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Published online 7 November 2007
Published in Crop Sci 47:2462-2472 (2007)
© 2007 Crop Science Society of America
677 S. Segoe Rd., Madison, WI 53711 USA
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CROP ECOLOGY, MANAGEMENT & QUALITY

Evaluation of the Effect of Density on Potato Yield and Tuber Size Distribution

Alvin J. Bussan*, Paul D. Mitchell, Michael E. Copas and Michael J. Drilias

Dep. of Horticulture and Dep. of Agriculture and Applied Economics, Univ. of Wisconsin-Madison, 1575 Linden Dr., Madison, WI 53706

* Corresponding author (ajbussan{at}wisc.edu).


   ABSTRACT
TOP
 NOTES
 ABSTRACT
INTRODUCTION
 MATERIALS AND METHODS
 RESULTS
 DISCUSSION
 REFERENCES
 
Potato (Solanum tuberosum) yield has been optimized for in-row spacings ranging from 15 to 40 cm depending on region, targeted market, variety, and other factors. Production goals require optimizing tuber size to maximize crop value. Our goal was to evaluate the effect of plant, stem, and tuber density on stem and tuber set, potato yield, tuber size distribution, and other quality factors. Research plots were established within a 20-ha commercial production field, and analysis was done with linear and nonlinear regression. Plant density decreased with increasing in-row plant spacing. Stem density increased linearly with increasing plant density, but response differed across years. Tuber density increased to a maximum of 190 tubers m–2 in response to plant and stem density, with stem density more accurately predicting tuber set. Yield was related to plant, stem, and tuber density using nonlinear regression, more accurately predicted by stem and tuber density than by plant density. A hyperbolic model was used to predict yield with estimated maximum yield of 86 Mg ha–1 when related to stem density. Average tuber size was related to stem and tuber density using the inverse yield law and estimated maximum average tuber size of >200 g. The distribution for tuber sizes was estimated as a Weibull probability density function that predicted changes in tuber size in response to stem and tuber density. The hyperbolic model accurately predicted tuber density and yield with the added benefit that estimated parameters have biological importance, unlike polynomial or other regression models used to predict crop yield. Modeling tuber size distribution over different stem densities provides a mechanism for future economic analysis to optimize management and conduct sensitivity analysis to determine the most important factors influencing crop value.

Evaluation of the Effect of Density on Potato Yield and Tuber Size Distribution

Alvin J. Bussan*, Paul D. Mitchell, Michael E. Copas and Michael J. Drilias

Dep. of Horticulture and Dep. of Agriculture and Applied Economics, Univ. of Wisconsin-Madison, 1575 Linden Dr., Madison, WI 53706

* Corresponding author (ajbussan{at}wisc.edu).

Potato (Solanum tuberosum) yield has been optimized for in-row spacings ranging from 15 to 40 cm depending on region, targeted market, variety, and other factors. Production goals require optimizing tuber size to maximize crop value. Our goal was to evaluate the effect of plant, stem, and tuber density on stem and tuber set, potato yield, tuber size distribution, and other quality factors. Research plots were established within a 20-ha commercial production field, and analysis was done with linear and nonlinear regression. Plant density decreased with increasing in-row plant spacing. Stem density increased linearly with increasing plant density, but response differed across years. Tuber density increased to a maximum of 190 tubers m–2 in response to plant and stem density, with stem density more accurately predicting tuber set. Yield was related to plant, stem, and tuber density using nonlinear regression, more accurately predicted by stem and tuber density than by plant density. A hyperbolic model was used to predict yield with estimated maximum yield of 86 Mg ha–1 when related to stem density. Average tuber size was related to stem and tuber density using the inverse yield law and estimated maximum average tuber size of >200 g. The distribution for tuber sizes was estimated as a Weibull probability density function that predicted changes in tuber size in response to stem and tuber density. The hyperbolic model accurately predicted tuber density and yield with the added benefit that estimated parameters have biological importance, unlike polynomial or other regression models used to predict crop yield. Modeling tuber size distribution over different stem densities provides a mechanism for future economic analysis to optimize management and conduct sensitivity analysis to determine the most important factors influencing crop value.


