# Phosphorus and Nitrogen Yield Response Models for Dynamic Bio-Economic Optimization: An Empirical Approach

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

^{−1}a

^{−1}) and clay soils (3.9 mg L

^{−1}a

^{−1}). For coarse soils, a higher steady-state P fertilization rate is required (21.7 kg ha

^{−1}a

^{−1}) compared with clay soils (6.75 kg ha

^{−1}a

^{−1}). The steady-state N fertilization rate was slightly higher for clay soils (102.4 kg ha

^{−1}a

^{−1}) than for coarse soils (95.8 kg ha

^{−1}a

^{−1}). This study shows that the iterative elimination of plausible functional forms is a suitable method for reducing the effects of structural uncertainty on model output and optimal fertilization decisions.

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Phase I: Conceptual Model and Data

^{−1}a

^{−1}. On clay soils, the corresponding ranges are 0–2, 2–3.5, 3.5–7, 7–14, 14–23, 23–40, and >40 mg l

^{−1}a

^{−1}[47]. STP was extracted from soil with acid ammonium acetate, pH 4.65 [48].

^{−8}) for coarse soils and correlation = −0.49 (p-value = 0.00027) for clay soils). The datasets were initially unbalanced panel datasets, implying that the datasets contained multiple observations for each cross-section unit in the sample, but the time dimension was not the same for each individual experiment. However, the observations in the applied datasets were averaged over the experimental years. Therefore, the observations from longer experiments were more reliable, since these data were less affected by random variation attributable to weather-driven events [50]. Notably, the observations within the datasets were serially correlated, since multiple measurements were obtained from the same site. Moreover, for model validation, two independent datasets were gathered, including 28 short-term and long-term NPK fertilizer field experiments conducted in Finland between 1964 and 1988 at 10 sites on clay and coarse-textured soils. All the eight applied datasets are provided in the supplemental material.

#### 2.2. Phase II: Model Development

^{−1}), $i\in \left\{1,\dots ,n\right\}$ is a particular fertilization experiment, ${y}_{P{0}_{i}}$ denotes the yield without added P (kg ha

^{−1}), ${\omega}_{{P}_{i}}\in \left(0,\infty \right)$ is a scaling factor for P fertilization, ${\omega}_{{N}_{i}}\in \left(0,\infty \right)$ is a scaling factor for N fertilization, ${f}_{l}$ is a given nonlinear function specified by a certain structure for a model element $l\in \left[1,3\right]$, ${\theta}_{l,j}$ indicates an unknown parameter with subscript $j$ denoting a parameter number, ${\epsilon}_{l,i}$ indicates the model residual error, and ${y}_{N{0}_{i}}$ denotes the yield without added N (kg ha

^{−1}). It is assumed that Equation (1) satisfies the typical properties of a production function [53]. The first element, defined by Equation (2), determines a yield without added P (kg ha

^{−1}) as a function of plant available soil P. The second element, defined by Equation (3), determines the yield response (%) to P fertilizer as a scaling factor that scales the first element up depending on the P application rate (kg P ha

^{−1}) and the STP level. The third element, defined by Equation (4), determines the yield response (%) to N fertilizer (kg N ha

^{−1}) as a scaling factor that scales the yield up or down depending on whether the N application rate is lower or higher than the average N rate in P experiments. The third model element is a decreasing function of yield without added N (kg ha

^{−1}). In model applications, ${y}_{N0}$ was treated as a parameter rather than a variable due to the non-availability of a suitable transition function describing the N dynamics in the soil. The absence of a model component describing the N stock development is recognized as a model limitation. We hypothesized that a yield without added N must be lower than that without added P because typically N has a stronger effect on yields compared with P or STP [30,54]. Therefore, ${y}_{N0}$ was fixed to a “reference-level” of 2/3${y}_{P0}$ (2148 kg ha

^{−1}and 2466 kg ha

^{−1}for coarse-textured and clay soils, respectively).

