Modeling wheat productivity under elevated CO₂ using fuzzy logic and mycorrhizal inoculation

Abstract

Background

Understanding the complex interactions between plants, Arbuscular Mycorrhizal Fungi (AMF), and elevated atmospheric CO₂ (eCO₂) is crucial for enhancing agricultural sustainability and productivity, particularly in the face of future climate change. While elevated CO₂ concentrations can influence AMF colonization development, AMF are known to benefit plants by improving nutrient uptake, especially phosphorus, enhancing drought tolerance, and increasing resistance to certain soil-borne pathogens. These beneficial effects of AMF can potentially mitigate some of the negative impacts of climate change on crop yields. This study explores the interplay between wheat (Triticum aestivum L.), AMF inoculation, and eCO₂ levels using the Mamdani Fuzzy Inference System (MFIS), a tool well-suited to handle uncertainties in modeling complex plant responses to environmental changes. By integrating fuzzy logic-based approaches, this research aims to elucidate how AMF inoculation can modulate wheat productivity under projected future elevated CO₂ levels, thereby providing insights into strategies for maintaining or improving crop yields in changing climatic conditions. The goal was to explore the relationship between CO₂ levels, AMF inoculation, and wheat yield, specifically investigating the potential of AMF to enhance wheat performance under elevated CO₂.

Results

Statistical analyses revealed that eCO₂ significantly increased ear length (p < 0.05), while AMF inoculation significantly enhanced the number of spikelets per ear (p < 0.05), number of grains per ear (p < 0.05), and weight of 1000 seeds (p < 0.05). The Mamdani Fuzzy Inference System (MFIS) models demonstrated that under eCO₂ conditions, the predicted 1000-seed weight stabilized around 40 g/plant in AMF-inoculated wheat, compared to approximately 37 g/plant in uninoculated plants. Similarly, ear length simulations showed stabilization at around 14 cm with AMF inoculation under eCO₂, versus 12.2 cm without AMF. These results highlight the synergistic effects of eCO₂ and AMF inoculation on key wheat productivity parameters.

Conclusion

This study underscores the importance of integrating fuzzy logic-based approaches into agricultural management strategies to optimize crop yields while minimizing environmental impacts. The findings encourage further research into refining experimental designs and expanding datasets to enhance our understanding of plant responses to changing environmental conditions.

Supplementary Information

The online version contains supplementary material available at 10.1186/s12870-025-06642-3.

Introduction

Arbuscular Mycorrhizal Fungi (AMF) represent a pivotal component of terrestrial ecosystems, forming symbiotic associations with a vast array of plant species. This intricate relationship between plants and AMF has garnered significant attention due to its profound impacts on plant growth, nutrient acquisition, and stress tolerance. Despite plants’ inherent defense mechanisms against environmental stresses, external applications of tolerance-inducing agents (e.g., AMF, plant hormones, or biostimulants) have emerged as a promising avenue to bolster plant resilience and productivity [1–4].

Elevated atmospheric CO₂ (eCO₂) concentrations, resulting from anthropogenic activities, have sparked considerable interest in their potential effects on plant physiology and agricultural productivity. Research has demonstrated that eCO₂ can stimulate photosynthesis, enhance carbohydrate accumulation, and confer resistance to nutrient deficiencies and toxic ions [5, 6]. Furthermore, eCO₂ levels have been linked to increased colonization of plant roots by AMF, attributed to enhanced plant growth and root exudation, facilitating mycorrhizal establishment and growth [7–10].

Wheat (Triticum aestivum L.), a cornerstone of global food security, is a staple crop highly sensitive to environmental changes, making it an ideal candidate to study the effects of eCO₂ and AMF symbiosis on productivity. This crop is cultivated across diverse climatic regions to meet the demands of a growing population. However, optimizing wheat production faces multifaceted challenges, including fluctuating environmental conditions, soil health and the availability of nutrients [11]. Understanding the intricate interplay between wheat, AMF, and eCO₂ is imperative for developing sustainable agricultural practices that ensure food security while mitigating environmental impacts [12–16].

