Economic considerations of in-season potassium applications to soybean using payoff matrices

Abstract

Potassium (K) deficiency is a common yield-limiting factor in Arkansas soybean (Glycine max (L.) Merr.) production that can be addressed with innovative supplemental fertilizer application. An established leaf sampling protocol and dynamic critical concentration allow accurate diagnosis of K deficiency with corresponding recommendations for corrective, in-season K fertilizer applications at site-specific rates and times to reach anticipated yield goals. However, the profitability of in-season K fertilizer applications to irrigated soybean remains unclear. Research was conducted in Arkansas from 2021 to 2023 to evaluate multiple rates of in-season applications of muriate of potash to soybean at 15 and 30 days after first flower (DAR1). The economic ramifications of in-season fertilizer applications were quantified by calculating yield averages, partial returns (PRs), and regret when comparing PR to not fertilizing, each assuming 5-year average prices for fertilizer and soybean grain. Significant yield responses to in-season potash fertilizer were found at both 15 and 30 DAR1 times. These yield increases translated to large increases in PR or lower regret when compared to using no fertilizer. Corrective applications of 74–112 kg K ha⁻¹ were often considered optimal, with risk assessments provided to allow informed decisions. Results were summarized by category of leaf-K concentration, and treatment averages were provided to calculate payoff matrices for 15 and 30 DAR1 times. The resulting payoff matrices can be used as a decision support tool with any grain and fertilizer price to facilitate informed management decisions that optimize profitability as soybean and fertilizer prices impact optimal outcomes.

Abbreviations

DAR1

days after first flower

MOP

muriate of potash

PR

partial return

RGY

relative grain yield

STK

soil test potassium

1 INTRODUCTION

Potassium (K) fertilizer prices have recently experienced extreme volatility, leading to additional financial stress for soybean (Glycine max L. Merr.) producers. Record high prices of muriate of potash (MOP; 50% K), the most used source of K fertilizer in Arkansas, were recorded in 2022 (Figure 1) (YCharts, 2023). High fertilizer input costs have directly impacted soybean production, increasing the economic risk associated with crop production. When the costs are considered, K fertilizer rate recommendations are often reduced to maximize profitability (Popp et al., 2020). However, when low rates of K fertilizer are applied to soils that measure low in plant-available K, the likelihood of K deficiency increases. Potassium deficiency in soybean can result in large potential yield loss with as much as 41% loss confirmed in Arkansas (Slaton et al., 2021). In addition to uncertain cost, a large range in soybean yields across varying levels of soil fertility warrants risk and relative profitability analyses of K fertilizer management in soybean.

FIGURE 1

Price histories of muriate of potash (MOP) fertilizer and soybean grain over the past 5 years, expressed as monthly averages (YCharts, 2023; Macrotrends, 2023).

Traditional K management involves a full-season, yield-maximizing rate of K fertilizer applied prior to planting (Slaton et al., 2013). Fertilization using this approach relies on recent soil samples collected and analyzed for plant-available K. Previous correlation and calibration research established the yield-maximizing fertilizer rate based on the soil test potassium (STK) values (Slaton et al., 2010). The build and maintain fertilization philosophy is currently implemented for the University of Arkansas soybean fertilizer K recommendations, resulting in K rate recommendations that meet the needs of the crop and include additional K fertilizer to build STK when STK is suboptimal and maintain STK at the optimum level (Olson et al., 1982).

Recently, an economic potash rate calculator was established to consider not only the STK value but also the current prices of fertilizer K, the cost of application, the value of soybean grain, and the anticipated yield potential (Popp et al., 2020). These crop inputs are used to compute a profit-maximizing K fertilizer rate for preplant K management. The potash rate calculator allows producers to compare the yield-maximizing K fertilizer rate recommendation to the profit-maximizing K fertilizer rate recommendation to make the best decision for specific field situations. Popp et al. (2020) reported that the standard agronomic K rate recommendations were often higher than profit-maximizing rates.

While the preplant fertilizer recommendations are reliable, in-season K deficiency may still occur and result in yield loss, especially if preplant rates were reduced or eliminated due to a lack of available capital. Potassium deficiency often shows no visible symptoms, known as hidden hunger, or symptoms may not appear until very late in the season when yield loss is permanent. Widespread hidden hunger was confirmed in production soybean fields in Arkansas, indicating yield loss from K deficiency is a common problem even in seemingly healthy soybean fields (Ortel et al., 2023). Therefore, proactive and routine tissue sampling may be the best way to monitor nutrient status and identify potential hidden hunger before significant yield loss is unavoidable.

Core Ideas

• Yield response to in-season K fertilizer had greater partial returns or least regret compared to no fertilizer.
• In-season, corrective K fertilizer application to soybean is profitable at 15 and 30 days after the first flower when K is deficient.
• Payoff matrices can be helpful decision support tools when considering in-season K management in soybean.

The recent development of a dynamic critical K concentration curve for soybean improves the diagnostic ability of in-season deficiency by providing the critical concentrations of leaf-K required to maintain 95%, 85%, and 75% relative grain yield (RGY) goals at any given point during the reproductive growth stages of soybean (Slaton et al., 2021). When a K deficiency is confirmed, an in-season application of granular MOP may correct the deficiency and minimize the yield loss (Slaton et al., 2020). The leaf-K concentration and sampling time can be used to determine the site-specific K fertilizer rate needed to correct the deficiency, using a calibration curve developed for corrective applications between early full flower (R2) and 30 days after first flower (DAR1). The critical window to correct K deficiencies is 20 DAR1 for severe K deficiency and extends out to 44 DAR1 for moderate K deficiency (Slaton et al., 2020). Information from in-season tissue analysis and the ability to interpret the results increases opportunities for in-season applications of granular K, as either a split application or a corrective application.