#### 2.3. Phase III: Verification and Validation

#### 2.4. Phase IV: Formulation of the Dynamic System Model

#### 2.5. Phase V: Model Application for Economic Optimization

#### 2.6. Phase VI: Uncertainty and Sensitivity Analysis

## 3. Results

#### 3.1. Model Estimation

#### 3.2. Verification and Validation Results

#### 3.3. Economically Optimal Nutrient Inputs and Uncertainty Analysis

^{−1}), is also minor. The relative (mean-scaled) effect of the structural uncertainty on model output was greater for coarse soils. Regardless, the relative cost of structural uncertainty was greater for clay soils.

## 4. Discussion

## 5. Conclusions

## Supplementary Materials

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A

#### Appendix A.1. Standardization of the Separate Datasets

#### Appendix A.2. Estimating the Yield Response Models

#### Appendix A.3. Estimated Models and the Yield Response Curves and Surfaces

**Table A1.**The estimated models and the summary of the model ranking results for coarse and clay soils

^{†}.

Soil Texture, Model Element ^{‡} | Model Structure ^{§} | $\mathit{R}\mathit{S}\mathit{S}$ | $\mathit{A}\mathit{I}\mathit{C}\mathit{c}$ | $\mathit{\Delta}$ | $\mathit{w}$ | ${\mathit{w}}_{1}/{\mathit{w}}_{\mathit{j}}$ |
---|---|---|---|---|---|---|

Coarse 1. Model element | ${\overline{y}}_{P0}=STP{\left({\theta}_{1}+{\theta}_{1}ST{P}^{1/2}\right)}^{-2}$ | 20.26 × 10^{6} | 317.0 | 0.00 | 0.42 | 1 |

${\overline{y}}_{P0}=\left({\theta}_{1}\mathrm{log}\left(STP+1\right)+1\right){\left(1+ST{P}^{1/2}\right)}^{-1/2}$ | 21.63 × 10^{6} | 318.5 | 1.51 | 0.20 | 2.13 | |

${\overline{y}}_{P0}={\theta}_{1}{\left(1+\mathrm{exp}\left({\theta}_{2}ST{P}^{-1/2}\right)\right)}^{-1/2}$ | 20.06 × 10^{6} | 319.2 | 2.18 | 0.14 | 2.97 | |

${\overline{y}}_{P0}={\theta}_{1}{\mathrm{STP}}^{{\theta}_{1}{}^{-1}}+{\theta}_{2}\mathrm{log}\left(STP+1\right)$ | 20.19 × 10^{6} | 319.4 | 2.34 | 0.13 | 3.22 | |

${\overline{y}}_{P0}={\theta}_{1}{\mathrm{STP}}^{{\theta}_{1}{}^{-1}}\left(1-{\theta}_{2}{}^{STP}\right)$ | 20.39 × 10^{6} | 319.6 | 2.56 | 0.12 | 3.59 | |

2. Model element | ${\overline{\omega}}_{P}={\theta}_{1}{P}^{1/2}{\left(1+\mathrm{exp}\left({\theta}_{2}ST{P}^{2}\right)\right)}^{-1}+1$ | 0.335 | −344.5 | 0.00 | 0.54 | 1 |

${\overline{\omega}}_{P}={\theta}_{1}\left(1-\mathrm{exp}\left(-{\theta}_{2}P\right)\right){\left(1+\mathrm{exp}\left({\theta}_{3}ST{P}^{2}\right)\right)}^{-1}+1$ | 0.3315 | −343.0 | 1.51 | 0.25 | 2.12 | |

${\overline{\omega}}_{P}={\theta}_{1}\mathrm{log}\left(P+1\right){\left(1+\mathrm{exp}\left({\theta}_{2}ST{P}^{2}\right)\right)}^{-1}+1$ | 0.3515 | −341.3 | 3.16 | 0.11 | 4.86 | |