   INTRODUCTION
TOP
 NOTES
 ABSTRACT
 INTRODUCTION
MATERIALS AND METHODS
 RESULTS
 DISCUSSION
 REFERENCES
 
Anumber of experiments have examined potato (Solanum tuberosum) yield response to planting population or in-row spacing to optimize plant population (Allen and Wurr, 1992; Arsenault et al., 2001; Creamer et al., 1999; DeBuchananne and Lawson, 1991; Love and Thompson-Johns, 1999; Lynch and Rowberry, 1977; Lynch et al., 2001; O'Brien and Allen, 1992; Rex, 1991; Sekhon and Singh, 1985; Strange and Blackmore, 1990; Wurr et al., 1990, 1992, 1993; Zebarth et al., 2006). Many of these experiments and others have also examined the effect of planting population on tuber size distribution (Allen and Wurr, 1992; DeBuchananne and Lawson, 1991; Love and Thompson-Johns, 1999; Rex, 1991; Sekhon and Singh, 1985; Strange and Blackmore, 1990; Wurr et al., 1992, 1993; Zebarth et al., 2006). The tuber size distribution is an important determinant of price. Different potato market classes have different pay scales across different tuber size categories. Contract-established prices for chip potatoes are higher for tubers with diameters between 5 and 10 cm, and processing russet potatoes received price premiums for tubers >285 g and disincentives for tubers <100 g (Creamer et al., 1999; DeBuchananne and Lawson, 1991; Love and Thompson-Johns, 1999; Schotzko et al., 1984). Fresh market potato prices are more variable over different types of potatoes and times of the year, but fresh market russets received premiums for tubers meeting specific sizes for case counts (i.e., 100–285 g), and red potatoes received price premiums for tubers <5 cm in diameter during different times of the year (Strange and Blackmore, 1990). Tighter margins in potato have required growers to maximize price through optimization of tuber size distribution to maximize crop value (Love and Thompson-Johns, 1999; Schotzko et al., 1984; Wurr et al., 1992, 1993).

Research on managing potato in-row spacing has focused on modeling the response of crop yield and tuber size distribution in response to crop density (Allen and Wurr, 1992; Arsenault et al., 2001; Bleasdale, 1965; Iritani et al., 1983; Knowles and Knowles, 2006; Lynch et al., 2001; Wurr et al., 1990, 1992, 1993). Polynomial regression and square root models related potato yield to crop density. Allen and Wurr (1992) identified the need for biologically meaningful models to predict crop yield response to crop density. A hyperbolic model derived from the inverse yield law was proposed as a means to predict yield response to competition (Spitters, 1983a,b; Weiner, 1982). The hyperbolic model has become a standard method of assessing the impact of weed competition on crop yield (Cousens et al., 1987; Holman et al., 2004; Jasieniuk et al., 2001). Recently, the hyperbolic model predicted wheat yield over multiple years across sites around the western United States (Holman et al., 2004; Jasieniuk et al., 2001). Yet this model has had little application in the prediction of crop yields. Potato yield was related to crop density with modified forms of the inverse yield law, but the hyperbolic function has not been used to predict potato yield response to crop density (Lynch and Rowberry, 1977).

Understanding and predicting the response of tuber size distribution to tuber, stem, or crop density has become increasingly important because of its effect on crop price. Average tuber size has been shown to decrease nonlinearly in response to increasing crop density (Allen and Wurr, 1992; DeBuchananne and Lawson, 1991; Iritani et al., 1983; Knowles and Knowles, 2006; Travis, 1987; Wurr et al., 1992, 1993; Zebarth et al., 2006). Similar to crop yield, quadratic and square root models were used to relate average tuber size and crop density in potato (Knowles and Knowles, 2006; Wurr et al., 1992, 1993). The inverse yield law predicted response of average yield or biomass per plant to increasing crop density (Harper, 1965; Holliday, 1960; Lynch and Rowberry, 1977; Spitters, 1983a; Weiner, 1982). Changes in average tuber size in response to increasing density had a similar pattern to average plant yield or biomass (Allen and Wurr, 1992, Knowles and Knowles, 2006; Lynch and Rowberry, 1977). The inverse yield law may predict average tuber size, with the added benefit that estimated parameters have biological significance, such as maximum average tuber size (Holliday, 1960; Weiner, 1982). However, potato crop price was influenced by the proportion of tubers in multiple size categories, not just the average tuber size. To predict crop value, methods of predicting changes in the tuber size distribution were necessary (Allen and Wurr, 1992; Knowles and Knowles, 2006; Love and Thompson-Johns, 1999; Rex, 1991; Strange and Blackmore, 1990; Wurr et al., 1990, 1992, 1993). Normal distributions have been used to estimate the distribution of tuber sizes to increasing potato density (Travis, 1987; Wurr et al., 1992). However, most potato size distributions do not have a normal distribution, limiting the utility of this approach for predicting tuber size distribution (Knowles and Knowles, 2006; Love and Thompson-Johns, 1999; Rex, 1991; Sekhon and Singh, 1985; Strange and Blackmore, 1990).