The Mamdani Fuzzy Inference System (MFIS) is a powerful approach to modeling uncertainty and imprecision in agricultural systems. It is based on fuzzy logic to represent human knowledge in a more flexible way, allowing better decision making in complex and variable environments, commonly found in agriculture [17, 18]. By considering linguistic and imprecise variables such as “high”, “medium” and “low” rather than just precise numerical values, the Mamdani system can capture nuances and exceptions that are crucial to optimizing agricultural processes [19, 20]. The MFIS is widely used for modeling uncertainty and imprecision in agricultural systems, as demonstrated in recent studies exploring its efficacy under variable environmental conditions [17].

This approach makes it possible to create more robust and adaptable models, capable of dealing with climate variations, soil conditions and other unpredictable variables, thus contributing to increasing efficiency and productivity in agriculture [21–23]. Furthermore, fuzzy models can work with less information than traditional crisp models [24].

Among the various studies that propose using fuzzy systems (logic) to predict wheat productivity, none of them consider CO₂ levels together with inoculation with Rhizophagus irregularis as input [25–28]. They are mostly based on forecasting using time series. Thus, we found that there is a gap in the study of predicting inoculated wheat productivity in an environment with elevated CO₂, when there are few samples to build the models.

In this work we propose the use of MFIS to model the relationship between the CO₂ level and variables related to wheat productivity, namely: Seed yield, Ear length, Number of spikelets per ear, Number of grains per ear and Weight of 1000 seeds. The models are also obtained for environments inoculated and not inoculated with AMF (Rhizophagus irregularis). Finally, using the models obtained from the experimental data, computational simulations are carried out to verify the predictions of each variable.

Methods

Experimental design and treatments

A commercially pure inoculum of Rhizophagus irregularis (MUCL 41833) was acquired from the Glomeromycota in vitro collection (GINCO) (www.mycorrhiza.be/ginco-bel) and added to pots (25 x 15 cm). Rhizophagus irregularis belongs to the Glomeromycotina subphylum, Glomeromycetes class, Glomerales order, and Glomeraceae family. Rhizophagus irregularis has a multinucleate mycelium that forms highly branched arbuscules within plant roots. It produces round, hyaline spores (50–150 µm), aiding in nutrient exchange and plant symbiosis, particularly for phosphorus uptake. Each pot received 10 g of trapped soil containing approximately 50 spores per 1 g of soil a standard inoculum concentration, measured using Hemocytometer Counting. Autoclaved inoculum was added in equal amounts to the control pots to ensure equivalent nutrient levels apart from mycorrhizal spores. The soil mixture consisted of sterilized Tref EGO substrates (30%) and sterilized sand (70%) at 68% soil water capacity. Half of the soil was inoculated with Rhizophagus irregularis MUCL 41833 (AMF), while the other half remained uninoculated. Wheat (Triticum aestivum L., cv Giza 112) seeds, obtained from the Agriculture Research Center, Giza, Egypt, were surface sterilized with 5% v/v sodium hypochlorite for 20 minutes. Plants were grown in pots (8 cm diameter x 10 cm depth) containing either AMF-inoculated or uninoculated soil.

The experimental soil initially contained 11.7 mg C, 14.8 mg nitrate-N, 1.1 mg ammonium-N, and 9.4 mg P/g air dry soil, with a pH of 7.56, EC of 3.4 dS/m, and K of 2.75 meq/L. Plants were subjected to ambient CO₂ (aCO₂) at 375 ± 17 ppm or eCO₂ at 645 ± 22 ppm conditions, reflecting IPCC-SRES B2-scenario predictions for the year 2100, these levels were chosen to represent an intermediate baseline (near early 21 st-century concentrations) and a higher CO₂ scenario based on IPCC projections for the year 2100 [29], ensuring relevance to future climate trajectories [29]. Aiming to mimic anticipated future atmospheric conditions CO₂ levels were monitored continuously using a CO₂ analyzer with automated control (WMA-4, PPSystems, Hitchin, UK) to ensure steady atmospheric conditions in the climate chamber. The plants were grown in a custom-built climate-controlled chamber with a 16/8 h day/night photoperiod, maintaining 60% humidity and 150 µmol PAR m⁻² s⁻¹ light intensity.