${\overline{\omega}}_{P}={\theta}_{1}{P}^{1/2}{\left(1+{\theta}_{2}ST{P}^{2}\right)}^{-1}+1$ | 0.3532 | −341.0 | 3.49 | 0.09 | 5.73 | |

3. Model element | ${\overline{\omega}}_{N}=\left({\theta}_{1}+{\theta}_{2}{\left(log\left(N+1\right)\right)}^{2}\right){\left(1+{\theta}_{3}{Y}_{N0}{}^{2}\right)}^{-1}$ | 0.4828 | −300.5 | 0.00 | 0.27 | 1 |

${\overline{\omega}}_{N}=\left({\theta}_{1}{N}^{{\theta}_{1}{}^{-1}}+1\right){\left({\theta}_{2}{y}_{N0}{}^{2}+{\theta}_{3}\right)}^{-1}$ | 0.4836 | −300.4 | 0.10 | 0.26 | 1.05 | |

${\overline{\omega}}_{N}=\left({\theta}_{1}{N}^{1/2}+1\right){\left({\theta}_{2}{y}_{N0}{}^{2}+{\theta}_{3}\right)}^{-1}$ | 0.4882 | −299.8 | 0.70 | 0.19 | 1.42 | |

${\overline{\omega}}_{N}={\theta}_{1}\left(1-\mathrm{exp}\left(-{\theta}_{2}{N}^{{\theta}_{2}}\right)\right){\left(1+{\theta}_{3}{y}_{N0}{}^{2}\right)}^{-1}$ | 0.4920 | −299.3 | 1.20 | 0.15 | 1.82 | |

${\overline{\omega}}_{N}={\theta}_{1}+{\theta}_{2}N+{\theta}_{3}{y}_{N0}+{\theta}_{4}{N}^{2}+{\theta}_{5}N{y}_{N0}+{\theta}_{6}{y}_{N0}{}^{2}$ | 0.4416 | −299.0 | 1.48 | 0.13 | 2.09 | |

Clay 1. Model element | ${\overline{y}}_{P0}=exp{\left({\theta}_{1}log{\left(STP+1\right)}^{2}+{\theta}_{2}\right)}^{1/2}$ | 7.16 × 10^{6} | 236.8 | 0.00 | 0.29 | 1 |

${\overline{y}}_{P0}={\left({\theta}_{1}log{\left(STP+1\right)}^{2}+{\theta}_{2}\right)}^{2}$ | 7.22 × 10^{6} | 236.9 | 0.06 | 0.28 | 1.03 | |

${\overline{y}}_{P0}={\theta}_{1}{\mathrm{STP}}^{{\theta}_{1}{}^{-1}}+{\theta}_{2}ST{P}^{1/2}$ | 7.23 × 10^{6} | 237.0 | 0.22 | 0.26 | 1.12 | |

${\overline{y}}_{P0}={\theta}_{1}\left(ST{P}^{{\theta}_{2}}+1\right){\left(1+ST{P}^{1/2}\right)}^{-1}$ | 7.52 × 10^{6} | 237.8 | 0.95 | 0.18 | 1.61 | |

2. Model element | ${\overline{\omega}}_{P}={\theta}_{1}\left(\mathrm{ln}\left(P+1\right)+1\right){\left(1+{\theta}_{2}ST{P}^{2}\right)}^{-1}+{\omega}_{P,min}$ | 0.0698 | −264.5 | 0.00 | 0.34 | 1 |

${\overline{\omega}}_{P}={\theta}_{1}\left(\mathrm{ln}\left(P+1\right)+1\right){\left(1+exp\left({\theta}_{2}STP+1\right)\right)}^{-0.5}+{\omega}_{P,min}$ | 0.0711 | −263.7 | 0.78 | 0.23 | 1.476 | |

${\overline{\omega}}_{P}={\theta}_{1}\left({P}^{1/2}+1\right){\left(1+{\theta}_{2}ST{P}^{2}\right)}^{-1}+{\omega}_{P,min}$ | 0.0720 | −263.2 | 1.28 | 0.18 | 1.90 | |