The importance of managing crop density to optimize potato yield and tuber size distribution should be evident, but the appropriate measure of crop density in potato has been debated. Plant density (in-row spacing) seems a logical measure of crop density. However, predictions of potato yield response to plant density have been limited due to variability in yield components within and across multiple experiments (Allen and Wurr, 1992; Bleasdale, 1965; Hammes, 1985; Lynch and Rowberry, 1977; Lynch et al., 2001; Wurr et al., 1990, 1992, 1993). Differences in potato seed production, seed size, and handling have affected stems and tubers per plant within a common variety and contributed to variability in crop yield response to plant density (Bleasdale, 1965; Iritani et al., 1983; Knowles and Knowles, 2006; Knowles et al., 1985; Lynch and Rowberry, 1977; Lynch et al., 2001; O'Brien and Allen, 1992; Sekhon and Singh, 1985; Strange and Blackmore, 1990; Wurr et al., 1990, 1992, 1993). Stem density predicted potato crop yield better than plant density, in part because stem density predicted tuber density more accurately than plant density (Allen and Wurr, 1992; Bleasdale, 1965; De la Morena et al., 1994; Lynch et al., 2001; Wurr et al., 1990; Zebarth et al., 2006). In addition, stems per plant have not been influenced by plant density but by physiological factors resulting from the management of the seed (Allen and Wurr, 1992; De la Morena et al., 1994; Iritani et al., 1983; Knowles and Knowles, 2006; Knowles et al., 1985; Love and Thompson-Johns, 1999; O'Brien and Allen, 1992; Rex, 1991; Wurr et al., 1990). As a result, research efforts have focused on identifying methods to manage stem density by seed manipulation and optimal crop planting rates based on anticipated stem production per seed piece. Tubers per plant changed with increasing plant or stem density, making management of tuber density dependent on the predicted response to increasing stem or plant density (Allen and Wurr, 1992; Bleasdale, 1965; De la Morena et al., 1994; Hammes, 1985; Iritani et al., 1983; Knowles and Knowles, 2006; Lynch et al., 2001; Love and Thompson-Johns, 1999; O'Brien and Allen, 1992; Rex, 1991; Wurr et al., 1990, 1992, 1993; Zebarth et al., 2006). In addition, assessing and hence managing tuber density requires destructive plant sampling, further increasing the difficulty of manipulation. Ultimately, the high correlation between stem and tuber density has allowed prediction of tuber density and ultimately yield with stem density (Allen and Wurr, 1992; Bleasdale, 1965; De la Morena et al., 1994; Hammes, 1985; Lynch et al., 2001; Knowles and Knowles, 2006; Wurr et al., 1990, 1992, 1993).

The goal of this project was to quantify effects of crop density on potato tuber yield and the tuber size distribution. Specific objectives were (i) to identify the influence of in-row spacing between seed pieces on stem and tuber density, (ii) to quantify the crop yield response of potato to plant, stem, and tuber density, and (iii) to quantify the response of average tuber size and the tuber size distribution to stem and tuber density.


   MATERIALS AND METHODS
TOP
 NOTES
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
RESULTS
 DISCUSSION
 REFERENCES
 
Field-scale experiments were conducted during 2002 and 2003 to evaluate the influence of changing potato planting population on stem and tuber set, potato yield, and tuber size distribution. Experiments were conducted on Coloma Farms, near Coloma, WI. Research fields were planted for production of ‘Russet Burbank’ potatoes and for their proximity to the University of Wisconsin–Madison, Hancock Agricultural Research Station (latitude: 44°8'23' N; longitude: 89°31'23'; elevation: 328 m). The soil type was Plainfield loamy sand (sandy, mixed, mesic, Typic Udipsamments). The experimental design was a randomized complete block with six replications. Treatments included three different planting rates. A commercial 12-row planter was preset to plant seed pieces at 30-, 40-, and 50-cm spacings within the row. The size of the experimental field was 16 ha each year of the experiment. Each plot was 18 m wide (two passes with the potato planter) and 400 m long. Subplots were established at four distinct locations across the length of the plot following emergence. Each subplot included five plants, and the distance between the first and fifth plant was recorded to allow for calculating density. Sprayer track lanes and center pivot irrigation wheel tracks had highly variable microclimates and severe compaction and were avoided when establishing subplots. Experimental measurements were collected within each subplot.

Field corn was the previous crop, and Russet Burbank was the potato variety planted across the entire field each year. Crop management strategies used at each site were based on recommendations developed at the University of Wisconsin–Madison (Boerboom et al., 2006; Kelling et al., 1998). Fields were prepared for planting by spring subsoil tillage before the 2002 experiment and fall subsoil tillage before the 2003 experiment. Final tillage was completed with a soil finisher. Seed tubers were machine cut into 65- to 75-g pieces and suberized for 3 to 5 d before planting. Plots were machine planted at a depth of 12 to 15 cm, and seed spacing was set according to treatment in rows spaced 75 cm apart on 20 Apr. 2002 and 25 Apr. 2003. Fertilizer applications were based on soil and plant tissue analysis following standard practices for Russet Burbank potatoes in central Wisconsin. Hilling was completed just before crop emergence with a disk hiller, resulting in standard flat hills in 2002. The hilling operation was completed with a rotating blade reservoir hiller in 2003. Supplemental irrigation was applied with a center pivot and typically started in early June and continued through early September following vine desiccation. Irrigation and timing were based on a scheduling program that used estimates of evapotranspiration and precipitation.