The experiment followed a randomized complete block design with five replicates per treatment, and pots were regularly watered with tap water. For the analysis of seed yield, ear length, number of spikelets per ear, number of grains per ear, and the weight of 1000 seeds, these measurements were taken after 12 weeks to ensure full seed maturation. We harvested the wheat after 12 weeks, as this period allowed the seeds to reach complete maturity, providing more accurate data on the final seed characteristics.

Fuzzy logic and the proposed approach

Fuzzy logic is a modeling paradigm that allows us to deal with uncertainty and imprecision in complex systems, such as those found in agriculture. Instead of binary or precise values, fuzzy logic works with fuzzy sets, where elements have varying degrees of membership to a set [17, 19, 21, 23]. This allows for a more natural and flexible representation of human knowledge, making it especially useful for systems where variables are difficult to quantify precisely, as is the case with CO₂ concentration.

The MFIS is a practical implementation of fuzzy logic, widely used in several areas, including agriculture [19, 20]. It consists of a structure composed of three main steps: fuzzification of input variables, application of fuzzy rules and defuzzification to generate a crisp output. Mamdani is particularly suitable for complex systems, as it allows the expression of imprecise knowledge in terms of linguistic rules. Membership functions play a fundamental role in fuzzy logic as they define how input values are mapped to fuzzy sets [22–25]. A common membership function is the trapezoidal function, which allows you to effectively model the smooth transition between low and high membership values. Details about this membership function and the modeling scheme are in the supplementary material. The fuzzy modeling proposed here is based on experimental data. To calculate the parameters of the input membership functions, we use the measured CO₂ values, with ambient CO₂ (aCO₂) ranging from 358 to 392 (375±17), and eCO₂ ranging from 623 to 667 (645±22).

Based on observations of experimental data and expert knowledge, for each output variable we constructed two rules, considering an input $x$ and an output $y$: R1: “If ($x$ is aCO₂) then ($y$ is Low)” and R2: “If ($x$ is eCO₂) then ($y$ is High)”. For defuzzification, the centroid method was used. Since the information about inoculation or not is a crisp value, we do not consider this information as input. Therefore, the models were designed separately for data from the inoculated and non-inoculated environment. The computational implementation and fuzzy modeling script were written using the GNU Octave Fuzzy Logic Toolkit package [30]. Figure 1 shows a flowchart of the proposed approach, in two stages. In the first, fuzzy models are built from the dataset using extreme values, means and standard deviations, according to the calculations specified in the previous paragraphs. In the second stage, simulations are carried out using the fuzzy models obtained in the first stage. These simulations aim to predict yield variables.

Fig. 1.

Flowchart of the proposed approach for obtaining fuzzy wheat yield models and for simulating predictions

Statistical analysis

Statistical analyzes were performed on the Google Colab [31] platform using Python and the Scipy and Statsmodel packages [32, 33]. The Kruskal-Wallis test was applied to calculate the H statistics and the p-value results were analyzed at a significance level of 5% for each of the treatments. It was used instead of analysis of variance (one-way ANOVA) as the variables did not pass the normality and homoscedasticity tests.

Results

Table 1 presents the results obtained by the non-parametric Kruskal-Wallis test for both treatments followed by the Tukey post-hoc test. Considering that the null hypothesis is that there is no mean difference between treatments, and a significance level of 5%, the significant variables in each treatment are highlighted with the symbols “^*” or “^#” for comparisons of p-values for Kruskal-Wallis and Tukey test, respectively. We also consider marginal p-values (values ​​very close to the established significance level, but which do not necessarily reach the exact limit), close to 5%, as significant. Table 1 also includes the coefficient of variation for each variable and the H statistic from the Kruskal-Wallis test.