${\overline{\omega}}_{P}={\theta}_{1}\left({P}^{{\theta}_{2}}+1\right){\left(1+{\theta}_{3}ST{P}^{2}\right)}^{-1}+{\omega}_{P,min}$ | 0.0684 | −263.0 | 1.49 | 0.16 | 2.110 | |

${\overline{\omega}}_{P}={\theta}_{1}\left(1+{\theta}_{2}exp\left(-{\theta}_{3}P\right)\right){\left(1+{\theta}_{2}ST{P}^{2}\right)}^{-1}+{\omega}_{P,min}$ | 0.0669 | −261.5 | 2.98 | 0.08 | 4.446 | |

3. Model element | ${\overline{\omega}}_{N}=\left({\theta}_{1}{N}^{1/2}+1\right){\left({\theta}_{2}+{\theta}_{2}{y}_{N0}{}^{1/2}\right)}^{-1}$ | 0.2945 | −250.3 | 0.00 | 0.58 | 1 |

${\overline{\omega}}_{N}=\left({\theta}_{1}{N}^{{\theta}_{1}{}^{-1}}+1\right){\left({\theta}_{2}+{\theta}_{2}{y}_{N0}{}^{1/2}\right)}^{-1}$ | 0.3041 | −248.7 | 1.61 | 0.26 | 2.239 | |

${\overline{\omega}}_{N}={\left({\theta}_{1}+{\theta}_{1}\mathrm{log}\left(N+1\right)+1\right)}^{2}{\left(1+{\theta}_{2}{y}_{N0}{}^{1/2}\right)}^{-1}$ | 0.3163 | −246.8 | 3.57 | 0.10 | 5.96 | |

${\overline{\omega}}_{N}={\theta}_{1}\left(1-\mathrm{exp}\left(-{\theta}_{2}{N}^{{\theta}_{2}}\right)\right)\left(1+{\theta}_{3}{y}_{N0}{}^{1/2}\right)$ | 0.3067 | −246.0 | 4.36 | 0.07 | 8.826 |

^{†}RSS is the sum of squared residuals, AICc is the second order variant of Akaike’s information criteria, Δ is the AIC difference, $w$ is the Akaike’s weight, and ${w}_{1}/{w}_{j}$ is the evidence ratio between the best model (1) and another model j, and the parameter ${\omega}_{P,min}$ is the minimum observation for yield response (0.96).

^{‡}The 1. Model element refers to Equation (2) in the main text describing the association between soil test P (STP) and yield without added P (${y}_{P0}{}_{i}={f}_{1}\left(ST{P}_{i};{\theta}_{1,j}\right)+{\epsilon}_{1,i}$), 2. Model element refers to Equation (3) in the main text describing the yield response to P fertilization (${\omega}_{P}{}_{i}={f}_{2}\left({P}_{i},ST{P}_{i};{\theta}_{2,j}\right)+{\epsilon}_{2,i}$), and the 3. Model element refers to Equation (4) in the main text describing the yield response to N fertilization (${\omega}_{N}{}_{i}={f}_{3}\left({N}_{i},{y}_{N0}{}_{i};{\theta}_{3,j}\right)+{\epsilon}_{3,i}$).

^{§}${\overline{y}}_{P0}$ indicates expected average yield without added P, STP indicates soil test P, ${\overline{\omega}}_{P}$ indicates the scaling factor for the average expected P response, ${\overline{\omega}}_{N}$ indicates the scaling factor for the average expected N response, and ${y}_{N0}$ indicates the yield without added N.

#### Appendix A.4. Model Combinations Applied in Analysis of the Structural Uncertainty

**Figure A1.**The data points and STP response curves for (

**a**) coarse and (

**b**) clay soils. The P response surfaces for (

**c**) coarse and (

**d**) clay soils, and the N response surfaces for (

**e**) coarse and (

**f**) clay soils provided by the best candidate models. For the third model element for clay soils, the random effects were set to their expected zero level.