Final harvest was coordinated with the cooperating grower, and all subplots were hand harvested before field digging operations. Plots were hand dug between the third and fourth week of September, depending on the timing of vine desiccation. The total number of stems and tubers was recorded for each five-plant subplot at harvest. The length of subplot and row width were used to calculate plant, stem, and tuber density. Hand harvesting was done to ensure accurate assessment of stem and tuber counts and to ensure harvest of all tubers. Harvested subplots were graded at the University of Wisconsin–Madison Hancock Agricultural Research Station for yield and tuber distribution. Tubers were washed and graded according to industry size categories: less than 113, 113 to 170, 170 to 284, 284 to 369, 369 to 454, and greater than 454 g. B-sized potatoes (tubers <4.75 cm in diameter) and cull potatoes (including rotted, off-shaped, growth cracked, sunburned, and green tubers) were removed and weighed separately before size grading. The average weight of B-sized tubers was 84 g as determined by four lots of 100 tubers.

Data Analysis
The data were subjected to linear and nonlinear regression analysis. Lack-of-fit tests were used to determine if nonlinear models improved predictability in the data compared with linear models. Data were then subjected to a modified Levene's test (Neter et al., 1996) to assess homogeneity of error variances between years before combining data. In addition, before combining data, parameter estimates were compared across years and F tests were completed to determine if analysis by year described more of the data than combined analysis. Linear regression was used to relate plant density to in-row spacing and stem density to plant density Tuber density was related to plant and stem density with nonlinear regression. Crop yield was related to plant, stem, and tuber density with nonlinear regression as well. A hyperbolic model was used to predict tuber density and crop yield (Eq. [1]) (Cousens et al., 1987; Jasieniuk et al., 2001):

Formula 1[1]
where Y is tuber density or yield; i is initial slope of the curve and represents the tubers or yield per plant, stem, or tuber at low density and in the absence of intraspecific competition; N is the density of plants, stems, or tubers per square meter; and a is the upper asymptote and represents the maximum tuber set or yield. Average tuber size was related to stem and tuber density with nonlinear regression. A modified version of the inverse yield law was used to predict average tuber size (Eq. [2] and [3]) (Holliday, 1960; Spitters, 1983a; Weiner, 1982):

Formula 2[2]

Formula 3[3]
where Formula 3t is the average tuber size, Rmax is the maximum tuber size in the absence of intraspecific competition, and b and c represent the change in average tuber size with change in density.

The distribution for the size of potato tubers was modeled as a probability density function with parameters depending on tuber or stem density. Parameters describing tuber size distribution were estimated by fitting a cumulative distribution function to observed accumulated proportion of tubers in each size category. A Weibull distribution was used because it was strictly positive and sufficiently flexible to be skewed left or right or to be symmetric (Evans et al., 2000). For a random variable Rt with a Weibull distribution, the probability density function f(Rt) and the cumulative distribution function F(Rt) were

Formula 4[4]

Formula 5[5]
where the parameters {alpha} and ß are strictly positive (Evans et al., 2000).

The cumulative distribution function (Eq. [5]) was fit to observed data using least squares to estimate the parameters {alpha} and ß. The effect of tuber density or stem density was captured by estimating the parameters {alpha} and/or ß as functions of these measures. Several equations for {alpha} and ß as functions of Nt, tuber density, or Nst, stem density, were estimated. Based on the R-squared, the best model directly estimated {alpha} and fit the following exponential model for ß:

Formula 6[6]
where N is either Nt for tuber density (tubers m–2) or Nst for stem density (stems m–2).


   RESULTS
TOP
 NOTES
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
 RESULTS
DISCUSSION
 REFERENCES
 
In-row spacing of plants effectively altered plant density across both years (Fig. 1 ). Plant density varied within treatments even though the planter was set to deliver predetermined in-row spacings of 30, 40, and 50 cm. The range in minimum to maximum density around each row spacing treatment was 2 plants m–2 centered on the average density targeted for the treatment. Total range in plant density resulting from the treatment was between 2.0 to 4.5 plants m–2 or 20,000 to 45,000 plants ha–1. Plant densities listed assume each hill was planted to a single seed piece and that two plants were not growing together. The grower used a mechanical pick planter to minimize the number of doubles in each hill. The number of doubles could not be confirmed without destructively sampling the hills after planting.


Figure 1
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  		Figure 1. Influence of in-row spacing on potato plant population near Coloma, WI, in 2002 and 2003.

 
Stem density was linearly related to potato plant density across the field landscape each year (Fig. 2 ). The linear response of the data indicated that the stems per plant were not influenced by plant density. Plant density ranged from 2.0 to 4.5 plants m–2 across years and maximum stem density approached 26 and 16 stems m–2 during 2002 and 2003, respectively. Maximum stem density was higher in 2002 than in 2003 because stems per plant were 60% higher in 2002, hence the differences in response between years. In 2002 each plant had approximately 4.5 stems per plant and only 2.9 stems per plant in 2003.