Table 1.

H statistics and p-values of the Kruskal-Wallis test and the p-adjusted of the Tukey pairwise comparison, and the coefficient of variation

Measure	Variables
	Seed yield	Ear length	Number of spikelets per ear	Number of grains per ear	Weight of 1000 seeds
CV (%)	13.7496	10.8334	11.5605	12.3375	8.0818
Treatment: CO2
H	1.6410	4.3333	2.0769	0.4102	2.0769
p-value	0.2001	0.0373*	0.1495	0.5218	0.1495
p-adjusted	0.1112	0.0281#	0.0776	0.3998	0.1985
Treatment: AMF
H	4.3333	0.1025	0.6410	3.6923	3.6923
p-value	0.0373*	0.7487	0.4233	0.0546*	0.0546*
p-adjusted	0.0315#	0.5226	0.1776	0.0370#	0.0508#

^* and ^# means value < 0.05 or close to it, for Kruskal-Wallis and Tukey tests, respectively

Based on the modeling scheme of the FIS input functions presented in the “Fuzzy logic and the proposed approach” section, Fig. 2 shows the input membership functions for both CO₂ levels (ambient and elevated). This input model is used for all variables as CO₂ levels are the same regardless of the variable. Tables 2, 3, 4, 5 and 6 illustrate the output membership functions for each variable. Since inoculation affects their values, these functions were modeled differently for the inoculated and non-inoculated environment. The modeling of all functions is based on experimental data. Furthermore, the tables also display the results of the simulations, considering the universe of discourse for each of the input and output variables.

Fig. 2.

Input membership function for level of CO₂ according to experimental data

Table 2.

Output membership functions and simulation results for variable ear length

Table 3.

Output membership functions and simulation results for variable number of grains per ear

Table 4.

Output membership functions and simulation results for variable seed yield

Table 5.

Output membership functions and simulation results for variable number of spikelets per ear

Table 6.

Output membership functions and simulation results for variable weight of 1000 seeds

Table 2 presents the membership functions and simulation results for the ear length variable, considering the scenarios with and without inoculation with AMF. The membership functions indicate the gradual transition between the “Low” and “High” fuzzy sets for ear length. In the treatment with AMF, the length varies from approximately 11 cm to 15 cm, with the transition occurring around 12.5 cm, while the “Low” set has a maximum membership degree close to 11 cm and the “High” set is maximized from 14 cm. In the treatment without AMF, the ear length varies from approximately 10.5 cm to 13 cm, with the transition occurring around 11.8 cm, and the “Low” set maximized at 10.5 cm and the “High” set around 12.5 cm. The simulations show that ear length increases significantly with increasing CO₂ concentration, especially from 500 ppm, stabilizing at around 13.8 to 14 cm for inoculated plants and around 12.2 cm for non-inoculated plants.

The effect of AMF inoculation on the number of grains per spike, under varying CO₂ concentrations, is detailed in Table 3, which includes both membership functions and simulation results. The simulations show that elevated CO₂ generally increases the number of grains per ear, particularly beyond 500 ppm. However, the extent of this increase differs between inoculated and uninoculated plants. Specifically, the number of grains per ear stabilizes around 52 to 54 for inoculated plants and 44 to 45 for non-inoculated plants. Examining the membership functions, we observe that for AMF-inoculated plants, the transition between “Low” and “High” fuzzy sets occurs between approximately 44 and 54 grains, with the transition point around 49. The “Low” set reaches maximum membership near 44 grains, and the “High” set is maximized from 52 grains onward. In contrast, for uninoculated plants, the number of grains per spike varies from approximately 38 to 46, with the transition occurring around 42. The “Low” and “High” sets reach their maximum membership at 38 and 45 grains, respectively. This comparison highlights the differing responses to CO₂ and AMF inoculation in terms of the number of grains per spike.