#### Appendix A.5. Sensitivity Analysis Results

**Table A2.**The sensitivity of the steady-state N and P fertilizer rates and yields to the economic parameters of the model.

Parameter | Value | Steady-State P Rate | Steady-State N Rate | Steady-State Yield | |||
---|---|---|---|---|---|---|---|

Coarse Soils | Clays Soils | Coarse Soils | Clay Soils | Coarse Soils | Clay Soils | ||

P price (€/kg) | 1 | 27.9 | 13 | 101.8 | 108.6 | 4123 | 3644 |

1.99 (baseline) | 21.7 | 6.78 | 95.8 | 102.4 | 3959 | 3509 | |

3 | 15.7 | 4.06 | 87.5 | 98.4 | 3699 | 3416 | |

N price (€/kg) | 0.5 | 25 | 10.5 | 207 | 344.6 | 4635 | 4868 |

0.91 (baseline) | 21.7 | 6.78 | 95.8 | 102.5 | 3959 | 3509 | |

2 | 17.6 | 4.57 | 32.1 | 20.9 | 3183 | 2626 | |

Barley price (€/kg) | 0.06 | 5.79 | 1.78 | 28.5 | 23.8 | 2693 | 2600 |

0.12 (baseline) | 21.75 | 6.78 | 95.8 | 102.4 | 3959 | 3509 | |

0.24 | 31.17 | 19.25 | 244.8 | 443.2 | 4898 | 5387 | |

Discount rate (%) | 1 | 24.37 | 8.1 | 97.1 | 102.3 | 4029 | 3526 |

3.5 (baseline) | 21.75 | 6.78 | 95.8 | 102.4 | 3959 | 3509 | |

6 | 19.85 | 6.41 | 94.9 | 102.6 | 3897 | 3504 |

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**Figure 1.**Schematic diagram of the modeling process. AIC stands for Akaike information criteria [36]. STP, soil test phosphorus.

**Figure 2.**The conceptual farm-level dynamic decision model (adapted from Hardaker et al. [49]), where t denotes a stage (one year), STP is the soil test P, which is the state of the system, P and N denote annual input decisions, R denotes the annual state return, r is the stage return function, y is the annual crop yield,

**p**denotes the prices of inputs and output, and $\vartheta $ denotes a transition function that determines the evolution of the system from one stage to the next.

**Figure 3.**Above the simulated yield N-P-response surfaces for the fair STP class for (

**a**) coarse and (

**b**) clay soils. Below the yield as a function of STP for various P and N rates for (

**c**) coarse and (

**d**) clay soils.

**Figure 5.**Steady-state yields for (

**a**) coarse-textured soils and (

**c**) clay soils determined by the different model candidates, and the associated economic losses resulting from incorrect model specification measured in differences in net present value (NPV) for (

**b**) coarse and (

**d**) clay soils. The dashed line indicates the level associated with the best model candidate.

**Figure 6.**The effect the initial STP level on paths for STP development and for P and N fertilization for two fertilization management strategies.

**Figure 7.**The relationship between the initial soil test P (STP) and net present value (NPV) for the cases where the fertilization paths were determined by economic optimization and by applying the present maximum permissible rates (100 kg N ha

^{−1}a

^{−1}) and declining maximum permissible P rates for the increasing STP classes.

**Table 1.**Parameters and the associated statistics for the models describing the yield response to soil test P (STP), P and N fertilization on coarse and clay soils as per Equations (15) and (16)

^{†}.

