Introduction

Potato (Solanum tuberosum L.) is one of the most important food crops in the world, grown in more than 95 countries and ranked fourth among all food crops in total production (FAO 2022). Due to its shallow rooting system, potato is very sensitive to water stress (Djaman et al. 2021). Therefore, irrigation is required for profitable commercial potato production in semi-arid and arid regions (King and Stark 2003). Even in humid regions, irrigation is used as a means to supplement natural precipitation to ensure enough water supply for potato cropping (Zebarth et al. 2019). Worldwide, the acreage of irrigated lands has been increasing steadily over the years, contributing to the ever-growing yield and overall production of potatoes despite the slight decline in harvested area for potatoes in the world (FAO 2021). However, there are still a large share of potato fields in the world being under rain-fed agriculture, especially in developing countries. In Canada, potato has been cultivated since the mid-1600s in the Atlantic Canada (AC) province of New Brunswick (PotatoPro 2022). Together with Prince Edward Island (PEI), the two AC provinces became the traditional potato production area in Canada. Starting from the 1990s, there is a fast increase in potato acreage in western Canada, especially in the provinces of Manitoba and Alberta (Agriculture and Agri-Food Canada Crops and Horticulture Division 2021). Now a days, Canada ranks as the world's 12th largest potato grower and potato accounts for about one third of all vegetable farm cash receipts, making it Canada's most important horticultural crop (Agriculture and Agri-Food Canada Crops and Horticulture Division 2021). In the Atlantic Canada provinces in particular, potato is one of the main cash crops. For example, in 2020, potatoes contributed over 42% of the total farm cash receipts in PEI, and 18% of the total farm cash receipts in New Brunswick (Agriculture and Agri-Food Canada Crops and Horticulture Division 2021).

Whereas potato fields in the semi-arid to arid regions in western Canada are mostly irrigated, those in humid regions, such as AC, are usually not irrigated due to the large differences in natural conditions, especially water availability, between humid and semi-arid to arid regions. In AC, average annual precipitation is in the range of 1100 mm to 1200 mm (ECCC 2021). This is generally enough for growing potato and, therefore, irrigation is not required to sustain a profitable potato production system. As a result, this area is dominated by rain-fed agriculture (DeMerchant 1983; Zebarth et al. 2019). However, this may not be the case with climate change. Mean annual temperature of AC provinces has already increased by 1.1 °C in the past 30 years (ECCC 2021). Projections indicate an anticipated increase of approximately 5 °C across AC, with some areas becoming warmer than others (Kang et al. 2022). Rising future temperature linked with increased CO2 concentration of the atmosphere will potentially add in more uncertainties (ECCC 2021). There is a high confidence that the variabilities of precipitation, temperature and CO2 in both space and time will increase due to climate change (New Brunswick Department of Agriculture, Aquaculture and Fisheries 2022). This means that there will be more extreme weather events such as intensive rain storms, floods, unanticipated dry spells as well as long drying periods (Lemmen et al. 2007; Stocker et al. 2013). Overall, it is likely that in AC, agriculture will experience severe climate change related stress in the future (Maqsood et al. 2020). Increasing temperature will likely shorten the phenological phases and impact crop development rates (Eyshi Rezaei et al. 2017), decrease soil moisture through increased evapotranspiration (Williams et al. 2016), and ultimately reduce crop yield (Adekanmbi, et al. 2023; Klink et al. 2014). As a result of warmer temperatures, the first fall frost in AC is becoming later and the onset of spring is advancing in the Northeast. This aspect of climate change will increase the number of frost-free days, subsequently lengthening the growing season (Wolfe et al. 2018). Adversely, increased heavy rains in future could delay planting, cause physical damage to the crop, degrade fields and increase potential disease pressure (Hatfield and Prueger 2015; Wolfe et al. 2018). Changing temperature and soil moisture conditions may also affect the spread of pests and diseases, as well as the erosivity of the rainfall and the erodibility of the soil, leading to water quality issues in the steams (Wall et al. 2002). Rising CO2 concentrations (a driving force of temperature increase in climate change scenarios) have been documented to positively impact plant biomass accumulation and yield (Finnan et al. 2005; Frumhoff et al. 2007), further reinforcing how complicated the climate change impacts may be.

