Wheat-fallow system (WF)

Wheat was planted on November 10 and harvested June 15. Diammonium phosphate (18-46-0) and anhydrous ammonia (82-0-0) were applied on October 15 at 109.77 and 30.24 kg ha⁻¹. Top-dress nitrogen (UAN 28%) was applied at 264.36 kg ha⁻¹ on February 10. The seeding rate was 67.21 kilograms of seeds ha⁻¹. Seeds were planted with a seed drill to a soil depth of 40 mm.

Herbicides were applied four times during the NT wheat-fallow rotation, once in June, August, October, and November. Herbicides applied were Axial XL® (wild oat/ryegrass control, 700.25 g ha⁻¹), Finesse® (broadleaf weed control, 17.51 g ha⁻¹), and glyphosate (broadleaf control; applied during the summer as a burndown application on fallow at 1080 g ha⁻¹). Dimethoate was applied to manage insects (876.23 mL ha⁻¹). For conventional till systems, the field was plowed and disked after harvest in June, a second till was performed in August and September, and a final till was performed in October before planting (Decker et al., Reference Decker, Epplin, Morley and Peeper2009; Epplin, Reference Epplin2007).

Wheat-soybean double crop system (WS)

Soybeans were planted on July 10 and harvested on October 20 at a seeding rate of 48.54 kg ha⁻¹. Wheat was planted with a seed drill at a soil depth of 40 mm. Soybeans were planted with a planter, also at a depth of 40 mm. No fertilizer was applied during the soybean cycle. Herbicides were applied four times per year under the wheat-summer crop no-till systems. Bicep Lite II Magnum® (for annual weeds and grasses, 1400.49 g ha⁻¹) and glyphosate (for broadleaf weeds and grasses, 2240.78 g ha⁻¹) were used on soybeans and Axial XL® and Finesse® were used on wheat (the same rate as the WF system). Wheat management followed the same procedures as the WF rotation, except that no summer herbicides or tillage operations were used in the wheat-summer crop management schedules. The field was disked prior to planting soybeans.

Wheat-sorghum double crop (WSrgh)

Sorghum was planted on July 10 and harvested on October 20. Anhydrous ammonia was applied to sorghum on July 1 at 58.61 kg ha⁻¹. The seeding rate was 3.73 kg ha⁻¹. Seeds were planted with a planter at a depth of 40 mm.

Bicep Lite II Magnum® (for broadleaf species and grasses, 2240.78 g ha⁻¹) and Atrazine (broadleaf species, 840.29 g ha⁻¹) were applied to sorghum to control weeds. Sivanto 200 SL® (385.13 g ha⁻¹) was used to protect the crop from insect damage.

Wheat management followed the previous WF rotation management specification, except for summer herbicide applications and tillage. One pre-emergent herbicide and one post-emergent herbicide were applied to double-crop wheat. The field was disked prior to planting sorghum.

Net returns

Yields, crop prices, and input costs were used to calculate each system’s per-hectare net returns (NR). Output prices are from 2020 to match production cost data from the United States Department of Agriculture National Agricultural Statistics Service (USDA-NASS, 2021). Production costs for each cropping system are from the 2020 Oklahoma State University Extension Service Enterprise Budgets (Sahs, Reference Sahs2021). Holding output prices and production costs constant is a simplifying assumption and a limitation of this research. Because prices and costs are constant, producers are assumed to decide on cropping systems based on current price and cost information. While it is likely that price changes and historical prices factor into a producer’s decision-making process, this simplifying assumption isolates the effects of a cropping system on yield while holding prices and costs constant. Net returns for the WF system in period t were calculated as

(1) $$NR_{WF(t)}=P_{W}\cdot Y_{W\left(t\right)}-\left(C_{W}+C_{F}\right)-R_{C}$$

where P _W is the price of wheat ($ Mt⁻¹), Y _W(t) is the wheat yield (Mt ha⁻¹), C _W is the cost of wheat production ($ ha⁻¹), C _F is the cost of field management during the fallow period ($ ha⁻¹), and R _C is the per-hectare cropland rental rate for Canadian County, Oklahoma (USDA-NASS, 2021). The cropland rental rate is subtracted from revenue because it is the opportunity cost of not planting a summer crop or renting out the land.