Financial risk is unavoidable in crop production, as there is risk associated with any management decision. In-season K management in soybean requires more intense management than traditional production systems, offering the potential for increased yields with an increased risk factor (Mourtzinis et al., 2023). A payoff matrix is a way of expressing the risks associated with different management options (Morrison, 2022; Popp et al., 2010; Tester et al., 2019). For in-season K applications to soybean, this is considering the likelihood of deficiency (measured by the leaf-K; Slaton et al., 2021) and then the subsequent likelihood of partial returns (PRs) and regret for each rate of K fertilizer. PR summarizes revenue from yield achieved less only those costs incurred that vary by treatment (common costs incurred across treatments are ignored as they would not impact relative profitability). Regret in this context is defined as the loss in PR compared to the optimal choice, or the PR associated with making a fertilizer rate decision at a particular point in time, that is, non-optimal when compared to the PR of alternative fertilizer application rates. The goal is to provide this information to growers in a simple format, allowing them to make informed decisions about the yield and profit risks involved with applying in-season K, as well as the potential risk of not applying a corrective in-season application.

In a payoff matrix, the PRs, in this case defined as revenue from soybean sales less costs that differed across treatment (i.e., cost for fertilizer and its application), were calculated to estimate relative profitability differences across fertilizer treatments with associated yield responses. Comparisons of yield and economic outcomes within ranges of leaf-tissue K and across fertilizer rates thus delineate the risks associated with each fertilizer rate choice across a range of leaf-tissue K levels observed in the study. Within this matrix, the optimal value from a producer perspective is highlighted to easily direct the users to the top choice using a range of evaluation techniques. The optimal fertilizer rate choice could be the one showing the maximum PR across fertilizer-rate choices or one that shows the minimum range in PR across LK levels for a particular fertilizer-rate choice, for example. These evaluation techniques are described in Section 2.

The large potential soybean yield loss associated with K deficiency and the volatile prices of K fertilizer may leave soybean producers uncertain about the economics of in-season K fertilization decisions. This necessitates a cost and risk analysis of fertilizer prices, and potential yield losses that expand to changes in sales as a function of soybean price, to provide beneficial insight when making management decisions. Therefore, the research objectives were (a) to calculate PR of in-season K management strategies in irrigated soybean, (b) to calculate the potential regret associated with in-season K management strategies across fertilizer rate choices in irrigated soybean, and (c) to provide this information in a useful way to allow producers the ability to make informed decisions when implementing in-season K management strategies taking varying fertilizer cost and soybean price into consideration.

2 MATERIALS AND METHODS

2.1 Site descriptions

Irrigated soybean response to in-season K application rate was evaluated in 10 field trials conducted from 2021 to 2023 on silt-loam soils with access to irrigation, located across the primary soybean-producing regions of Arkansas. Varying levels of STK were selected between sites in anticipation that the soybean would express different levels of K deficiency, allowing more robust conclusions to be drawn from the results. Each trial was a randomized complete block design that contained four blocks. Within each block, three plots were all untreated controls that received no K fertilizer and were considered one treatment. Individual plots were four rows wide, measuring between 3.65 and 3.84 m wide depending on row spacing, each 9.14 m long. A composite soil sample (0- to 10-cm depth) was collected (n = 8–10 cores) in each block prior to planting. The soil was oven dried, ground to pass a 2-mm sieve, and mixed prior to analysis for pH (1:2 v/v soil/water mixture; Sikora & Kissel, 2014), organic matter by weight loss on ignition (LOI; Zhang & Wang, 2014), and Mehlich-3 extractable nutrients (Zhang et al., 2014) (Table 1). No K fertilizer was applied at preplant, while 19.2 kg P ha⁻¹ as triple superphosphate (198 g P kg⁻¹) was applied preplant to fields that measured lower than 40 mg kg⁻¹ of soil test phosphorus (P) to ensure that P was not yield limiting. Treatments were applied in-season within a 2-day window of either 15, 30, or 45 DAR1 at rates of 0, 37.2, 74.4, 111.6, and 149 kg K ha⁻¹ applied as granular MOP. Each treatment was broadcast by hand onto dry soil and furrow irrigation was initiated within 24–48 h to incorporate the fertilizer and facilitate plant uptake. Except for K fertilization, soybean management closely followed recommended guidelines for irrigated full-season soybean production as outlined by the University of Arkansas Cooperative Extension Service (Ross, 2000). The R1 date was predicted using the SoyStage program (dos Santos et al., 2019) and was confirmed with in-field observations (Fehr et al., 1971).

TABLE 1. Select soil physical and chemical characteristics for each site-year.