To overcome the challenges posed by climate change on potato production in AC, there is an urgent need to quantify the impacts of each individual climate factors as well as their interactions so that Beneficial Management Practices (BMPs) can be developed to reduce the negative impact and take advantage of the positive effects. While some effects can be investigated with well designed field experiments, conducting field experiments with multiple changing factors remains to be difficult, if not completely impossible, and costly. A more efficient way is to use models to simulate the responses of potatoes under different scenarios. While there are plenty of models in the literature for potato growth simulation, not all models, for instance, PotatoSoilWat or LINTUL1, can reflect the most relevant factors for potato production under specific sets of conditions. For example, under rain-fed conditions in AC, the most relevant models should be related to the supply of water and plant nutrients and enough temperature to offer resistance to frost damage. One of the applications for these models is adjusting for potato crop management based on model predictions. The selected models are assumed to guide where management can be feasibly implemented in a goal-based way. The highest potato tuber yields are obtained in areas with temperate climates while in tropical and subtropical climates tuber yields are lower and less stable (Kooman 1995). Proper model calibration and parameter estimation may produce equally good simulation results in temperate or tropical regions (Utset et al. 2000). Nevertheless, given the strong interactions between potato crop and climate, and the instability of the nature of such interactions between geographic regions, it is desirable to rely on a few models for AC’s potato production. Ideally, these groups of models should be rapidly clusterable at a detailed level, i.e., with exploration and confirmation based on identified factors. Therefore, the objectives of this review are to 1) analyse and summarize the genesis and functionalities of existing potato growth models; 2) evaluate their suitability to simulate potato growth in different climate regions; and 3) identify key crop growth models for rain-fed potato production systems in Atlantic Canada that can best reflect the influence of climate change.

Potato Model Functionalities

Potato models are used to simulate potato growth and potato yield under real world growing conditions. To date, more than 33 potato crop growth models have been developed (Table 1). Conceptually, the development of potato models can be categorized into four stages based on the functionalities simulated by the models (Fig. 1). As shown in Fig. 1, at the actual stage, models simulate the actual production. At this stage, yield reducing factors such as weeds, pests, diseases, pollutants, adverse temperature, drought etc. must be estimated for a defined geographical unit and time frame. At the intermediary stages, i.e., the water-nutrient-environment limited stage and the management stage, potato plants’ growth and yield are simulated based on limiting factors (water, nutrients, soil properties etc.; and run off, leaching, erosion, greenhouse gas etc.) and management (BMPs) to simulate the production under specific conditions. Management factors such as the BMPs may alter production through direct changes in water and nutrient availability or farmers’ adaptations in terms of planting dates, planting densities, cultivar maturities, tillage and crop rotation practices. These factors need to be quantified for individual fields for a given year, or for larger areas over longer time periods (several years) while taking into account their spatio-temporal variabilities (Ewert et al. 2011). At the potential stage, models are developed to simulate potential growth and yield of potato based on defining factors related to climate, crop environment and crop characteristics.

Table 1 List of potato plant growth models examined in this study
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Fig. 1
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Stages of potato plant growth model development: potential, management limited, nutrient-water-environment limited and actual; based on yield defining, limiting, reducing factors and management factors (adapted from van Ittersum et al., 2013)

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The actual stage models consider factors including climatic and biological factors that can negatively impact potato growth thus leading to reduced yield. Climatic variables play a key role in plant growth and potato production. Tuber induction and yield in potato crops is dependent on a number of factors among which temperature plays a major role in triggering the process (Jackson 1999). Most potato models describe tuber induction and development as a function of temperature (Ingram and McCloud 1984; Mackerron and Waister 1985). Earlier developed models such as the one developed by Sands et al. (1979) used the same functions of temperature for all growth and development phases, hence, did not give reliable estimates of potato emergence. In more improved models, such as, Light Interception and Utilization-Potato-Decision support system, LINTUL-Potato-DSS, emergence is estimated from a sprout growth rate of 0.7 mm with a base temperature of ‘zero’-degree C. The SUCROS model of Van Keulen et al. (1982) calculates crop growth from light extinction in the canopy, photosynthesis and respiration. Griffin et al. (1993) observed that the SUBSTOR-Potato v2.0 model performs poorly when simulating leaf area index (LAI) which potentially results in over-estimation of yield. Phenological development in potato and how the SUBSTOR-Potato incorporates these developmental stages into the modeling scheme were also given by Griffin et al. (1993). Five growth phases: (i) pre-planting, (ii) planting to sprout germination, (iii) sprout germination to emergence, (iv) emergence to tuber initiation, and (v) tuber initiation to maturity, are considered in SUBSTOR-Potato. All growth stages are affected by temperature: the temperature factor affects either vine growth or root and tuber growth. Tuber initiation in potato is a function of daylength and temperature and is also modified by plant N status and soil water deficit. SUBSTOR-Potato model assumes that tuber initiation by 'early' cultivars is less sensitive to high temperature and/or long daylengths than tuber initiation by 'late' cultivars (Ewing 1981; Wheeler and Tibbetts 1986). This effect is simulated in the model by a cultivar specific coefficient for critical temperature (TC), above which potato tuber initiation is inhibited. A higher TC (lower sensitivity to high temperature) corresponds to 'early' cultivars. The TC together with daily mean temperature determine the relative temperature factor for tuber initiation. SUBSTOR-Potato and EXpert-N-SPASS begin the calculation of the tuberization rate when emergence is reached and when photoperiod and temperature requirements are met. However, the final tuber yield is often subjected to plant disease or insect attacks. The actual stage models also consider biotic factors including weeds, diseases and pests that can significantly reduce potato production compared to the attainable and potential levels (Fig. 1). The decision support system TUBERPRO can be integrated to the model EPOVIR in which forecasts of tuber yields graded by size and the infection of the tubers by PVY (potato virus Y) and PLRV (potato leaf role virus) can be modeled (Nemecek et al. 1994). It also supports optimization of haulm killing (killing potato vine) dates in the seed potato production. The system calculates expected tuber yield and the probability that virus infection remains below the tolerance limit. All these models offer a wealth of information on different aspects of the crop, management and growing conditions.