Net returns for a wheat-summer double-crop system in period t were calculated as

(2) $$NR_{WD(t)}=\left(P_{S}\cdot Y_{S\left(t\right)}+P_{W}\cdot Y_{W(t)}-\left[C_{S}+C_{W}\right]\right)\cdot$$

where P _S is the price of the summer crop ($ Mt⁻¹), Y _S(t) is the summer crop yield (Mt ha⁻¹), and C _S are production costs for the summer crop ($ ha⁻¹). The rental rate (ha⁻¹) was subtracted from wheat-fallow systems to incorporate the opportunity cost of forgoing summer crop production by the landowner or a renting entity. The rental rate (ha⁻¹) was not included in wheat-double crop systems because the field was used to produce a summer crop.

Equipment cost calculations include the use of two tractors. The 95-horsepower tractor was used to fertilize, plant, and apply herbicides and pesticides. The 160-horsepower tractor was used for conventional till operations. The certified wheat seed cost was $0.42 kg⁻¹ ($28.40 ha⁻¹) for wheat production in all systems. Certified seeds are more expensive than uncertified seeds due to the extra costs of cleaning seeds.

Manufacturer-recommended rates were programmed into ALMANAC’s management files to control weeds. Herbicides and pesticides included in wheat budgets were Axial XL® ($0.04 g⁻¹) ($29.39 ha⁻¹), Finesse® ($0.58 g⁻¹) ($10.18 ha⁻¹), and dimethoate ($0.009 mL⁻¹) ($8.12 ha⁻¹). Anhydrous ammonia (82-0-0) ($0.53 kg⁻¹) ($16.00 ha⁻¹), diammonium phosphate (18-46-0) ($0.42 kg⁻¹) ($45.98 ha⁻¹), and 28% urea ammonium nitrate ($0.26 kg⁻¹) ($69.94 ha⁻¹) were used to fertilize wheat. Total fuel, lubrication, and repair costs for conventional till and NT wheat were $140.42 and $78.10 ha⁻¹, respectively.

Soybean and sorghum seed were $2.25 and $8.06 kg⁻¹, respectively ($109.06 ha⁻¹ for soybean seed, $30.10 ha⁻¹ for sorghum seed). Sorghum seed was treated with Gaucho® to protect against chinch bugs. Anhydrous ammonia (82-0-0) was used to fertilize sorghum at $0.53 per kilogram ($31.01 ha⁻¹). Bicep Lite II Magnum® ($0.03 g⁻¹) ($37.94 ha⁻¹), Atrazine ($0.004 g⁻¹) ($3.56 ha⁻¹), and Sivanto 200 SL® ($0.09 g⁻¹) ($34.37 ha⁻¹) were used on sorghum to control for weeds and insects, respectively. Bicep Lite II Magnum® ($0.03 g⁻¹) ($47.86 ha⁻¹) and glyphosate ($0.005 g⁻¹) ($5.33 ha⁻¹) were used on soybeans to control weeds. Total fuel, lubrication, and repair costs were $61.90 ha⁻¹ for NT soybean and $102.53 ha⁻¹ for NT sorghum. Total fuel, lubrication, and repair costs for conventional till soybean and sorghum were $97.52 and $133.65 ha⁻¹, respectively.

Comparison of cropping system yield performance and net returns

Net returns and yield performance of cropping systems are analyzed graphically and empirically. A two-sample Kolmogorov-Smirnov (KS) test is used to determine if each production system’s yields and net return distributions were different (Kolmogorov, Reference Kolmogorov1933; Smirnov, Reference Smirnov1939). The null hypothesis is that the two distributions are indistinguishable. The p-value associated with each pairwise comparison is used to determine if the null hypothesis is rejected. KS tests were performed using the ks.test function of the stats package in Rstudio (v. 2.1.461) (RStudio Team, 2020).

Stochastic dominance (SD, Anderson, Reference Anderson1974; Mas-Colell et al., Reference Mas-Colell, Whinston and Green1995) was used to determine if risk preferences affected the decision to use a particular cropping system. SD analysis uses two criteria to compare challenger and defender technologies: (1) First-degree stochastic dominance (FDSD) and (2) second-degree stochastic dominance (SDSD). The first criterion assumes that producers generally prefer higher to lower returns. The second criterion assumes that risk-averse producers prefer to avoid low-valued outcomes but tolerate upside variability in net returns (Lambert and Lowenberg-DeBoer, Reference Lambert and Lowenberg-DeBoer2003). In the generic case of any generic expected utility (EU) function, FDSD implies that the utility function is monotonically increasing (u′(x)> 0) in net returns, meaning that individuals prefer more to less. SDSD implies that the second derivative of a generic EU function is decreasing (u″(x)<0) and monotonically increasing implies risk aversion (Dillon and Anderson, Reference Dillon and Anderson1990).

