intercropping and skip strip monocropping.

The stem/leaf AI ratio increased with increasing planting densities during the early stage (0-49 days after soybean sowing), but exhibited the opposite trend during the later stage of soybean ( Figures 8A, B ). Moreover, the trends of the stem sucrose/leaf sucrose assimilate ratio (AR) were consistent with the stem/leaf AI ratio ( Figures 8C, D ). Additionally, the value of the stem sucrose/leaf sucrose assimilate ratio (AR) decreased with increasing planting densities in both strip intercropping and skip strip monoculture ( Figures 8C, D ). Furthermore, the value of the stem sucrose/leaf sucrose AR ratio was significantly lower for intercropped soybean compared to soybean in monoculture ( Figures 8C, D ).

The two-year field experiments demonstrated that, at the R6 stage, the lodging percentage (LP) of strip intercropped soybean exhibited a significant positive correlation with planting densities ( Figure 9A ). Conversely, the lodging resistance index (LRI) of strip intercropped soybean displayed a significantly negative correlation with planting densities ( Figure 9B ). Furthermore, the correlations between LP and LRI for soybean in strip intercropping were significantly negative at both the V3 and R6 stages, regardless of planting densities ( Figure 9C ). Ultimately, the LRI serves as a crucial index for measuring the lodging resistance of soybean, irrespective of planting patterns or densities.

The correlations between LRI and stem/leaf ratio (dry matter and sucrose) in strip intercropping were significantly positive, regardless of planting densities ( Figures 10A, B ). Likewise, significant positive correlations were observed between LRI and the accumulation of lignin and cellulose in strip intercropped soybean across different planting densities ( Figures 10C, D ).

Lodging is one of the important factors that influences plant morphogenesis and yield formation (Hayes et al., 2017; Lyu et al., 2021). It is affected by various internal and external factors, including plant varieties and planting patterns (Board, 2001; Esechie et al., 2004). The benign external environment was conducive to plant growth and development with stronger stem strength, which in turn lowered lodging risk (Sher et al., 2018). Soybean stem morphology primarily develops during the early growth stages, whereas shading of soybean in the maize-soybean relay strip intercropping system by maize occurs during the seeding stage (Yang et al., 2014; Deng et al., 2016). Reduced photosynthetic active radiation in the soybean canopy within the maize-soybean relay strip intercropping system influenced soybean morphogenesis, resulting in slender and weak stems that are prone to lodging and ultimately leading to a decline in soybean yield (Cooper, 1971; Liu et al., 2017). However, our previous research has found that lodging also occurred in strip intercropped soybeans during the middle and later growth stages when they are shaded (Cheng et al., 2020). A previous study revealed that within the maize-soybean strip intercropping system, the photosynthetically active radiation (PAR) in the soybean canopy experienced a rapid decrease around 40 days after sowing, after which it stabilized around 60 days after sowing, which aligns with our research ( Figure 1B ; Liu et al., 2017). Furthermore, our previous research uncovered that intercropped soybeans at high densities exhibited lodging around 40 days after sowing. As shading intensified, soybeans at medium to low densities gradually encountered lodging in the intercropping system (Cheng et al., 2020). This indicates that soybean lodging is influenced by both interplant shading and intra-plant shading. Soybean in the strip intercropping or higher planting density could increase the shading or self-shading of plants, resulting in higher plant height, lower stem diameter and easier lodging (dos Santos Scholz et al., 2018; Cheng et al., 2020). A significantly negative correlation between LRI and LP was observed ( Figure 9C ), indicating that LRI can serve as an effective index for assessing the lodging resistance of soybean stems. The LRI was calculated by the main stem length, the shoot biomass fresh weight and the stem bending force (Hussain et al., 2020a). Previous research has demonstrated that soybean under shaded conditions exhibits elongation of the main stem and reduced stem bending force (Liu et al., 2019; Yang et al., 2020). In this study, we observed a significant negative correlation between LRI and plant density ( Figure 9B ), with soybean in monoculture and low plant density exhibiting higher LRI compared to soybean in the intercropping system and at high plant density ( Figure 2 ).