Site no.	Year	Coordinates	Primary soil series^a	Primary taxonomic class^b	pH	LOI^c	P^d (mg kg^−1)	K^d,e, ^d,e (mg kg^−1)
1	2021	36.096718, −94.171635	Pembroke	Fine-silty, mixed, active, mesic Mollic Paleudalfs	6.0	1.0	40	79 ± 9.3
3	2021	34.729512, −90.734227	Convent	Coarse-silty, mixed, superactive, nonacid, thermic Fluvaquentic Endoaquepts	6.4	1.4	29	40 ± 3.2
4	2022	36.098444, −94.174728	Captina	Fine-silty, siliceous, active, mesic Typic Fragiudults	7.3	1.3	48	70 ± 3.4
5	2022	36.100122, −94.166433	Captina	Fine-silty, siliceous, active, mesic Typic Fragiudults	6.5	1.2	45	149 ± 6.9
6	2022	35.661846, −90.712054	Henry	Coarse-silty, mixed, active, thermic Typic Fragiaqualfs	5.4	1.8	35	44 ± 2.4
7	2022	35.135344, −90.939141	Calhoun	Fine-silty, mixed, active, thermic Typic Glossaqualfs	7.4	2.4	34	85 ± 16.1
8	2022	35.135667, −90.940178	Calloway	Fine-silty, mixed, active, thermic Aquic Fraglossudalfs	7.3	2.5	40	164 ± 16.9
9	2023	35.662048, −90.712755	Henry	Coarse-silty, mixed, active, thermic Typic Fragiaqualfs	5.4	1.8	44	47 ± 11.4
11	2023	35.111903, −90.940775	Calloway	Fine-silty, mixed, active, thermic Aquic Fraglossudalfs	7.2	2.1	24	59 ± 3.7
12	2023	35.134803, −90.939767	Calhoun	Fine-silty, mixed, active, thermic Typic Glossaqualfs	7.4	2.2	21	54 ± 6.0

• ^a Soil Survey Staff et al. (n.d.).
• ^b Soil Survey Staff (2019).
• ^c LOI: loss on ignition (Zhang & Wang, 2014).
• ^d Zhang et al. (2014).
• ^e The average and standard deviation of each site soil test K. The optimum soil test K in Arkansas is 131–175 mg K kg⁻¹ (Slaton et al., 2013).

At each predetermined in-season treatment application timing (15, 30, and 45 DAR1), a composite sample of 12–16 trifoliolate leaves (no petiole) was collected from the uppermost fully expanded trifoliate leaves within the middle rows of each plot receiving a treatment at that time, as well as all no-K fertilizer control plots. The leaves were dried, ground, and digested with concentrated HNO₃ and 30% H₂O₂ (Jones & Case, 1990) and analyzed by inductively coupled plasma optical emission spectrophotometry for K concentration. At maturity, the middle two rows were harvested, and the seed yields were adjusted to 13% moisture.

The measurement of leaf-K concentration was used to describe the level of K deficiency experienced by the soybean and categorized into four uncertain states of nature. Categories of deficiency were determined using the dynamic critical tissue-K concentration values known for each leaf sampling time and each of 95%, 85%, and 75% expected RGY (Slaton et al., 2021). The categories for each leaf sampling time are marked by these thresholds to describe leaf-K concentrations <75% sufficiency, between 75% and 85% sufficiency, between 85% and 95% sufficiency, and above 95% sufficiency.

RGY was calculated by comparing the measured yield of each plot to the highest yielding treatment average at each site-year, multiplied by 100 to convert to a percentage, and capped at a value of 100% RGY (Pearce et al., 2022). RGY responses were evaluated within each site-year and fertilizer application time individually, using a quadratic regression analysis to interpret each site and fertilizer application time. An alpha value of 0.10 was used to classify each site-year and fertilizer application time as either responsive (p ≤ 0.10) or unresponsive (p > 0.10) to in-season K fertilizer applications. Additionally, all sites were combined and the raw yield response to in-season K fertilizer rate was considered as a mixed effect ANOVA within each leaf-K category. The raw yield response was analyzed as a randomized complete block design using the lme4 (Bates et al., 2015), lmerTest (Kuznetsova et al., 2017), and emmeans (Lenth, 2023) packages in R 4.0.2 (R Core Team, 2021) and RStudio (Posit Software, PBC v 2023.3.1.446). Outliers were identified as values that exceeded 1.5 times the interquartile range (IQR) for each measurement. When identified, the outlier was removed and the analysis was repeated.

2.2 Payoff matrix calculations

The likelihood of occurrence for each of the defined leaf-K categories was calculated from the data collected across the 10 site-years of research. The probability of each leaf-K category allows the payoff matrix to be implemented as a decision support tool even without knowing the exact leaf-K value. Consider a producer taking a leaf-K measurement at a specific time after DAR1. Not knowing a priori what category of K deficiency results will be observed but understanding the likelihood of different K deficiencies given these study results, the producer can analyze multiple fertilizer rate actions across a range of K deficiency category outcomes to make an informed decision. The range of yield outcomes within each of the sites varied due to a wide range of potential environmental factors, as well as the differing amounts of plant-available K in the soil.

The PR were calculated for each treatment and within each uncertain state of nature across all site-years of research. It can be defined as revenue earned by the soybean yield minus the fertilizer expenses as described in Equation (1):

$$$$\begin{equation}{\mathrm{P}}{{{\mathrm{R}}}_{ij}} = \left( {{{Y}_{ij}} \cdot {{P}_s}} \right) - \left( {{{F}_i} \cdot {{c}_K}} \right) - {\mathrm{FA}}\ \forall \ i\ {\mathrm{and}}\ j\end{equation}$$$$(1)

where Y are average yields (kg ha⁻¹) observed for i fertilizer rates (F) (kg K ha⁻¹) as observed in different leaf-K categories j, and P_S and c_K are soybean price ($ kg⁻¹) and fertilizer cost ($ kg⁻¹ K), respectively. Fertilizer application costs (FA) ($ ha⁻¹) are charged to treatments where fertilizer was applied with no such charge incurred in the no-K control treatments.