Graphically, FDSD occurs when a challenger’s empirical cumulative density function (ECDF) of net returns is always to the right of a defender technology. The second assumption – that people prefer to avoid variability in low net returns but tolerate upside variability – characterizes SDSD. Graphically, the empirical distributions of net returns from competing technologies cross at least once. A challenger with the lowest net return can never FDSD- or SDSD-dominate a defender. However, suppose the total area of the crossover areas for a challenger with a larger minimum net return exceeds the total area of crossover areas favoring a defender. In that case, the challenger dominates the defender by the second-degree criterion. The stochastic dominance analysis was conducted using a spreadsheet created by Lowenberg-DeBoer et al. (Reference Lowenberg-DeBoer, Krause, Deuson and Reddy1990) and Hien (Reference Hien1997). In FDSD and SDSD, a prevailing practice consistently yields a higher average return.

The stochastic dominance (SD) framework was used in this analysis because, unlike the stochastic efficiency (SE) framework (Hardaker et al., Reference Hardaker, Lien, Anderson and Huirne2015), SD requires fewer assumptions about risk preferences; namely, choosing an expected utility function and risk aversion parameter is not required. The rankings identified by SD are driven by the data and not necessarily by the expected utility function selected. SD ranks a set of actions or technologies by progressively comparing one alternative to the next until all comparisons are completed (Anderson, Reference Anderson1974). An ordering of FDSD or SDSD technologies remains, including indeterminate cases where neither FDSD nor SDSD can be established.

Sensitivity: cropping systems and weed pressure

A cost-to-go dynamic programming (DP) procedure (Bertsekas, Reference Bertsekas2000) is used to determine which cropping system maximizes net returns over a planning horizon. Earlier applications of cost-to-go programming include Burt and Allison (Reference Burt and Allison1963)’s study of rainfed WF systems in Kansas. Similar DP applications have examined state-contingent optimal cropping strategies (El-Nazer and McCarl, Reference El-Nazer and McCarl1986; Farquharson et al., Reference Farquharson, Cacho, Mullen and Schwenke2008; Jones et al., Reference Jones, Cacho and Sinden2006; Livingston et al., Reference Livingston, Fenandez-Cornejo, Unger, Osteen, Schimmelpfennig, Park and Lambert2015; Patino et al., Reference Patino, Pucheta, Fullana, Schugurensky and Kuchen2004; Periera et al., Reference Periera, Fontes, Ferriera, Pinho, Oliveira, Costa and Silva2013; Tongtiegang et al., Reference Tongtiegang, Zhao, Lei, Wang and Wu2017; Van Kooten et al., Reference Van Kooten, Weisensel and Chinthammit1990). Optimal policies are determined by choosing a cropping system that maximizes the sum of discounted net returns over each weed density state, measured as plants m⁻².

A producer maximizes the expected present value of net returns ha⁻¹ by choosing a cropping system conditional on some level of observed weed pressure and on the tillage method employed. The system used by the producer impacts weed pressure. For example, some practices may be more effective in controlling weed seed banks over multiple periods than others. The producer’s control variable is the cropping system j= WF, WS, or WSrgh. These systems are evaluated separately for conventional till and NT systems, assuming that a producer would unlikely switch back and forth between either tillage method from one season to the next. Some NT farmers might switch to conventional tillage to manage weeds. Zhou et al. (Reference Zhou, Roberts, Larson, Lambert, English, Mishra, Falconer, Hogan, Johnson and Reeves2016) found this to be the case for a limited number of cotton producers. Tillage methods are indexed by k = NT, conventional till. Stages, or periods, are years and denoted as t = 1, 2, …T. The state variable is the level of weed pressure and indexed as i = low (“L”), medium (“M”), and high (“H”). Details explaining these categories and the likelihood of their occurrence follow in the “Weed Pressure Transition Probabilities” section (below).

Net returns for wheat-double-crop systems in period t are the sum of the winter wheat and summer crop net returns. For the WF system, period net returns are from winter wheat less the costs of managing the field during the summer months (e.g., chiseling and herbicide applications) and the opportunity cost of renting the land ($ ha⁻¹). Expected (average) net returns from a system are used for comparisons, which means we assume that a producer makes decisions based on expected yields and prices. The expected net return associated with each crop decision (j), tillage type (k), and state i is π _j^ik (Table 2).