The water-nutrient-environment limited stage models are best represented by models that take into account Soil nitrogen (N) supply. Nitrogen is an important nutrient that influences potato tuber growth, development, quality, and yield (Ojala et al. 1990). Potato cultivars have specific patterns of N uptake (Bélanger et al. 2001; Giletto and Echeverría 2015). For N uptake and potato biomass predictions, the DAISY model was successfully calibrated and used in Europe (Abrahamsen and Hansen 2000; Heidmann et al. 2008). Jiang et al. (2011) simulated N leaching under a potato crop with the LEACHN (Leaching and Chemistry for Nitrogen) model. In Atlantic Canada, combined leaching of N and potassium (K) under a potato crop was simulated by the LS-SVM model (Fortin et al. 2015). The model is very good at simulating N and K leaching, however lacks potato plant growth simulation. Timlin & Pachepsky (1997) evaluated the 2DSPUD model to predict potato yield and N leaching under different N fertilizer management and irrigation regimes. They reported a close relation between the simulated vs. measured potato yields, N uptake, and N leaching. Gayler et al. (2002) validated another potato model, SPASS (Soil Plant Atmosphere System Simulation), to simulate the N uptake of early and late potato varieties, their growth, tuber yields in Europe. They demonstrated that the modified SPASS model adequately described the N uptake and plant growth under different N fertilizer applications. Their study also showed that the model predicted N uptake and tuber yields corroborate quite well in comparison with the actual measured responses for both early and late potato varieties. The LINTUL POTATO model also simulates N uptake (Kooman and Haverkort 1995), while the modified version, POTATOS, accounts for the effects of an elevated CO2 concentration (Wolf 2002). LINTUL POTATO was later on adapted to LINTUL-POTATO-DSS, and was successfully used for simulating production environments, climate change analysis, and tuber size distribution (Haverkort et al. 2015). LINTUL-Potato-DSS calculates attainable yields under water-limiting conditions and also estimates soil water holding characteristics such as field water capacity and wilting point based on the soil silt and clay content. The SUBSTOR and SIMPOTATO models accept multiple variable interactions at both regional and field scales to calculate attainable yields (Fortin et al. 2010; Hoogenboom et al. 2012).

A few integrated water-nutrient-environment limited stage models can simultaneously simulate potato plant growth, biomass accumulation, tuber yield and soil processes. An example of model integration is the integration of Expert-N, a model for simulation of daily fluxes of water, carbon, and nitrogen, (Engel and Priesack 1993; Priesack et al. 1999; Stenger et al. 1999) with SPASS, a process-oriented model for simulation of crop growth and uptake processes. Hodges et al. (1992) developed SimPotato model to simulate soil processes and potato plant growth. Later, Han et al. (1995) interfaced SimPotato with a GIS (geographic information system) software to study potato yield and N leaching distributions. Their study showed that areas of high N leaching corresponded to areas with high water and N applications. Model outputs were used by Han et al. (1995) for guiding sprinkler irrigation uniformity to reduce N leaching and increase potato yields. Organic matter in soils have different pools and different decomposition rates, and their simulation also varies across models. For instance, LINTUL-NPOTATO (pools: crop, manure, and wastewater solids) and EXpert-N-SPASS (pools: litter, manure, and humus) have three different pools, whereas, DAISY model has two pools (pools: soil organic matter and added organic matter). The plant N and carbon (C) balances, as well as the phenology and growth of potato, are simulated by CSPotato model.