The impact of AMF inoculation on seed yield per plant, under varying CO₂ concentrations, is detailed in Table 4, which includes both membership functions and simulation results. The simulations reveal that elevated CO2 generally enhances productivity, particularly beyond 500 ppm. However, the extent of this enhancement differs between inoculated and uninoculated plants. Specifically, seed yield stabilizes at approximately 2.6 g/plant with AMF inoculation and 1.9 g/plant without it. Examining the membership functions, we observe that for AMF-inoculated plants, the transition between “Low” and “High” fuzzy sets for seed yield occurs between 1.5 and 2.5 g/plant, with peak “Low” membership at 1.5 g and peak “High” membership at 2.5 g. In contrast, for uninoculated plants, this transition happens between 1.4 and 1.8 g/plant, with maximum “Low” membership at 1.4 g and maximum “High” membership at 1.8 g. This comparison underscores the differing responses to CO₂ and AMF inoculation in terms of seed yield.

Table 5 presents the fuzzy logic simulation results for the “Number of spikelets per ear” in wheat, comparing AMF-inoculated (“With”) and uninoculated (“Without”) conditions under varying CO₂ levels. The membership function graphs illustrate the fuzzy sets “Low” and “High” for the number of spikelets, showing how the model defines these categories. The simulation results demonstrate a clear trend: as CO₂ levels increase, the number of spikelets per ear also increases in both AMF-inoculated and uninoculated treatments. However, the magnitude of the increase is more pronounced in the AMF-inoculated plants, indicating a potential synergistic effect between elevated CO₂ and AMF on spikelet development.

Table 5 presents the fuzzy logic simulation results for the “Number of spikelets per ear” in wheat, comparing AMF-inoculated (“With”) and uninoculated (“Without”) conditions under varying CO₂ levels. The membership function graphs illustrate the fuzzy sets “Low” and “High” for the number of spikelets, showing how the model defines these categories. The simulation results demonstrate a clear trend: as CO₂ levels increase, the number of spikelets per ear also increases in both AMF-inoculated and uninoculated treatments. However, the magnitude of the increase is more pronounced in the AMF-inoculated plants, indicating a potential synergistic effect between elevated CO₂ and AMF on spikelet development.

Discussion

Statistical analysis (Table 1) was conducted using the Kruskal-Wallis test, followed by pairwise treatment comparison with Tukey’s p-value adjustment, in addition to calculating the coefficient of variation for each variable of interest. The results revealed significant differences in some measures, while others were not affected by the treatments. When analyzing the coefficient of variation (CV), it can be observed that data variability was relatively low for most variables, with values ranging from 8.0818% to 13.7496%. This suggests consistency in the data obtained under different experimental conditions. This low variability can be attributed to the rigorous control of the experimental conditions, including standardization of the soil substrate, precise application of CO₂ and AMF treatments, and maintenance of controlled climatic conditions in the growth chamber. According to [31, 32], in agricultural studies, CVs between 10% and 20% are often considered acceptable, reflecting the natural variation inherent in biological systems. However, the CVs observed in this study are even lower, indicating exceptional precision.

When examining the results of the Kruskal-Wallis test for the CO₂ treatment, it was found that there was a significant difference (p-value = 0.0373) in Ear length that could be related to increased photosynthesis at eCO₂ levels [10]. On the other hand, other variables did not show statistically significant differences. This may be due to the complex interaction between environmental factors and wheat physiology. However, after Tukey’s p-value adjustment, Ear length remained significantly different between CO₂ treatments (p-adjusted = 0.0281), indicating that CO₂ increase had a positive impact on this specific measure. On the other hand, the AMF treatment showed statistically significant differences in several variables. The number of spikelets per ear (p-value = 0.0373), the Number of grains per ear (p-value = 0.0546), and the Weight of 1000 seeds (p-value = 0.0546) were all significantly affected by the AMF treatment. After Tukey’s p-value adjustment, the Number of spikelets per ear and the Number of grains per ear remained significantly different between AMF treatments, with p-adjusted of 0.0315 and 0.0370, respectively. These results suggest that inoculation with AMF may promote an increase in seed production under certain conditions. This result is confirmed in another study which reported that AMF can improve P absorption, contributing to promoting plant growth [34]. And also, through the literature review study, where the authors found that AMF inoculation in wheat fields resulted in a significant increase in above-ground biomass, grain yield and harvest index [35].

