Optimization of fertilizer recommendations using response surface methodology and crop simulation models

Introduction

Here's the challenge facing agronomists: blanket fertilizer recommendations, often based on regional averages, routinely over-or under-supply nutrients at the field level, wasting money and polluting waterways [1] . Site-specific nutrient management promises better outcomes, but generating field-specific recommendations through conventional multi-year trials is prohibitively slow and expensive [2] . Two tools can accelerate this process. RSM is a statistical technique that uses efficient experimental designs (like Box-Behnken) to model the response surface of yield as a function of multiple nutrient inputs, identifying the combination that maximizes yield or profit with far fewer plots than a full factorial trial [3] . Crop simulation models like DSSAT-CERES can then validate RSM predictions under different weather scenarios, adding robustness to single-season empirical results [4] . Separately, both tools have been widely applied in agronomy. But their integration using RSM to define the response surface and DSSAT to stress-test the optimum across weather variability remains uncommon, particularly for European wheat systems [5] . This research aimed to demonstrate this integrated framework for optimizing N-P-K rates for winter wheat in southwestern Germany [6,7] .

Theoretical background

Response surface methodology models the relationship between a response variable (yield) and multiple input factors through a second-order polynomial: Y = β0 + ΣβiXi + ΣβiiXi² + ΣβijXiXj + ε, where Xi are the coded nutrient levels, β coefficients represent linear, quadratic, and interaction effects, and ε is the error term [3] . The Box-Behnken design was chosen because it requires fewer runs than a central composite design while still estimating all second-order terms. The DSSAT-CERES wheat model simulates daily crop growth using weather, soil, and Management inputs, calibrated against local phenology and yield data. By running the DSSAT model at the RSM-optimized nutrient rates across 20 years of historical weather, we tested whether the optimum held across seasons or was an artifact of single-year conditions [8,9] .

Materials and Methods

Materials

The field experiment was conducted over two seasons (2021-2022 and 2022-2023) at the University of Hohenheim's Ihinger Hof Research Station, Stuttgart, Germany (48°44′N, 8°55′E, elevation 478 m). The soil was a Luvisol (haplic) with pH 6.8, organic C 1.4%, and texture of silt loam. Winter wheat variety 'Reform' was sown at 380 seeds/m² in mid-October each year. N was applied as calcium ammonium nitrate, P as triple superphosphate, and K as muriate of potash. The Box-Behnken design generated 15 treatment combinations (plus 3 centre points), each replicated 3 times across 54 plots per season [10] .

Methods

Individual plot size was 6 m × 1.5 m. N was split 40% basal and 60% at GS30 (stem elongation). P and K were applied entirely at sowing. Grain yield was harvested at 14% moisture. RSM analysis was conducted in Design-Expert v13, and the desirability function was used to maximize yield while constraining total fertilizer cost below EUR 200/ha. DSSAT v4.8 was calibrated using 2021-2022 phenology and yield data, then validated against 2022-2023 observations. The RSMoptimized rates were input to DSSAT and run across 20 years (2003-2023) of Hohenheim weather data to assess yield stability. Economic analysis used 2023 German market prices: wheat EUR 230/t, N EUR 1.20/kg, P₂O₅ EUR 1.80/kg, K₂O EUR 0.90/kg [11,12] .

Results

Table The RSM model was highly significant (p<0.001) with R² = 0.913. Nitrogen was the dominant factor, accounting for 62.4% of yield variance. The N × P interaction was significant (p = 0.003), indicating synergistic effects between these nutrients. The RSM-optimized rates were N: 168, P₂O₅: 72, K₂O: 54 kg/ha, predicting a yield of 6.47 t/ha.

Comprehensive interpretation

The RSM-optimized rates delivered 10.8% higher net returns than blanket recommendations and 36% more than farmer practice, while using 32 kg/ha less N than the farmer's typical rate. DSSAT validation across 20 weather-years confirmed yield stability (CV = 8.3%), indicating the optimum isn't weatheryear-specific.

Discussion

The dominance of N in the response surface (62.4% of variance) confirms what agronomists have long known: nitrogen is the primary yield-limiting nutrient for wheat in western European soils [13] . The significant N × P interaction suggests that P availability modulates the crop's ability to convert applied N into grain a finding consistent with Vance et al. [14] . The close agreement between RSM prediction (6.47 t/ha) and DSSAT simulation (6.31 t/ha, 2.5% deviation) validates the integrated approach and gives confidence that the optimum is physiologically sound, not merely a statistical artifact [15] . Farmer practice applied 200 kg N/ha but only 60 kg P₂O₅/ha, creating a nutritional imbalance that the RSM optimization corrected by reducing N to 168 and increasing P to 72 kg/ha. This rebalancing improved both yield and profitability a counterintuitive result for farmers accustomed to equating more N with more grain [16,17] .

Performance evaluation

The DSSAT model was calibrated using 2021-2022 data (R² = 0.94, RMSE = 0.21 t/ha for phenology and yield) and independently validated against 2022-2023 observations (R² = 0.88, RMSE = 0.34 t/ha). The 20-year simulation at RSMoptimized rates yielded a mean of 6.24 t/ha (range: 5.38-6.97), with a CV of 8.3%. In only 2 of 20 years did simulated yield fall below 5.5 t/ha, both associated with spring drought periods. The desirability function achieved a composite score of 0.87 (maximum = 1.0), balancing yield maximization and cost constraint simultaneously [18,19] .

Limitations

The RSM model is empirical and site-specific extrapolation to soils with different textures or pH levels requires recalibration.

The Box-Behnken design, while efficient, limits the factor space to three levels per nutrient, potentially missing optima outside the tested range. DSSAT's soil organic matter module has known limitations for long-term nutrient cycling predictions.

And the economic analysis used fixed 2023 prices; fertilizer and grain price volatility could shift the optimal rates in other years [20, 21] .

Conclusion

The integrated RSM-DSSAT framework identified an optimal N-P-K combination (168:72:54 kg/ha) for winter wheat at Hohenheim that increased net returns by 10.8% over blanket recommendations while reducing nitrogen input by 12 kg/ha. The approach is reproducible, data-efficient, and adaptable to other crops and locations. Adoption by extension services could improve both farm profitability and environmental outcomes by replacing one-size-fits-all fertilizer advice with site-calibrated prescriptions.

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