The PR value was then used when considering the regret (R), which is calculated for each leaf-K concentration category (j) as described across fertilizer rate options (i) in Equation (2) in a manner similar to Tester et al. (2019), Popp et al. (2010), and Morrison (2022):

$$$$\begin{equation}{{R}_{ij}} = \mathop {\max }\limits_i {\mathrm{P}}{{{\mathrm{R}}}_{ij}} - {\mathrm{P}}{{{\mathrm{R}}}_{ij}}\ \forall \ i\ {\mathrm{and}}\ j\end{equation}$$$$(2)

Regret represents the regret a producer would experience in a particular leaf-K category j by comparing PR of a particular fertilizer rate (i) to the best performing of fertilizer rate choices $$( { \def\eqcellsep{&}\begin{array}{@{}*{1}{c}@{}} {{\mathrm{max}}}\\ i \end{array} } )$$. Hence, the best-performing fertilizer rate had a regret of 0 in each uncertain leaf K category (j), and the best fertilizer rate could vary across leaf K categories. The PR was also used to calculate the expected value of each individual fertilizer rate treatment by calculating a weighted average of PR across all leaf K categories using Equation (3) for each fertilizer rate choice (i):

$$$$\begin{equation}\overline {{\mathrm{P}}{{{\mathrm{R}}}_i}} = \frac{{\sum_{{\mathrm{\ j = 1}}}^{\mathrm{4}} {\mathrm{P}}{{{\mathrm{R}}}_{{\mathrm{ij\ }}}} \cdot {\mathrm{LO}}{{{\mathrm{C}}}_{\mathrm{j}}}}}{{\mathrm{4}}}\,\forall \,i\end{equation}$$$$(3)

where the likelihood of occurrence (LOC) for a leaf-K category was determined as the ratio of observations falling within a leaf-K category to the total number of leaf-K observations at DAR1 across all plots considered.

Similarly, the expected regret for each fertilizer rate option was calculated using Equation (4):

$$$$\begin{equation}\overline {{{R}_i}} = \frac{{\sum_{j = 1}^4 {{{\mathrm{R}}}_{ij\ }} \cdot {\mathrm{LO}}{{{\mathrm{C}}}_j}}}{4}\,\forall \,i\end{equation}$$$$(4)

The optimal choice for expected regret is the fertilizer rate (i) with the least expected regret.

To further assist with management decision making, optimal outcomes involving the maximax, the maximin, the minimum range, and the maximum regret algorithms are identified in bold text in the payoff matrix using conditional formatting in Excel (Microsoft) as soybean price and fertilizer cost impact PR outcomes. The maximax algorithm is an optimist's approach. The fertilizer rate choice with the highest possible PR across leaf-K categories for a particular fertilizer rate choice is assumed to occur. Ultimately, assuming for each fertilizer rate, the leaf-K category with the optimum PR will occur, the maximax decision maker chooses the fertilizer rate that maximizes PR across all leaf-K category by fertilizer rate combinations. In that sense, this criterion is different from expected PR and expected regret as the comparison is not only within each fertilizer rate choice but across all outcomes.

The maximin approach is the pessimist's approach, again encompassing all potential outcomes. The decision maker assumes the worst outcome across leaf-K categories for each fertilizer rate choice and chooses the fertilizer rate choice with the highest of the worst outcomes. The minimum range algorithm calculates the potential PR difference across leaf-K categories for each fertilizer rate choice. The decision maker chooses the fertilizer rate choices with the least range of outcomes. Finally, the maximum regret algorithm identifies the fertilizer rate choice with the least maximum regret across all leaf-K categories for each fertilizer rate applied in-season. The latter two algorithms offer in-depth risk analysis of uncertain leaf-K categories for each fertilizer rate choice and are then compared across fertilizer rate decisions.

All the above calculations were computed to build a payoff matrix for 15 and 30 DAR1 timings. Each payoff matrix can be used as a decision support tool to allow producers to adjust the price of fertilizer and value of soybean grain to match their situation, providing a simple method of assessing the risks of in-season K applications and the economic ramifications with a multitude of comparisons across all combinations of outcomes captured by the different algorithms. The producer can then either choose the fertilizer rate option with the largest number of positive signals across decision algorithms or pick the fertilizer rate options that most aptly fit his or her situation (i.e., a risk-averse producer may focus more on maximin, minimum range, and least of maximum regret).

3 RESULTS AND DISCUSSION

Significant (p ≤ 0.10) yield responses occurred in seven of the 10 sites at 15 DAR1 (Figure 2), six of the 10 sites at 30 DAR1 (Figure 3), and five of the 10 sites at 45 DAR1. These results were used to build calibration curves for 15 and 30 DAR1, providing site-specific K fertilizer rate recommendations based on the leaf-K. The 45 DAR1 results did not produce a significant calibration model, likely because this is too late to consistently correct a K deficiency in-season. Therefore, the economic analysis was only performed for the 15 and 30 DAR1 application timings.