These results indicate that both CO₂ and inoculation with AMF may have significant effects on different measures related to seed production. However, it is important to note that the impact of these treatments may vary depending on the variable considered, highlighting the complexity of interactions between plants and their environment.

According to the meta-analysis [35], the main factors that influenced the response of wheat to AMF inoculation are the concentration of organic matter, pH, total N and the concentration of P available in the soil, in addition to the climate and the inoculated species of AMF.

The FIS input membership function, Fig. 2, maps CO₂ levels to “Ambient” and “Elevated” linguistic variables. Due to the characteristic of the trapezoidal function, the degree of pertinence of one function increases as the other decreases. The intersection occurs at the value 510 ppm (the average of the central values), where the degrees of pertinence are the same. This model is suitable for dealing with uncertainties and nuances related to CO₂ variability, which constitutes an intrinsic attribute of fuzzy models [19], using little information from the environmental system for its modeling [24].

The results of the simulations obtained using defuzzification and the centroid method in the trapezoidal membership function models shown in Tables 2, 3, 4, 5 and 6 show the iteration between the increase in CO₂ when inoculated with AMF. The results show that the values predicted by the FIS models (the output variables) increase when inoculation occurs. This finding was observed in other studies, which showed that eCO₂ significantly improves AMF colonization [7], one of the causes being better plant growth and metabolism, as it leads to a greater allocation of sugars [10]. And AMF contributes to plant growth and productivity by providing essential inorganic nutrients to the host [35]. These findings align with the broader understanding of AMF’s role in enhancing plant growth and nutrient acquisition under various conditions [4]. Similar results were observed in studies focusing on AMF inoculation under variable environmental conditions, where increased nutrient acquisition was directly linked to enhanced plant productivity [36].

Still on the results of the simulations, it is observed that an abrupt transition occurred in the prediction of output variables when CO₂ goes from “Ambient” to “Elevated”. This result is due to having considered only two linguistic variables to model the output, that is, “Low” and “High”. This is a consequence of the number of small samples obtained in the experiment, which made more accurate modeling impossible. If we had access to a larger sample size, we could have an intermediate linguistic variable like “Middle”, which would make the output results transition more smoothly. However, this is acceptable in computational models that only represent an approximation of reality [37]. Therefore, in future research we intend to design the experiment in order to better approximate the model to reality.

We can also see that the simulations reflected the results of the statistical tests. From Table 1 we note that the variables Ear length and Number of spikelets per ear did not present statistical significance in relation to inoculation with AMF, with p-values above 5%. With emphasis on the Ear length variable with the highest p-value.

The consistent results observed in the fuzzy model simulations for variables such as “Ear Length” and “Spikelets per Ear” (Tables 2 and 5) can be attributed to several factors. Firstly, statistical analysis revealed a lack of significant differences in these variables in response to AMF inoculation, indicating a similar or negligible effect of AMF across varying CO₂ levels. Consequently, the fuzzy models, grounded in experimental data, mirrored this lack of significant variation by producing comparable output patterns regardless of inoculation. Secondly, both raviables exhibited a similar response to changes in CO₂, which significantly influenced “Ear Length.” Given that the fuzzy model input is uniform across variables, reflecting CO₂ levels, this shared response naturally led to similar output trends. Additionally, the model’s sensitivity to CO₂, which had a pronounced impact on “Ear Length” and a potentially related effect on “Spikelets per Ear,” suggests that the simulations primarily reflected this CO₂-driven influence. Furthermore, the inherent relationship between “Ear Length” and “Spikelets per Ear,” as related morphological traits, likely contributed to similar simulation patterns.