Figure 2
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  		Figure 2. Response of potato stem density to potato plant density in commercial production field during 2002 and 2003 near Coloma, WI.

 
Tuber density increased with plant and stem density across years (Fig. 3 ). Initial analysis showed that tuber density was slightly higher in 2002 than 2003, but on deeper investigation the differences in tuber density were due to slightly higher stem densities in 2002 (Fig. 2). Nonlinear models did not describe data any better than linear regression (data not shown). However, tubers per plant decreased with increasing plant and stem density in 2002, suggesting that slope was not constant as was predicted by linear regression. Therefore, nonlinear models were used to describe response of tuber density to plant and stem density. A single nonlinear model was fit across both years because describing tuber density response independently each year did not explain any more of the data based on an F test (data not shown). In addition, parameter estimates were consistent across both years. Plant density described 27% of the variation in the tuber density data across both years. Tuber set was 30 tubers per plant in the absence of intraspecific competition (initial slope of the curve) and the maximum tuber set was 187 tubers m–2. Stem density described 55% of the tuber density across years. Tuber set was 7.6 tubers per stem in the absence of intraspecific competition, and the maximum was estimated at 196 tubers m–2.


Figure 3
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  		Figure 3. Potato tuber density response to plant and stem density in a commercial potato production field during 2002 and 2003 near Coloma, WI.

 
Yield increased with plant, stem, and tuber density and was predicted with nonlinear models (Fig. 4 ). In predicting yield, nonlinear regression described more of the data based on plant, stem, and tuber density than linear regression based on lack-of-fit tests. Because yield was similar across years and parameter estimates did not differ, the data were combined for quantifying yield response to plant, stem, and tuber density. Plant density explained only 6% of the variability in yield, but the parameter estimates were highly significant (P < 0.01) with an estimated maximum yield of 82 Mg ha–1 and yield per plant at low densities of 88 Mg ha–1. The estimate of yield per plant at low densities was significant (P = 0.002) but had a large error due to the lack of data collected at low densities. Stem density described 19% of the variability in the yield data, with each stem contributing 20 Mg ha–1 at low densities and a maximum estimated yield of 86 Mg ha–1. Tuber density provided the greatest explanatory power, predicting 22% of the variation in yield. Potato yield was 2.7 Mg ha–1 per tuber at low densities, with maximum yield estimated at 100 Mg ha–1.


Figure 4
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  		Figure 4. Relationship between potato yield and plant, stem, and tuber density in commercial field trials during 2002 and 2003 near Coloma, WI.

 
Average tuber size decreased with increasing stem and tuber density (Fig. 5 ). Data were combined due to similar parameter values across years and lack of improved fit with individual analysis each year. Potato tubers were smaller in 2002 due to higher stem and tuber density compared to 2003. Inverse yield law described more of the data than linear regression in large part due to response at higher tuber densities in 2002. Stem density described 26% of the average tuber size variability with a maximum of 210 g tuber–1. Tuber density described 26% of the variability in average tuber size as well, but the maximum was 245 g tuber–1. Estimates of maximum tuber size were not different when describing the response to tuber versus stem density.


Figure 5
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  		Figure 5. Average tuber size in response to stem and tuber density in commercial field trials during 2002 and 2003 near Coloma, WI.

 
The potato tuber and stem density also appeared to affect the distribution of tuber size (Fig. 6a ). Examples from plots selected from extreme tuber or stem densities were presented to illustrate the range in tuber size distributions. The most limiting property of the data was that tuber size was strictly positive, requiring a probability density with a strictly positive range, thus precluding the popular normal (Gaussian) density. In addition, the tuber size distribution was skewed right or left, or symmetric for different plots, requiring a flexible probability density function able to capture these possible shapes. The Weibull distribution predicted the proportion of tubers falling into each size category and improved estimation relative to a normal distribution (data not presented). The Weibull distribution can be estimated using the data for an individual plot (Fig. 6b). However, statistical analysis (not shown) indicated that pooling the data across treatments and years was appropriate. In addition, the tuber size distribution and its response to changing stem or tuber densities were successfully estimated by directly estimating {alpha} and using Eq. [6] for ß (Tables 1 and 2 ).


Figure 6
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  		Figure 6. (a) Proportion of tubers fitting into each size category from three plots selected with high, medium, and low density, and (b) observed versus estimated proportion of tubers within each size category for a select plot from commercial field trials near Coloma, WI.

 

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  		Table 1. Least squares estimated parameters for the Weibull distribution describing the distribution of potato tuber sizes as a function of tuber density.

 

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  		Table 2. Least squares estimated parameters for the Weibull distribution describing the distribution of potato tuber sizes as a function of stem density.