FIGURE 2

Yield response predicted using a regression equation for each of the significantly (p ≤ 0.10) responsive sites when K fertilizer was applied at 15 days after first flower (DAR1).

FIGURE 3

Yield response predicted using a regression equation for each of the significantly (p ≤ 0.10) responsive sites when K fertilizer was applied at 30 days after first flower (DAR1). Sites 2 and 10 were identified as outliers and removed from analysis.

At 15 and 30 DAR1, the actual grain yield response to in-season K fertilizer rates applied was considered within each leaf-K category (Table 2). At 15 DAR1, a significant yield response occurred when leaf-K measured below 9.7 g K kg⁻¹ (p < 0.0001) and when leaf-K measured between 9.7 and 13.27 g K kg⁻¹ (p = 0.0088). Once leaf-K measured above this 85% RGY threshold at 15 DAR1, there was not a significant yield response (Table 2). At 30 DAR1, a significant yield response occurred when leaf-K measured below 9.12 g K kg⁻¹ (p < 0.0001). When leaf-K measured above this 75% RGY threshold at 30 DAR1, there was not a significant yield response when all sites were considered together (Table 2).

TABLE 2. Significance level of the actual grain yield response to K fertilizer rate as a mixed effect model for each leaf-K (LK) category at each sampling time.

Timing	Leaf-K (g K kg^−1)	Predicted RGY^a	Response model p value
15 DAR1	0xx.0LK < 9.7xx	<75%	<0.0001xxx
	xx9.7 ≤ LK < 13.27	75%–85%	0.0088
	13.27 ≤ LK < 18.96	85%–95%	0.9658
	0xx.0 < LK ≥ 18.96	≥95%	0.4248
30 DAR1	0xx.0 LK < 9.12	<75%	<0.0001
	x9.12 ≤ LK < 12.32	75%–85%	0.7969
	12.32 ≤ LK < 17.28	85%–95%	0.7058
	0xx.0 < LK ≥ 17.28	≥95%	0.5225

• Abbreviations: DAR1, days after first flower; LK, leaf-K concentration; RGY, relative grain yield.
• ^a Dynamic critical concentration used to delineate leaf-K categories to align with <75%, 75%–85%, 85%–95%, and above 95% sufficiency when left untreated (Slaton et al., 2021).

The volatility of MOP fertilizer prices and the value of soybean grain experienced in the past 5 years complicates the ability to select a representative price in an economic analysis (Figure 1). An average year will experience price fluctuations in both fertilizer and grain prices, but the magnitude of recent price fluctuations was extreme, corresponding to the recent global economy. Therefore, the 5-year average prices for both MOP fertilizer ($0.49 kg; YCharts, 2023) and soybean grain ($0.45 kg; Macrotrends, 2023) were used to build the payoff matrices for 15 and 30 DAR1, with the ability to build a decision support tool for each timing to consider input-specific prices at the time a decision is made (Supporting Information). As prices change, the most profitable management decisions will often change as well, emphasizing the importance of adjusting these factors to reflect relevant prices (Popp et al., 2010).

3.1 Profitability of applications at 15 days after first flower

Across all locations at 15 DAR1, there was a 79.7% chance of a yield-limiting K deficiency, indicated by leaf-K values less than the critical concentration of 18.96 g K kg⁻¹ (Table 3) (Slaton et al., 2021). Leaf-K concentrations ranged from 5.8 to 26.1 g K kg⁻¹ across all site-years at 15 DAR1. However, this may not be representative of producer-managed fields, as these treatments did not receive any preplant K fertilizer. The nutrient status of producer-managed fields is captured in soil samples, with 33% of the samples collected following soybean and submitted to the University of Arkansas System Division of Agriculture's Marianna Soil Testing Laboratory in 2021 measuring in the low or very low STK categories (DeLong et al., 2022). Soybean grown in low or very low STK sites have a 92% or greater chance of a significant yield response to K fertilizer (Slaton et al., 2013). Even when K fertilizer is applied preplant, widespread hidden hunger has been confirmed in producer-managed fields (Ortel et al., 2023), emphasizing the importance of in-season K management in soybean to achieve yield responses and improved profitability.

TABLE 3. Payoff matrix of partial returns (PRs) for soybean which received corrective applications of K fertilizer at 15 days after first flower (DAR1), divided as categories of leaf-K concentrations (LK) and considering the 5-year average prices of muriate of potash fertilizer ($0.49 kg⁻¹) and soybean grain ($0.45 kg⁻¹).