Table 2. Net returns, simulated cropping system and weed pressure ($ ha⁻¹)

Notes: Yields used to generate net returns were simulated using ALMANAC (Kiniry et al., Reference Kiniry, Williams, Gassman and Debacke1992). CV, coefficient of variation; Std. Dev., standard deviation.

^a Low = 10–20 Palmer amaranth and 5–10 horseweed plants m⁻²; Medium = 21–30 Palmer amaranth and 11–20 horseweed plants m⁻²; High = 31–40 Palmer amaranth and 21–30 horseweed plants m⁻².

^b N = simulated periods.

An optimal policy determines which cropping system maximizes the sum of expected net present values of returns, conditional on a weed pressure state. Bellman (Reference Bellman1957)’s return function is maximized to determine the sum of expected net returns over a T-period planning horizon. Let v _t (i) denote the discounted expected return for a t-period decision sequence under state i. The return function is

$v_{T}(i,k)=\max _{j\in (\mathit{WF},\mathit{WS},\mathit{WSrgh})} \{\pi _{j}^{ik}+\delta \cdot \sum _{i'=1}^{M}p_{ii'}^{jk}\cdot v_{T-1}(i)\}$ , subject to

(3) $$v_{T+1}\left(i,k\right)=0$$

where t = 0, …, T; i = low, medium, high, i′ aliases i, indexes weed pressure states; and [ $p_{ii'}^{jk}$ ] is a matrix of transition probabilities governing the likelihood that weed pressure will increase or decrease from one period to the next (discussed below). The discount factor is δ and is calculated as one divided by one plus the interest rate. The interest rate is 1.44 percent, the 2020 average long-term US bond interest rate (OECD, 2022). The discount factor represents the producer’s preference to enjoy net returns sooner rather than later. The constraint is a terminal condition and implies that the value function is zero in the final period (that is, returns are zero once farming the field ends). Equation 1 was solved using Miranda and Fackler (Reference Miranda and Fackler2004)’s ddpsolve package (Miranda and Fackler, Reference Miranda and Fackler2004) in MATLAB (R2020a, Update 7; MathWorks Inc., 2020).

Weed pressure transition probabilities

Transition probabilities govern the evolution of weed pressure from state i to i′ between periods, given the implementation of the (j, k)th cropping system. Each cropping system and conventional till-NT tillage method has a unique transition probability matrix (six matrices). Weed density is observed at the beginning of a planting cycle (fall) in state i = low (“L”), medium (“M”), and high (“H”). Weed density levels were discretized by calculating the number of weeds m⁻² from ALAMANAC output and then categorizing the counts into discrete groups based on classifications reported in the literature (Crose et al., Reference Crose, Manuchehri and Baughman2020; Mahoney et al., Reference Mahoney, Jordan, Hare, Leon, Roma-Burgos, Vann, Jennings, Everman and Cahoon2021; Shyam et al., Reference Shyam, Chahal, Jhala and Jugulam2020; Swanton et al., Reference Swanton, Shrestha, Clements, Booth and Chandler2002; Van De Stroet and Clay, Reference Van De Stroet and Clay2019). Under the low weed pressure state, the number of horseweed and Palmer amaranth plants m⁻² were five to 10 and 10 to 20, respectively. Weed counts m⁻² for the medium density state was 11 to 20 and 21 to 30 plants m⁻² for horseweed and Palmer amaranth, respectively. High weed pressure counts were 21 to 30 and 31 to 40 plants m⁻², respectively, for horseweed and Palmer amaranth.

Weed biomass at wheat planting was used to benchmark the weed pressure categories. Cumulative horseweed and Palmer amaranth biomass were recorded at wheat harvest and converted to plants m⁻². Palmer amaranth biomass was assumed to be 35 g plant⁻¹ (Mahoney et al., Reference Mahoney, Jordan, Hare, Leon, Roma-Burgos, Vann, Jennings, Everman and Cahoon2021). Horseweed biomass was assumed to be 4.3 g plant⁻¹ (Nandula et al., Reference Nandula, Poston, Kroger, Reddy and Reddy2015). Weeds m⁻² counts were then assigned to L, M, or H states. Each system had 20 weed biomass measurements, one for each simulation year.