 
Using either the stem or tuber density provided R2 values of 0.93 for fitting the cumulative proportion of tubers by size (Tables 1 and 2). The surface response of the cumulative proportion of tubers by size was presented across different stem densities (Fig. 7 ). The surface was similar across tuber densities (data not shown). The plotted surface was the cumulative proportion of tubers by size category as reported in Eq. [5], with constant value for {alpha} and the value for ß calculated as a function of stem density Nst using Eq. [6] and the estimated ß0 and ß1 (Table 1).


Figure 7
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  		Figure 7. Observed and estimated cumulative proportion of potato tubers with increasing tuber size across different stem densities in commercial field trials near Coloma, WI, across 2002 and 2003. Individual data points were provided ({circ}) to illustrate fit of the predicted response.

 
Examination of the proportion of tubers by size as predicted by the probability density function (Fig. 8 ) seemed more intuitive, rather than the cumulative proportion of tubers by size category as predicted by the cumulative distribution function (Fig. 7). The implied tuber size distribution in Fig. 8 was conditional on stem density. The plotted surface was the tuber size distribution reported in Eq. [4], with constant value for {alpha} and ß calculated as a function of stem density Nst using Eq. 6 and the estimated ß0 and ß1 (Table 1). As the stem density increased, the estimated tuber size distribution shifted so that the proportion of large tubers decreased and the proportion of small tubers increased. The estimated proportion of tubers around 150 g remained relatively constant across all stem densities, while the proportion of small tubers (generally <150 g) increased with the stem density and the proportion of large tubers (generally >150 g) decreased with stem density.


Figure 8
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  		Figure 8. Estimated tuber size distribution across different stem densities estimated with data from commercial field trials from 2002 and 2003 near Coloma, WI.

 

   DISCUSSION
TOP
 NOTES
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
 RESULTS
 DISCUSSION
REFERENCES
 
Variability in plant density in response to in-row spacing was high, but this may be attributable to the mechanical planting operation (Fig. 1). Previous research in potato has reported variation in plant density due to the efficacy of mechanical planting operations (Pavek and Thornton, 2005; Schotzko et al., 1984). Field corn (Zea mays L.) has seen similar variations in stand establishment across fields (Lauer and Rankin, 2004). Small-plot research trials that were mechanically planted have seen similar variation in wheat (Triticum aestivum L.), sunflower (Helianthus annuus L.), and canola (Brassica rapa L.) stands due to variability in the operation of mechanical planting (Holman et al., 2004). Sampling error associated with the data collection method may also have contributed to variability in plant establishment in response to mechanical planting. Samples of five plants were collected from each subplot, which allowed for detailed measurements of stem, tuber, tuber size, and other parameters. This may have increased the error of true mean within each area of the field where the sample was collected. Each sample plot was 1 to 3 m of row, which is a common area assessment for determining crop status during the year. Variability in stand and other potato data may have been influenced by sampling error, but the variation was within previously reported estimates in planting operation (Pavek and Thornton, 2005). This error did not influence various relationships or the interpretation of the data. Furthermore, the sample size permitted extensive sampling within the trial, allowing for ample data points to fit the different regression functions. Another source of error may have been the development of double drops. Double drops occur where more than one potato seed piece has been placed in a single hill and would have resulted in six instead of five plant samples in this research (Pavek and Thornton, 2005). Double drops were unavoidable but would have only adversely influenced relationships with plant density. Stem and tuber density responses should not have been affected by double drops other than to potentially increase stem and tuber densities. Finally, sample variation could have been influenced by spatial variability within the field. This experiment was conducted in a 16-ha field, and sample-to-sample variation may have affected results. However, many of the models were accurately predicted despite the spatial variability, further emphasizing their biological importance in determination of potato crop response.

Stems per plant was constant across the range of densities observed within these trials, resulting in a linear relationship between stem and plant densities (Fig. 2). Previous research has demonstrated that stem density is rarely influenced by intraspecific competition in potato (Allen and Wurr, 1992; Bleasdale, 1965; De la Morena et al., 1994; Iritani et al., 1983; Knowles and Knowles, 2006; Knowles et al., 1985; Love and Thompson-Johns, 1999; Lynch and Rowberry, 1977; O'Brien and Allen, 1992; Rex, 1991; Strange and Blackmore, 1990; Wurr et al., 1990, 1992, 1993). Year-to-year variability in stem set per plant (slopes from linear models in Fig. 2) occurred, leading to different relationships between stems per meter and plant density. Recent research has illustrated the influence of seed tuber size, physiological age, and handling on the stem set of potato seed of the same cultivar (Iritani et al., 1983; Knowles and Knowles, 2006; Knowles et al., 1985; O'Brien and Allen, 1992). The seed was from different sources and produced under different growing conditions during 2002 and 2003, likely resulting in different response across years. In addition, many experiments evaluating the influence of seed on stem set commonly used seed tubers from common size categories, but commercial seed was used in this study that included whole seed as well as cut seed from tubers 70 to 280 g. Even though 90% of seed pieces were 65 to 75 g (data not shown), the current research suggests the seed could produce variable stem per plant if derived from different-sized seed tubers. Allen and Wurr (1992) suggested that stem density should be assessed during early plant development, around the time of tuber initiation. New stems seldom initiate after tuber set, and few have senesced that early in the growing season. Stems were counted at harvest in this trial to allow for destructive harvest due to the propensity of potato to branch under ground. Accuracy of stem counts may have been compromised to allow for destructive sampling at harvest, but this was the only way to avoid sampling error in the determination of stem density.