On the opposite side, the simulation of the outputs for the Seed yield variable shows two very different forms. And, in Table 1 we see that this variable is the one that presents the lowest p-value and p-adjusted in response to inoculation with AMF. This highlights the suitability of the proposed fuzzy model, despite the approximations made, as it is an intrinsic feature of Mamdani systems which are seen as a form of approximate reasoning [38].

The symbiosis with arbuscular mycorrhizal fungi (AMF) can enhance the uptake of phosphorus (P), an essential nutrient for grain development [7, 8, 10]. Under eCO₂ conditions, wheat exhibits increased photosynthetic activity, resulting in greater carbohydrate production. However, this growth is often limited by the availability of nutrients, particularly P. Consequently, carbohydrates are transferred to AMF, optimizing fungal growth and expanding their hyphal network in the soil. This expansion increases the soil exploration area, allowing plants to access greater volumes of nutrients—especially phosphorus—and water, which is particularly beneficial in nutrient-poor or degraded soils. These findings are consistent with studies reporting AMF-mediated phosphorus mobilization under eCO₂ conditions [7, 8, 10] and AMF’s ability to improve nutrient uptake and overall plant growth [4].

In addition to phosphorus, AMF also positively influence nitrogen metabolism in plants. According to Alsherif et al. [39] AMF inoculation in maize under drought stress significantly increased the levels of amino acids such as arginine, glutamine, and glutamic acid, suggesting enhanced nitrogen assimilation. This response is associated with the fungi’s efficiency in acquiring both organic and inorganic nitrogen, as well as their ability to modulate key nitrogen metabolism enzymes, such as nitrate reductase [39, 40]. Thus, AMF not only improve nutrient uptake but also promote the accumulation of nitrogen-containing compounds essential for osmoregulation and antioxidant defense, such as proline and arginine.

Additionally, AMF act as elicitors, activating plant defense responses even before the onset of environmental stress and enhancing plant resilience to these stresses [41]. This priming effect strengthens the antioxidant system and modulates primary and secondary metabolism, rendering plants more resilient to adverse conditions such as drought or eCO₂ [2, 8, 12]. Such priming is critical for enabling plants to adapt more rapidly to the challenges imposed by climate change.

Therefore, the observed increases in grain yield components under AMF inoculation suggest a nutrient- and defense-mediated synergy, in which the plant supplies carbohydrates to the AMF, which in turn enhance phosphorus, nitrogen, and other nutrient uptake rates while promoting a physiological state better prepared to withstand environmental stresses—thereby maximizing the benefits of CO₂-driven photosynthesis.

These results demonstrate the significant positive impact of eCO₂ and AMF inoculation on wheat yield parameters, notably increasing ear length, spikelet number, grain number, and 1000-seed weight. For agricultural policymakers and practitioners, these findings underscore the potential of leveraging both enhanced CO₂ environments and AMF inoculation to bolster wheat productivity, especially in the face of predicted climate change scenarios. Implementing strategies that promote AMF colonization, such as reduced tillage and crop rotation, alongside optimizing CO₂ levels in controlled environments, could lead to substantial yield improvements. These practical implications highlight the need for integrating these factors into sustainable agricultural practices to ensure food security and maximize crop output, particularly in regions facing environmental challenges.

Conclusion

In this study, we investigated the intricate interplay between elevated atmospheric CO₂ (eCO₂), AMF inoculation, and wheat productivity, using the MFIS to model their relationships. Our findings shed light on the dynamic responses of wheat to changing environmental conditions and mycorrhizal symbiosis, offering valuable insights for sustainable agricultural practices. Statistical analyses revealed significant effects of both CO₂ levels and AMF inoculation on various measures related to wheat seed production. While CO₂ increase positively influenced ear length, AMF inoculation notably enhanced the number of spikelets per ear, number of grains per ear, and weight of 1000 seeds. These results align with previous research highlighting the role of AMF in promoting plant growth and nutrient acquisition, particularly phosphorus absorption, contributing to increased productivity.