			Controllable action
Payoff matrix (15 days after first flower)			0 kg K ha^−1	37 kg K ha^−1	74 kg K ha^−1	112 kg K ha^−1	149 kg K ha^−1
Uncertain state of nature	Leaf-K^a (g K kg^−1)	Probability of LK occurrence (%)	Yield least square means in kg ha^−1^b
	LK < 9.7	17.3	2272b	3060a	3247a	3676a	3463a
	9.7 ≤ LK < 13.27	18.2	2759b	3332a	3576a	3552a	3511a
	13.27 ≤ LK < 18.96	44.2	3752a	3718a	3919a	3730a	3710a
	LK ≥ 18.96	20.3	4107a	3903a	4365a	4198a	4426a
			Partial return in $ ha^−1
	LK < 9.7	17.3	$1022	$1340	$1406	$1581	$1467
	9.7 ≤ LK < 13.27	18.2	$1242	$1463	$1554	$1525	$1488
	13.27 ≤ LK < 18.96	44.2	$1688	$1636	$1709	$1605	$1578
	LK ≥ 18.96	20.3	$1848	$1720	$1909	$1816	$1900
			Regret (high–low within row) in $ ha^−1
	LK < 9.7	17.3	$558	$240	$174	$0	$114
	9.7 ≤ LK < 13.27	18.2	$313	$92	$0	$29	$66
	13.27 ≤ LK < 18.96	44.2	$20	$72	$0	$104	$131
	LK ≥ 18.96	20.3	$61	$190	$0	$94	$9
Decision algorithms^c	Expected value		$1524	$1570	$1669	$1629	$1608
	Maximax		$1848	$1720	$1909	$1816	$1900
	Maximin		$1022	$1340	$1406	$1525	$1467
	Minimum range		$826	$379	$503	$291	$433
	Maximum regret		$558	$240	$174	$104	$131
	Expected regret		$175	$129	$30	$70	$91

• ^a Dynamic critical concentration used to delineate leaf-K categories to align with <75%, 75%–85%, 85%–95%, and above 95% sufficiency (Slaton et al., 2021).
• ^b Letter separation indicates statistically significant differences within each LK category. Model p values are listed in Table 2.
• ^c See Section 2.2 in text for a description. Bold numbers indicate optimal choice for each algorithm.

The average yields achieved from soybean with leaf-K less than 9.7 g K kg⁻¹ at 15 DAR1 were less than the Arkansas state average soybean yield (3497 kg ha⁻¹), except when 112 kg K ha⁻¹ was applied at 15 DAR1 as a corrective application (Table 3) (USDA, 2023). This leaf-K category describes severe K deficiencies and is highly likely to respond to an in-season application of K fertilizer within 20 DAR1 (Table 2) (Slaton et al., 2020). When the 5-year averages were considered for soybean grain and K fertilizer, the PR ranged from $1022 to $1581, with the highest regret when no K fertilizer was applied, and the K deficiency remained severe (Table 3). The high level of regret can be attributed to the large potential yield losses from severe K deficiency and the ability to increase the average yield by 38% with a corrective application of 112 kg K ha⁻¹ at 15 DAR1 (Table 3).

Meanwhile, the average yields achieved from soybean with leaf-K of 18.96 g K kg⁻¹ or greater consistently measured above the state average yield regardless of the K fertilizer rate applied (USDA, 2023). The higher leaf-K category describes K-sufficient situations that measure at or above the critical concentration (Slaton et al., 2021). However, there was a profit-maximizing 6% yield increase when 74 kg K ha⁻¹ was applied, compared to the untreated control. When the 5-year average prices were used, this resulted in an increased PR of $61 ha⁻¹ (Table 3). There was no financial regret when 74 kg K ha⁻¹ was applied to soybean with leaf-K measuring above 18.96 g K kg⁻¹; in fact, this was true for all soybean leaf-K concentrations above 9.7 g K kg⁻¹ (Table 3).

Across all leaf-K categories, the expected value of a soybean crop managed at 15 DAR1 was the highest when 74 kg K ha⁻¹ was applied, valued at $1669 ha⁻¹ (Table 3). The decision to apply 74 kg K ha⁻¹ also had the lowest expected regret compared to all other fertilizer rate options and would be selected by an optimist, as it produced the highest overall PR (Table 3). The management decision to apply 112 kg K ha⁻¹ may also be a sound decision and would be selected by the pessimist as it produced the highest of the minimum PR among fertilizer rates. Similarly, 112 kg K ha⁻¹ also measured the lowest minimum range and least maximum regret, indicating a reduced risk factor associated with this fertilizer rate selection (Table 3). Ultimately, these calculations are appropriate when the leaf-K category is unknown (or shown to fall into leaf-K categories similar to what was experienced across fields in this analysis). Ideally, producers could collect a leaf sample to measure the leaf-K and can then use their cost of MOP and the expected value of soybean grain to make an informed decision within a leaf-K category, although spatial variation in leaf-K can still exist when taking one leaf-K sample by collecting leaves across a field.

When 149 kg K ha⁻¹ was applied to a soybean crop at 15 DAR1, the raw yield was statistically similar to those achieved with lower rates of K fertilizer across all leaf-K categories (Table 3). However, the cost of potash has increased and there is no guarantee the additional K will remain available for a subsequent crop. Loss of K fertilizer via runoff is more likely when large volumes of fertilizer are applied to a field (Daniels et al., 2023). Soybean is a luxury consumer of K and will continue to take up plant-available K from the soil when the plant already has sufficient levels of K nutrition. Soybean seed is a sink for excess K and a portion of K that is luxury consumed will be removed from the cropping system at harvest (Parvej et al., 2016). Therefore, caution is advised when considering applying high rates of K fertilizer to soybean in-season because of the evidence of no statistical yield increase combined with the potential for losses from the cropping system, either by runoff or crop removal.