The accumulation of lignin and cellulose in stems is crucial for LRI of strip intercropped soybean, and it is influenced by planting densities, planting patterns, and their interaction (Liu et al., 2016; Hussain et al., 2020a). Numerous studies have demonstrated that higher planting density and maize-soybean strip intercropping systems increase the susceptibility of soybean stems to lodging due to the reduced accumulation of lignin and cellulose (Liu et al., 2016; Cheng et al., 2020; Raza et al., 2020). As a precursor of lignin and cellulose synthesis, the accumulation of sucrose in stem affects the lignin and cellulose accumulation in stems (Boerjan et al., 2001; Coleman et al., 2009; Ruan, 2014). A significant positive correlation was found between LRI and accumulation of lignin, cellulose and sucrose in stems regardless of planting densities and patterns ( Figure 10 ). Beta and logistic growth models were utilized to analyze the accumulation of carbohydrates (e.g., sucrose, lignin, and cellulose), revealing lower contents of these carbohydrates in strip intercropped soybean compared to skip strip monoculture. Furthermore, the contents decreased with increasing planting densities ( Figure 5 ). High planting density intensifies the species and interspecific competition for light energy among strip intercropped plants, particularly during the symbiotic period, potentially impacting the distribution of sucrose among soybean organs (Huang et al., 2018; Zhang et al., 2020). The w_max , c_max , t_e and t_c of carbohydrates were affected by planting densities and patterns, and the shade in high planting density inhibited maximum accumulation rate and shortened continuous accumulation time of soybean in both strip intercropping and skip strip monoculture ( Figure 6 ; Supplementary Table 2 ). Soybean intercropped at high density exhibited a higher rate of sucrose accumulation in leaves, while the rate of sucrose accumulation in the stems remained consistently low ( Figures 6A, C ). This suggests that shading also influenced the transport of sucrose from the leaves to the stems. In this study, we conducted further analysis of the stem/leaf sucrose accumulation rate ratio (stem/leaf sucrose AR) and observed that high planting density promoted sucrose accumulation in leaves while decreasing accumulation in stems, thus impeding the synthesis of lignin and cellulose in the stems ( Figures 8C, D ). Furthermore, several studies have indicated that the relationship between supply and demand of sucrose in different plant organs changes under higher planting density or shade stress. Sucrose synthesis in leaves increases rapidly and is promptly transported to the stems to support plant elongation and growth, thereby capturing more solar energy (Tobias et al., 1999; Cheng et al., 2020). However, at high plant density, the cells in stems exhibited more longitudinal growth and less lateral growth, which aggravated the occurrence of stem lodging (Coleman et al., 2009). Furthermore, it has been demonstrated that optimum nitrogen fertilization boosts soybean stem lodging resistance by modulating the lignin metabolism in the maize-soybean intercropping system (Raza et al., 2023).

The distribution of assimilates among different soybean organs was influenced by planting densities, planting patterns, and their interaction. Numerous studies have shown that shading leads to stem thickness, resulting in stem lodging (Kang et al., 2015; Liu et al., 2016; Song et al., 2016). In this study, the relative content of assimilates in stems to leaves of strip intercropped soybean was closely related to LRI ( Figures 10A, B ). Higher assimilate content in stems is beneficial for maintaining stronger stem strength (Hussain et al., 2020a). Our data indicated that higher planting density not only reduced the growth rate in various organs ( Figure 4 ), but also shortened the time to reach maximum dry matter and maximum growth rate for these organs ( Supplementary Table 1 ; Figure 6 ) in both strip intercropping and skip strip monoculture. Ultimately, the accumulation of assimilates in various organs reduced ( Figure 3 ). Interspecific and intraspecific shading stress significantly restricts the assimilate production of intercropped soybeans, thus, its distribution among various organs determines its stem lodging resistance and ultimately influences yield formation. Light intensity significantly affected the distribution of C-assimilates in the stems and leaves of plants (Hocking and Steer, 1994). We ulteriorly analyzed the ratio of stem assimilate partitioning index to leaf assimilate partitioning index (stem/leaf AI ratio) in this study and observed that increasing planting density accelerated the distribution of assimilates into stems rather than leaves of intercropped soybean during the early growth stage, contributing to enhanced plant height rather than stem diameter of soybean (Cheng et al., 2020; Raza et al., 2020). Increasing the planting density of soybean increased the dry matter allocation index (AI) of stems, but it lowered the AI of leaves, branches and pods in strip intercropped and skip strip monocropped systems, which affected the final yield formation ( Figure 7 ). The most essential reason for the above might be that the unbalanced source-sink relationship of the strip intercropped soybean plants in the vegetative growth period induced by high planting densities and maize shading. In other words, interspecific and intraspecific shading stress resulted in a source-limited soybean in a longer period of symbiosis, which, in turn, leaded to a sink-limited soybean growth. This directly affected the morphogenesis of soybean and yield formation ( Table 1 ; Borrás et al., 2004).