Soil Texture | Parameter | Estimate | SD * | t-Value | P (>$|\mathbf{t}|$) |
---|---|---|---|---|---|

Coarse | ${\theta}_{1,1}$ | 0.0133 | 4.535 × 10^{−4} | 29.37 | <2.2 × 10^{−16} |

${\theta}_{2,1}$ | 0.0797 | 0.0126 | 6.33 | 2.747 × 10^{−8} | |

${\theta}_{2,2}$ | 0.0095 | 0.0031 | 3.09 | 0.0029 | |

${\theta}_{3,1}$ | 0.727 | 0.0742 | 9.787 | 4.791 × 10^{−14} | |

${\theta}_{3,2}$ | 0.0269 | 0.00269 | 10.01 | 2.023 × 10^{−14} | |

${\theta}_{3,3}$ | 3.653 × 10^{−8} | 6.694 × 10^{−9} | 5.46 | 9.699 × 10^{−7} | |

Clay | ${\theta}_{1,1}$ | 0.317 | 0.150 | 2.12 | 0.0493 |

${\theta}_{1,2}$ | 65.80 | 1.248 | 52.74 | <2.2 × 10^{−16} | |

${\theta}_{2,1}$ | 0.0372 | 0.0041 | 9.08 | 2.93 × 10^{−11} | |

${\theta}_{2,2}$ | 0.010 | 0.0045 | 2.232 | 0.0313 | |

${\theta}_{3,1}$ | 0.0799 | 0.021 | 3.81 | 6 × 10^{−4} | |

${\theta}_{3,2}$ | 0.0377 | 0.0048 | 7.91 | <2.2 × 10^{−16} |

^{†}The standard errors are heteroscedasticity-and-autocorrelation-consistent (HAC) for the coarse soils. In the case of first model element for the clay soils, the parameters were estimated via bootstrapping since the data sample was small. For the second model element, the standard errors were not HAC since the distribution of the residual errors was not significantly different from a normal one. Nonlinear mixed-effect (nlme) estimation was applied to the third model element.

**Table 2.**Evaluation of the goodness-of-fit between the yield response models and the initial datasets

^{†}.

Soil Texture | Number of the Model Element ^{‡} | $\mathit{r}$ | NSE | RMSE | MB | MAE |
---|---|---|---|---|---|---|

Coarse | (1) | 0.62 | 0.36 | 939 (29%) | −65 (2.0%) | 811 (25.2%) |

(2) | 0.8 | 0.63 | 0.07 (6.5%) | 0.01 (0.9%) | 0.05 (4.6%) | |

(3) | 0.87 | 0.76 | 0.09 (9.2%) | 0.01 (1.0%) | 0.07 (7.2%) | |

Clay | (1) | 0.46 | 0.21 | 630 (2.3%) | 32.28 (0.8%) | 458.1 (12%) |

(2) | 0.76 | 0.54 | 0.04 (3.8%) | 0.005 (4.7%) | 0.03 (2.9%) | |

(3) | 0.94 | 0.89 | 0.08 (8.5%) | 0.01 (0.6%) | 0.06 (6.4%) |

^{†}Pearson’s product-moment correlation coefficient between measured and simulated values is indicated by r, NSE is Nash−Sutcliffe efficiency, RMSE is root mean squared error, MB is the mean bias, and MAE is the mean absolute error. The statistics are a percentage of the observed mean values, in brackets.

^{‡}Model Element (1) refers to Equation (2), Model Element (2) refers to Equation (3), and Model Element (3) refers to Equation (4).

**Table 3.**Evaluation of the goodness-of-fit between the model simulations and the observations in the validation datasets

^{†}.

Soil Texture | r | NSE | RMSE | MAE | MB |
---|---|---|---|---|---|

Coarse | 0.81 | 0.66 | 452 (12%) | 338 (9.2%) | 21.1 (0.6%) |

Clay | 0.76 | 0.54 | 706 (21%) | 564 (17%) | 132 (4%) |

^{†}The mean-normalized statistics are shown in brackets. Pearson’s product-moment correlation coefficient between measured and simulated values is indicated by r, NSE is Nash−Sutcliffe efficiency, RMSE is root mean squared error, MAE is the mean absolute error, and MB is the mean bias.