3.2 Profitability of applications at 30 days after first flower

When soybean remained untreated with K fertilizer until 30 DAR1, there was a 91.8% chance of a yield-limiting K deficiency, indicated by the leaf-K values less than the critical concentration of 17.28 g K kg⁻¹ (Table 4) (Slaton et al., 2021). When leaf-K measured above this critical concentration, the soybean is K-sufficient and not yield-limited by K. Leaf-K concentrations ranged from 6.4 to 22.5 g K kg⁻¹ across all site-years at 30 DAR1. Across all leaf-K categories and possible fertilizer rates, the maximum PR was achieved when 74 kg K ha⁻¹ were made to K-sufficient soybean, identified as the maximax choice (Table 4). The data, averaged across all sites, agree with these findings, as the highest yield and PR were achieved when no K deficiency occurred and 74 kg K ha⁻¹ K fertilizer was applied (Table 4), indicating there was no regret when a corrective application was applied with the increased input cost that was more than offset by the yield increase even when leaf-K was non-yield limiting.

TABLE 4. Payoff matrix of partial returns (PRs) for soybean which received corrective applications of K fertilizer at 30 days after first flower (DAR1), divided as categories of leaf-K concentrations (LK) and considering the 5-year average prices of muriate of potash fertilizer ($0.49 kg⁻¹) and soybean grain ($0.45 kg⁻¹).

			Controllable action
Payoff matrix (30 days after first flower)			0 kg K ha^−1	37 kg K ha^−1	74 kg K ha^−1	112 kg K ha^−1	149 kg K ha^−1
Uncertain state of nature	Leaf-K (g K kg^−1)^a	Probability of LK occurrence (%)	Yield least square means in kg ha^−1^b
	LK < 9.12	9.8	2234d	3077c	3430bc	3923a	3833ab
	9.12 ≤ LK < 12.32	28.6	3217a	3405a	3259xa	3549a	3469xa
	12.32 ≤ LK < 17.28	53.4	3542a	3734a	3727xa	3867a	3770xa
	LK ≥ 17.28	8.2	4265a	2596a	4816xa	4324a	4295xa
			Partial return in $ ha^−1
	LK < 9.12	9.8	$1005	$1348	$1489	$1692	$1633
	9.12 ≤ LK < 12.32	28.6	$1448	$1496	$1412	$1524	$1469
	12.32 ≤ LK < 17.28	53.4	$1594	$1644	$1622	$1667	$1605
	LK ≥ 17.28	8.2	$1919	$1132	$2112	$1872	$1841
			Regret (high–low within row) in $ ha^−1
	LK < 9.12	9.8	$687	$344	$203	$0	$59
	9.12 ≤ LK < 12.32	28.6	$76	$28	$112	$0	$54
	12.32 ≤ LK < 17.28	53.4	$73	$23	$44	$0	$62
	LK ≥ 17.28	8.2	$193	$981	$0	$240	$271
Decision algorithms^c	Expected Value		$1521	$1530	$1589	$1645	$1588
	Maximax		$1919	$1644	$2112	$1872	$1841
	Maximin		$1005	$1132	$1412	$1524	$1469
	Minimum range		$914	$512	$701	$349	$372
	Maximum regret		$687	$981	$203	$240	$271
	Expected regret		$144	$134	$76	$20	$76

• ^a Dynamic critical concentration used to delineate leaf-K categories to align with <75%, 75%–85%, 85%–95%, and above 95% sufficiency (Slaton et al., 2021).
• ^b Letter separation indicates statistically significant differences within each LK category. Model p values are listed in Table 2.
• ^c See Section 2.2 in text for a description. Bold numbers indicate optimal choice for each algorithm.

When K deficiency did occur at 30 DAR1, 112 kg K ha⁻¹ resulted in the highest expected PR, was the pessimist's choice, had the least range in PR across leaf-K categories, and the least expected regret across all deficient leaf-K categories (Table 4). When leaf-K measured less than 9.12 g K kg⁻¹, K deficiencies were severe at 30 DAR1, and a significant (p < 0.0001) yield increase was measured when all sites were averaged together, and 112 kg K ha⁻¹ was applied (Tables 1 and 3).

An application of 112 kg K ha⁻¹ was a profitable management choice across all leaf-K categories at 30 DAR1, averaging $478 ha⁻¹ in profit after considering the estimated 5-year average cost of $1167 ha⁻¹ for growing glyphosate-resistant soybean in a furrow-irrigated system with preplant fertilizer K application (Watkins, 2023). Overall, these results support the conclusion that 112 kg K ha⁻¹ is a sound decision when managing K-deficient soybean at 30 DAR1. Producers should again rely on a leaf sample to identify the leaf-K concentration and confirm when hidden hunger is occurring. Visual symptoms should not be exclusively used to diagnose K deficiency in soybean because of hidden hunger; tissue sampling is the only scientific option for confirmation of a hidden yield-limiting K deficiency (Ortel et al., 2023). When visual K-deficiency symptoms are present in the reproductive growth stages of soybean, an irreversible yield loss may have occurred (Slaton et al., 2020). Additionally, relevant prices should be considered instead of 5-year averages, when possible, to make the most informed decision for each situation.