Tuber per plant decreased with increasing plant and stem densities observed within these trials, resulting in nonlinear relationship between tuber set and density (Fig. 3). Interspecific competition has been observed in other research to reduce tubers per plant as potato density increased (Allen and Wurr, 1992; Bleasdale, 1965; De la Morena et al., 1994; Iritani et al., 1983; Knowles and Knowles, 2006; Knowles et al., 1985; Love and Thompson-Johns, 1999; Lynch and Rowberry, 1977; Lynch et al., 2001; O'Brien and Allen, 1992; Rex, 1991; Strange and Blackmore, 1990; Wurr et al., 1990, 1992, 1993). The hyperbolic model (Eq. [1]) accurately predicted tuber density in response to plant and stem density. Somewhat surprising was that the response in tuber density to stem density was similar across the 2 yr of the study and that stem density explained more than 50% of the variation in tuber density. This occurred despite potential differences across years, seed, and spatial variability over the field landscape. Tuber set per stem has been shown to be highly conserved across years in previous experiments, but under tightly controlled conditions (Knowles and Knowles, 2006; Knowles et al., 1985; Love and Thompson-Johns, 1999; O'Brien and Allen, 1992). In contrast, Lynch et al. (2001) reported variable tuber per stem affected by factors independent of plant density. This research indicated that the relationship between stem and tuber density held true in field-scale production. One item to note was that maximum tuber density within this trial was measured to be 120 tubers m–2 and estimated at 180 tubers m–2. The measured tuber density for Russet Burbank was three times as high as reported in research from the western United States (Love and Thompson-Johns, 1999). Discrepancies in tuber set per stem by common varieties across different growing regions may require specific seed management and crop planting rates to optimize tuber densities within each region.

Yield increased with potato crop density, and the response to plant, stem, and tuber density was described with nonlinear relationship (Fig. 4). Yield response to crop density was not surprising as several other researchers have documented similar relationships to potato density (Allen and Wurr, 1992; Bleasdale, 1965; De la Morena et al., 1994; Iritani et al., 1983; Lynch et al., 2001; Knowles and Knowles, 2006; Knowles et al., 1985; Love and Thompson-Johns, 1999; Lynch and Rowberry, 1977; O'Brien and Allen, 1992; Rex, 1991; Strange and Blackmore, 1990; Wurr et al., 1990, 1992, 1993). De la Morena (1994) identified stem density as the most relevant predictor of potato crop yield through path analysis. Other researchers have agreed that stem density was the most appropriate measure of plant density for predicting potato yield (Allen and Wurr, 1992; Bleasdale, 1965; Iritani et al., 1983; Love and Thompson-Johns, 1999; Wurr et al., 1990, 1992, 1993). Our data agrees in that stem density explained much more of the variability in yield than did plant density in large part due to variability in stem density per plant. Stem density described nearly 20% of the data across years, landscape, and seed. Tuber density described slightly more of the variation in yield, but managing tuber density poses more challenges than stems because it is not easily quantified on field inspection. In addition, stem density explained slightly more than half of the variability in tuber density.

The hyperbolic model (Eq. [2]) used to predict tuber set and yield has not been previously used to predict potato yield response to crop density (Fig. 4). The model was successfully fit to yield data across different density measurements and described more of the variability in yield than linear regression. The primary benefit of the hyperbolic model was the fitting of biologically significant parameter estimates, per plant yield at low density and maximum yield or yield potential (Cousens et al., 1987; Holman et al., 2004; Jasieniuk et al., 2001). Quadratic or square root models included parameter estimates with minimal biological meaning, especially if nonzero intercepts were fit (Allen and Wurr, 1992; Lynch et al., 2001; Knowles and Knowles, 2006; Wurr et al., 1990, 1992, 1993). The relative ability of different models to accurately predict data can be debated, but the hyperbolic model required fitting of only two parameter estimates compared with three for quadratic or square root models (Cousens et al., 1987). A weakness of the hyperbolic model was that it will not fit decline in crop yield at extremely high crop densities, but that has rarely been observed in data of potato yield (Knowles and Knowles, 2006; Knowles et al., 1985; Love and Thompson-Johns, 1999; Lynch and Rowberry, 1977; O'Brien and Allen, 1992; Rex, 1991; Strange and Blackmore, 1990; Wurr et al., 1990, 1992, 1993). The challenge of fitting the hyperbolic model was the need for data at the density extremes, both low and high. Estimates of i and a were equal when predicting potato yield response to plant density, and i may have been overestimated when predicting yield response to tuber and stem density due to lack of data at low densities. Similarly, maximum tuber density (Fig. 3) was likely overestimated, due to lack of high enough densities to determine tuber response at the extremes. However, the model explained tuber set and yield response to potato density within the range of data collected, and parameter estimates were determined that were different than zero. Most statistical software can easily complete nonlinear regression analysis, and the hyperbolic estimation of yield response to density seems suitable, especially for potato where negative yield response at extremely high densities has not been observed.