The MFIS models provided a robust framework for capturing the complexities of CO₂ and AMF effects on wheat productivity, despite the inherent uncertainties and limited experimental data. Simulations demonstrated the synergistic relationship between elevated CO₂ and AMF inoculation, with predicted output variables consistently increasing under combined treatments. Although the transition from ambient to elevated CO₂ levels exhibited abrupt changes in predictions, reflecting the limitations of small sample sizes, the models effectively approximated the reality of wheat responses to environmental effects.

The fuzzy logic approach offers a flexible framework that can be readily adapted and expanded for other crops and environmental conditions. To enhance its applicability, future adaptations could incorporate a wider array of input variables, such as temperature, humidity, soil nutrient levels, and specific stress factors relevant to different crops. For instance, when modeling rice cultivation, flood depth and water salinity could be included. Expanding the linguistic variables beyond “Low” and “High” to include “Medium” or “Optimal” ranges would allow for a more nuanced representation of crop responses. Additionally, integrating dynamic fuzzy logic systems capable of adapting to real-time environmental changes would provide a more accurate and responsive model for precision agriculture. Finally, incorporating crop-specific physiological data and expert knowledge into the fuzzy rule base would tailor the model to the unique requirements of various cultivations, ensuring its effectiveness across diverse agricultural scenarios.

Our study underscores the importance of considering both CO₂ levels and AMF symbiosis in agricultural management strategies aimed at enhancing wheat productivity. By leveraging fuzzy logic-based approaches like MFIS, we can navigate the inherent uncertainties of agricultural systems, making informed decisions to optimize crop yields while minimizing environmental impacts.

These findings highlight the importance of incorporating CO₂ enrichment and AMF inoculation into future agricultural practices, especially in areas where environmental changes due to climate change are expected. By optimizing these factors, wheat productivity may be improved, contributing to global food security.

Future research endeavors should focus on refining experimental designs and expanding datasets to further refine and validate fuzzy models, ultimately advancing our understanding of plant-microbe-environment interactions in agroecosystems. Specifically, future studies should aim to increase sample sizes to allow for the incorporation of more linguistic variables in the fuzzy models, such as ‘Middle’ or ‘Moderate,’ to create smoother transitions in output predictions and better reflect the nuanced responses of wheat to varying CO₂ levels. Additionally, exploring a wider range of CO₂ concentrations and AMF inoculation rates would provide a more comprehensive understanding of their interactive effects. Incorporating time-series data to track dynamic changes in plant growth and physiological responses over the entire growth cycle would also enhance model accuracy. Furthermore, integrating soil microbiome analyses and detailed nutrient cycling studies would offer valuable insights into the mechanisms underlying AMF’s beneficial effects. Finally, validating these models across diverse wheat cultivars and environmental conditions, including field trials, would ensure their robustness and applicability in real-world agricultural settings.

Authors’ contributions

RLS, BRO, AMZ, MCMTF: conceptualization, methology, investigation and writing; ACMF, TZ, MKO, IAA, YAA: methology, investigation; YAH, AK, SS, MSS, MA, AA: Writing the first draft of the paper— review & editing the final version of this manuscript. RLS, BRO: supervision, conceptualization, writing — review & editing, methology, resources, funding acquisition.

Funding

The authors extend their appreciation to the Ongoing Research Funding Program (ORF-2025-176), King Saud University, Riyadh, Saudi Arabia.

Data availability

The data and computational scripts will be provided by simple request to the corresponding authors, with justification for the request.

Declarations

Ethics approval and consent to participate

Not applicable.

Consent for publication

Not applicable.

Competing interests

The authors declare no competing interests.