The probability of K deficiency increased as the soybean matured, from 79.7% at 15 DAR1 to 91.8% at 30 DAR1 (Tables 2 and 3). Similarly, the chance of hidden hunger increased in producer managed soybean fields when samples were collected at R2 and then again at R4 (Ortel et al., 2023). However, the degree of K deficiency was lower, likely because of preplant K fertilizer applied to the fields. Mild K deficiencies can be described as leaf-K concentrations between 13.27 and 18.96 g K kg⁻¹ at 15 DAR1 and 12.32 and 17.23 g K kg⁻¹ at 30 DAR1, ultimately measuring between the critical concentration thresholds expected to produce 85% and 95% relative yields (Slaton et al., 2021). These were leaf-K concentration values that fell in the most likely category, with an expected value of $20 ha⁻¹ and $73 ha⁻¹ of regret if left untreated at 15 and 30 DAR1, respectively (Tables 3 and 4). While severe K deficiency was less likely to occur, regret was as high as $687 when left untreated. When extrapolated across a producer scale, this is an enormous loss with the potential to limit the success of the whole farm operation.

3.3 Additional management costs

Proper nutrient management begins with soil sampling and following the recommended rates to reduce the chances of yield-limiting nutrient deficiencies. Repeating this analysis to include a preplant fertilizer treatment may provide additional valuable information regarding the likelihood of leaf-K categories for in-season management. In-season K management in soybean necessitates implementing a leaf sampling protocol to collect a composite sample that represents each management zone within the field (Ortel et al., 2023). Incorporating proactive tissue sampling involves increased labor to collect the sample as well as a laboratory fee for each sample analysis. These costs may vary across producer and could be added to the fertilizer application charge in the payoff matrix already accounted for in Tables 3 and 4. Granular MOP must be incorporated into the soil for plant uptake, either through timely rain or irrigation event. Each of these management factors add up, resulting in increased costs of soybean production.

An analysis of the past 5 years of the Arkansas Crop Enterprise Budgets for glyphosate-resistant, furrow-irrigated soybean (Watkins, 2023) identified a trend. Total specified expenses have increased by 138% in the past 5 years due to the rising prices of crop management. Therefore, proper nutrient management is vital to maximize efficiency and profitability. Proper management begins with preplant nutrient management. When no preplant K fertilizer was applied, there was only an 8.2% chance of K sufficiency when leaf-K was measured at 30 DAR1. The remaining 91.8% of the soybean were considered K deficient, but not all sites had significant yield responses to in-season applications of K fertilizer (Figures 2 and 3). If no yield-limiting K deficiency is present, the addition of K fertilizer may still improve profitability (Tables 3 and 4) (Quinn & Steinke, 2019), although soybean grain price and fertilizer cost will play a large role. Careful consideration of the financial risks associated with in-season fertilizer applications is required to ensure a profitable outcome, as intensifying management does not ensure an increased yield or profitability (Orlowski et al., 2016).

4 CONCLUSIONS AND PRACTICAL APPLICATIONS

In-season applications of K fertilizer are not only successful at correcting a deficiency and minimizing the yield loss but can also be profitable. To the best of our knowledge, this is the first manuscript to provide economic insight to in-season K management in soybean when granular fertilizer is used to correct K deficiencies. Payoff matrices can be a valuable tool in any economic climate that considers the PR and regret of management choices. Significant yield responses were found at both the 15 and 30 DAR1 sampling times, with treatment averages measuring 38% and 43% yield increases when K fertilizer was applied to severely K-deficient soybean, respectively. The yield increase from K fertilizer application translated to large increases in PR, indicating the additional input cost of K fertilizer was a sound investment.

Results of several site-years of research were summarized using treatment averages when applied to soybean by categorizing leaf-K concentrations to represent various levels of deficiency. Across the 15 and 30 DAR1 times, it was consistently profitable to correct in-season K deficiency when they occurred. Producers could collect leaf samples to measure the leaf-K concentration and use this information to make the best possible management decision, as this is the only scientific way to confirm a yield-limiting nutrient deficiency. The fertilizer rate choice is impacted by relevant prices for both K fertilizer and soybean grain to facilitate the best decision possible. When 5-year averages for soybean price and fertilizer cost were considered, 74–112 kg K ha⁻¹ were the most profitable management decisions at 15 DAR1 and 30 DAR1. When higher K rates were applied, no significant yield response was observed at any time for level of deficiency, but an increase in fertilizer costs impacted the profitability. The most profitable scenario of all was when 74 kg K ha⁻¹ was applied in-season.

AUTHOR CONTRIBUTIONS

C. C. Ortel: Conceptualization; data curation; formal analysis; investigation; methodology; visualization; writing—original draft; writing—review and editing. T. L. Roberts: Conceptualization; data curation; formal analysis; funding acquisition; investigation; methodology; project administration; resources; software; supervision; writing—review and editing. M. Popp: Conceptualization; formal analysis; investigation; methodology; software; supervision; validation; visualization; writing—review and editing. W. J. Ross: Conceptualization; investigation; methodology; resources; writing—review and editing. N. A. Slaton: Conceptualization; investigation; methodology; writing—review and editing. M. R. Parvej: Conceptualization; methodology; writing—review and editing.

ACKNOWLEDGMENTS

The authors thank the staff of the several Agricultural Experiment Stations for assistance in establishing, managing, and harvesting research trials. Funding for this project was provided from the Arkansas Soybean Checkoff Program administered by the Arkansas Soybean Promotion Board, Fertilizer Tonnage Fees administered by the Arkansas Soil Test Review Board, University of Arkansas System Division of Agriculture, and Agricultural Experiment Station funding related to Hatch Project 2701.

CONFLICT OF INTEREST STATEMENT

The authors declare no conflicts of interest.