Transition probabilities were estimated using a maximum likelihood procedure (Amemiya, Reference Amemiya1985), which yields a closed-form solution for each probability:

(4) $$p_{ii'}^{jk}={m_{ii'} \over \sum _{i'=1}^{3}m_{ii'}}$$

where m _ii′ is the number of times over the 20 periods that weed pressure was observed in state i, period t, and was followed by the weed pressure moving to state i′ in time t+ 1. Transition probabilities were estimated using the markovchain package (Spedicato, Reference Spedicato2017) in RStudio (v.1.4.1106; RStudio Team, 2020).

As the number of periods increases, the transition probabilities converge to a stationary distribution (Isaacson and Madsen, Reference Isaacson and Madsen1976). In other words, in the long term and no matter what the beginning state is, steady-state transition probabilities proxy the proportion of time the system will be observed in state i as π _i . The stationary distribution of transition probability matrix ${\bf P}$ is some vector ${\bf \pi\ }$ such that ${\bf P'}{\bf \pi\ }={\bf \pi }$ . The vector ${\bf \pi\ }$ is constant for all states and implies that, over the longer term, no matter what state of weed pressure the cropping area started in, the proportion of time the cropping area spends in weed pressure state i is approximately π _i for all i. Steady-state transition probabilities were solved as a system of linear equations using Miranda and Fackler (Reference Miranda and Fackler2004)’s procedure.

A drawback of using the transition probabilities in Eq. 4 is that weed pressure transitions are assumed to be constant over time. This is a simplifying assumption. Weed populations have variable lifecycles. The transition probabilities could differ between older weed populations with a higher probability of natural die-offs and newly established populations (Torell et al., Reference Torell, McDaniel and Williams1992). Information regarding the age of the weed population at the time of biomass measurement was unavailable, and this simplifying assumption is a caveat.

Medium- and long-term net returns

Net returns from each system are calculated for medium- and long-term planning horizons. The medium-term evaluation is a five-year planning horizon. There is no terminal value for the long-term planning horizon (e.g., equation 2 is omitted), and land used to produce crops is assumed to continue indefinitely. Burt and Allison (Reference Burt and Allison1963) and Kennedy (Reference Kennedy1986) show that, as the number of periods in a planning horizon increases, the recurrence relation of Eq. 1 converges to an invariant decision rule according to the properties of Markov systems. In the long term, an optimal policy corresponding with a specific weed pressure state is invariant no matter the period. Thus, for a long-term planning horizon, whenever a producer observes weed pressure state i before planting wheat, the optimal decision rule is to implement cropping system j.

Backward induction cannot be used to solve infinite-horizon weed management scenarios. An approximation method is required to find an optimal policy that maximizes discounted net returns. A policy iteration procedure is used here, which solves optimal cropping systems, given a prevailing state of weed pressure (Miranda and Fackler, Reference Miranda and Fackler2004).

Sensitivity: net returns, weed pressure dynamics, and stochastic yields

Monte Carlo methods were used to evaluate the performance of each system under yield uncertainty. The Monte Carlo simulation included the following steps for m = 1,…M iterations. First, the means and variances of each ALMANAC-simulated crop yield a = W, Soy, Srgh, for cropping system j= WF, WS, and WSrgh and tillage method k= conventional till and NT under weed pressure i= L, M, and H were calculated as ${\hat u_{ia\left( j \right)k}}$ and $\hat\sigma _{ia\left( j \right)k}^2$ , respectively. Afterward:

1. 1. Yields from each cropping system were randomly drawn from a truncated normal distribution, $Y_{ia\left( j \right)k}^m \sim N\left( {{{\hat \mu }_{ia\left( j \right)k}},\hat \sigma _{ia\left( j \right)k}^2} \right);Y_{ij\left( a \right)k}^* \in {{\mathbb R}^ + }$ .

2. 2. Randomly drawn yields, $Y_{ia\left( j \right)k}^m$ , were used to re-calculate net returns for each cropping/tillage system and weed pressure state (denoted as $\hat \pi _{ijk}^m$ ).

3. 3. Equation 1 was re-solved, given $\hat \pi _{ijk}^m$ .

4. 4. Optimal cropping decisions under each weed pressure state were recorded as a (0,1) variable at each iteration. For example, if NT-WF net returns were optimal under weed pressure state L, then “1” was recorded for NT-WF, and “0” was assigned to challengers.

5. 5. The probability that a cropping system was profitable under each state is the sum of the (0,1) outcomes divided by M = 1,000 iterations. For example, suppose NT-WF net returns were optimal 200 times when the weed pressure state was low. In that case, there is a 0.20 likelihood that this system generated the highest net returns, given stochastic yields and the weed pressure state.