Soil Texture | Test | Test Statistic | p-Value |
---|---|---|---|

Coarse | Shapiro−Wilk normality test | 0.95 | 0.124 |

t-test for coefficient $\alpha $ in Equation (5) | −35.9 | 0.96 | |

t-test for coefficient $\beta $ in Equation (5) | 1.004 | 3.66 × 10^{−6} | |

Paired t-test | −0.124 | 0.902 | |

Clay | Shapiro−Wilk normality test | 0.98 | 0.55 |

t-test for coefficient $\alpha $ in Equation (5) | −713 | 0.119 | |

t-test for coefficient $\beta $ in Equation (5) | 1.17 | 7.155 × 10^{−13} | |

Paired t-test | −0.87 | 0.39 |

**Table 5.**Steady states obtained by applying economically optimal fertilization rates and the maximum permissible fertilizer rates of the voluntary agro-environmental scheme in Finland

^{†}.

Soil Texture | Variable ^{‡} | Values Characterizing the Steady State | |
---|---|---|---|

Values Obtained by Dynamic Optimization | Values Obtained by Simulating the Model Using the Present Maximum Permissible Fertilization Rates ^{§} | ||

Coarse | N rate (kg N ha^{−1} a^{−1}) | 95.8 | 100 |

P rate (kg P ha^{−1} a^{−1}) | 21.7 | 16 | |

STP (mg P l^{−1} a^{−1}) | 9.88 | 7.85 | |

Yield (kg ha^{−1} a^{−1}) | 3958 | 3795 | |

Profit (€ ha^{−1} a^{−1}) | 345 | 333 | |

NPV (€ ha^{−1}) | 9614 | 9446 | |

Clay | N rate (kg N ha^{−1} a^{−1}) | 102.4 | 100 |

P rate (kg P ha^{−1} a^{−1}) | 6.75 | 16 | |

STP (mg P l^{−1} a^{−1}) | 3.86 | 6.81 | |

Yield (kg ha^{−1} a^{−1}) | 3509 | 3603 | |

Profit (€ ha^{−1} a^{−1}) | 314.3 | 309.6 | |

NPV (€ ha^{−1}) | 9006 | 8809 |

^{†}The exemplary fixed prices are 0.91€ kg

^{−1}, 1.99€ kg

^{−1}, and 0.12€ kg

^{−1}, for N, P, and barley, respectively. The discount rate is 3.5% and the initial STP levels are 7.5 mg L

^{−1}and 4.5 mg L

^{−1}for coarse and clay soils, respectively.

^{‡}STP indicates soil test P and NPV indicates net present value.

^{§}The present maximum permissible N rate is 100 kg ha

^{−1}and that of P is 34 kg ha

^{−1}for the poor STP class, 26 kg ha

^{−1}for the rather poor STP class, 16 kg ha

^{−1}for the fair STP class, 10 kg ha

^{−1}for the satisfactory STP class, 5 kg ha

^{−1}for the good STP class, and 0 kg ha

^{−1}for the high and excessive STP classes [69].

**Table 6.**Effect of the structural uncertainty on the robustness of the economic optimization results

^{†}.

Soil Texture | Variable ^{‡} | Average Steady-State Value (95% CI) | Range (min, max) | SD/Average |
---|---|---|---|---|