Average tuber size decreased with increasing stem or tuber density (Fig. 5). Plant density was not used to estimate tuber size because of variability seen in the estimation of crop yield, suggested plant and tuber densities were more suitable management factors. The modified inverse yield law (Eq. [3]) was used to predict the change in average tuber size with increasing plant density (Holliday, 1960; Weiner, 1982). Potato yield per plant was predicted with different versions of the inverse yield law (Lynch and Rowberry, 1977), but it has not been used to describe changes in tuber size in response to increasing density. Average tuber size has been predicted with square root models or polynomial regression (Allen and Wurr, 1992; Wurr et al., 1992). As discussed earlier, this leads to the development of parameter estimates with little biological meaning and requires the fitting of three parameter models in the case of square root and quadratic functions compared with two parameters for the inverse yield model. The inverse yield law predicted theoretical maximum size unit, in this case, maximum average tuber size. The inverse yield law accurately estimated the response in average tuber size to increasing stem and plant density across years and over the field landscape. Stem and tuber density were similar in the proportion of the variability of the data that they described, providing further evidence in support of using stem rather than tuber density as the unit of management due to ease in counting stems relative to tubers (Allen and Wurr, 1992). The inverse yield law requires a broad range of densities for accurate parameter estimation, especially Rmax, similar to fitting the hyperbolic yield model above (Weiner, 1982).

Potato and other vegetable crops are commonly priced differently across a range of product sizes (tubers in the case of potato). Predicting average tuber size alone does not provide enough information on the distribution of tuber sizes to predict crop value. Tuber size data have commonly been reported across six or more categories independently (Arsenault et al., 2001; Love and Thompson-Johns, 1999; Zebarth et al., 2006). However, this has limited utility in quantifying the response of tuber size over a continuous variable such as crop density or for estimating the proportion of tubers that are of a size not directly measured. To address this issue, average tuber size and the tuber size distribution has been estimated with a normal distribution that allowed for evaluation of management factors on each size category (Travis, 1987; Wurr et al., 1992). Average tuber size and spread were influenced by management, including reductions in both as stem density increased. However, this approach had limited utility in potato because average tuber size did not always have a normal distribution (Love and Thompson-Johns, 1999; Zebarth et al., 2006). A Weibull distribution with three parameters effectively predicted the cumulative distribution function of potato tuber size across a range of stem and tuber densities, years, and the field landscape (Tables 1 and 2). The estimated cumulative distribution function allowed for prediction of the proportion of tubers in any size category across the range of densities evaluated within this trial (Fig. 7). Furthermore, the associated probability density function estimated the change in the proportion of tubers by size as a continuous function of tuber size (Fig. 8). The modeling of potato yield and the tuber size distribution will facilitate improved economic evaluations of potato management practices and crop value. Future research will focus on refining estimation of the tuber size distribution as well as using the model to predict crop value and economic return in response to management. Sensitivity analysis will allow the identification of variables most important for maximizing the profitability of potato production.

This research illustrates the utility of several nonlinear models for explaining potato yield response to increasing crop density. The hyperbolic model and inverse yield model provided biologically meaningful parameter estimates that will provide new insights in the optimization of potato crop productivity. Acceptable commercial crop yield of 55 t ha–1 was achieved with minimum densities of eight stems and 50 tubers m–2. Doubling crop density only increased predicted yield by 7 t ha–1 but reduced average tuber size by 20% and increased the proportion of undersized tubers by 10%. In addition, these models should provide insights into understanding the influence of other crop management practices on potato. For example, documenting the influence of weed interference on potato productivity should focus on tuber set per plant, tuber size, and tuber size distribution to understand yield responses to interspecific competition. Furthermore, the prediction of tuber size distribution will allow for economic assessments of crop value that were previously limited.


   ACKNOWLEDGMENTS
 
Funding for this research was provided by the Wisconsin Potato and Vegetable Growers Association. The authors appreciated the cooperation of Coloma Farms in the completion of this research. In addition, the authors acknowledge the contributions of the staff at the University of Wisconsin–Madison Hancock Agricultural Research Station and the numerous student hourly employees who contributed to the completion of this research.


   NOTES
TOP
 NOTES
ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
 RESULTS
 DISCUSSION
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Received for publication January 17, 2007.


   REFERENCES
TOP
 NOTES
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
 RESULTS
 DISCUSSION
 REFERENCES
 





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