Coarse | N rate (kg N ha^{−1} a^{−1}) | 103 (73.7, 132) | (82.8, 125) | 0.139 |

P rate (kg P ha^{−1} a^{−1}) | 22.8 (19.6, 26) | (19.6, 25.7) | 0.07 | |

STP (mg P l^{−1} a^{−1}) | 10.1 (8.96, 11.2) | (9.08, 10.9) | 0.055 | |

Yield (kg ha^{−1} a^{−1}) | 4100 (3790, 4410) | (3852, 4421) | 0.037 | |

Profit (€ ha^{−1} a^{−1}) | 353 (342, 364) | (343, 366) | 0.016 | |

NPV (€ ha^{−1}) | 9846 (9503, 10190) | (9568, 10200) | 0.017 | |

$\u2206$NPV (€ ha^{−1}) | −31.8 (−86.9, 23.3) | (−92.3, 0) | −0.85 | |

Clay | N rate (kg N ha^{−1} a^{−1}) | 95.2 (84.9, 105) | (86.2, 103) | 0.053 |

P rate (kg P ha^{−1} a^{−1}) | 5.96 (1.01, 10.9) | (1.71, 9.36) | 0.41 | |

STP (mg P l^{−1} a^{−1}) | 3.79 (2.64, 4.93) | (2.81, 4.64) | 0.15 | |

Yield (kg ha^{−1} a^{−1}) | 3412 (3221, 3602) | (3241, 3546) | 0.027 | |

Profit (€ ha^{−1} a^{−1}) | 311 (303, 319) | (304, 317) | 0.013 | |

NPV (€ha^{−1}) | 8914 (8709, 9119) | (8730, 9068) | 0.011 | |

$\u2206$NPV (€ ha^{−1}) | −44.25 (−168, 79.54) | (−171.3, 0) | −1.37 |

^{†}The number of the model candidates was 30 for coarse soils and 32 for clay soils.

^{‡}STP indicates soil test P, NPV indicates net present value, and $\u2206$NPV indicates the difference in NPV between the best model candidate and any other model candidate.

**Table 7.**Summary of the Monte Carlo analysis results used for examining the effect of the parametric uncertainty on model output (the barley yield), associated with individual model elements and the entire model for coarse and clay soils

^{†}.

Soil Texture | Model Element Subject to Parametric Uncertainty | Average Yield (95% CI) | SD (95% CI) | 95% CI (for Mean Values) | SD/Average |
---|---|---|---|---|---|

kg ha^{−1} a^{−1} | |||||

Coarse | First model element | 3947 (+/−5) | 270 (+4, −3) | (3397, 4497) | 0.068 |

Second model element | 3938 (+/−2) | 101 (+2, −1) | (3732, 4144) | 0.026 | |

Third model element | 3936 (+/−6) | 303 (+/−4) | (3319, 4553) | 0.077 | |

Entire model | 3955 (+7, −9) | 421 (+/−6) | (3099, 4811) | 0.106 | |

Clay | First model element | 3518 (+5, −6) | 281 (+/−4) | (2967, 4069) | 0.08 |

Second model element | 3509 (+/−1) | 41(+/−0.6) | (3429, 3589) | 0.012 | |

Third model element | 3567 (+/−12) | 636 (+/−11) | (2320, 4814) | 0.178 | |

Entire model | 3578 (+/−14) | 702 (+/−12) | (2201, 4954) | 0.196 |

^{†}During Monte Carlo analysis, the values for N, P, and STP were fixed to their economically optimal steady state levels as shown in Table 5. The asymptotic 95% CI for the average yield and its standard deviations illustrate the uncertainty of the statistic.

^{‡}The first model element refers to Equation (2), the second model element refers to Equation (3), and the third model element refers to Equation (4).

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**MDPI and ACS Style**

Sihvonen, M.; Hyytiäinen, K.; Valkama, E.; Turtola, E. Phosphorus and Nitrogen Yield Response Models for Dynamic Bio-Economic Optimization: An Empirical Approach. *Agronomy* **2018**, *8*, 41.
https://doi.org/10.3390/agronomy8040041

**AMA Style**

Sihvonen M, Hyytiäinen K, Valkama E, Turtola E. Phosphorus and Nitrogen Yield Response Models for Dynamic Bio-Economic Optimization: An Empirical Approach. *Agronomy*. 2018; 8(4):41.
https://doi.org/10.3390/agronomy8040041

**Chicago/Turabian Style**

Sihvonen, Matti, Kari Hyytiäinen, Elena Valkama, and Eila Turtola. 2018. "Phosphorus and Nitrogen Yield Response Models for Dynamic Bio-Economic Optimization: An Empirical Approach" *Agronomy* 8, no. 4: 41.
https://doi.org/10.3390/agronomy8040041