At the management stage, the potato models are used to simulate how different management practices in potato production can, on one hand improve tuber yield while on the other hand, affect the environment. Management practices used in potato production can have some critical environmental impacts which are particularly triggered during the growing season (Jiang et al. 2012; Kang et al. 2022). This includes emission of greenhouse gases and leaching of nutrients. During land preparation, it is the use of fertilizers particularly mineral fertilizers and ploughing that combined trigger greenhouse gas emission (Wichelns and Qadir 2015). During the growing season, the inappropriate or excessive use of chemical fertilizers, especially to supply nitrogen, phosphorous and potassium, increase environmental concern as a nonpoint source of water pollution. Crop modeling platforms like DSSAT offers the functionality to address these challenges while simulating leaching losses and heavy metal contamination caused by agricultural fertilizers. Process based models also have the potential to restrict nutrient use in agricultural systems, offering both potato producers and scientists tools to consider additional alternatives to improve nutrient use efficiency.

The potential stage models are signified in the LINTUL group of models. These models were the first deviation from the more complex, photosynthesis-based models of the “De Wit school” of crop modelling, also called the “Wageningen school” (Bouman et al. 1996). In the LINTUL models, total dry matter production is calculated in a comparatively simple way, using the Monteith approach (Monteith 1969; 1990). In this approach, crop growth is calculated as the product of interception of (solar) radiation by the canopy and a fixed light-use efficiency (LUE; Russell et al. 1989). This way of estimation reflects daily biomass production in the LINTUL models. For regional studies, LINTUL-type models have the advantage that data input requirements are drastically reduced and model parameterization is facilitated (Bouman et al. 1996). LINTUL model and its later generations, LINTUL-FAST (Angulo et al. 2013), LINTUL-NPOTATO (Van Delden et al. 2003) and LINTUL-POTATO (Kooman and Haverkort 1995), have been successfully applied to potato crop in potential water-limited, climate change or nitrogen-limited situations. Evaluation of the AquaCrop model by Gebremedhin et al. (2015) showed that the model well simulated tuber yield, total biomass and water use efficiency of potato. The model suggested early supplemental irrigation application after cessation of rainfall was enough to obtain good tuber yield and biomass. The most critical stage of physiological water stress of the crop to be supplied with deficit irrigation was preferable method, and then, the additional water used in full supplementary irrigation should be saved and invested in additional land productivity. Utset et al. (2000) determined the SWACROP root-water uptake function for potatoes on a Ferralsol in Havana, west Cuba. The function reveals the maximum pressure-heads under which water is optimally extracted by potato roots.

Potato Model Genesis and Developmental Interaction

The development of potato models in general follows the four stages mentioned above. However, this is not always the case. In Fig. 2, potato models are arranged according to their developmental period, exchange information and developmental linkage. With regards to the genesis of potato models, there are models developed for specific application in potato production but in most cases, potato models were adapted from models developed for other crops. In the adaption process, modelers often incorporate ideas, mechanisms or formulae from multiple sources, resulting significant developmental interactions. The evolution and biology of potato plant depend on the conditions of a relatively short growing season. The plant is particularly sensitive to the environmental cues associated with the coming of winter: cool temperatures and shortening day length. These cues are the primary factors controlling tuber formation and have therefore been extensively incorporated in model development (Lorenzen and Ewing 1990; Prange et al. 1990). Temperature variations and seasonality in day length can affect the biological development of the plant and consequently tuber development. Photosynthesis, dry matter production, partitioning of photosynthate away from leaves and toward tubers are dependent on these conditions (Reynolds et al. 1990; Midmore and Prange 1992). Nonetheless, potato contains considerable genetic variability and plasticity, providing a basis for modeling and simulation to optimize production with respect to adaptation to climate.

Fig. 2
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Potato plant growth models are arranged horizontally based on their developmental period (publication data). For the long-dash box, models having the same colour exchange information or have a developmental linkage; for the same colour, models in italics contribute to the development of models in non-italics; underlines are for information or module used by models having same colour. For the dotted box, models do not share a developmental linkage with another model. For the solid line box, all models share a developmental linkage and common modules

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Early crop modeling techniques using mathematical and statistical methods, have been applied to plant studies since the 1950s. Simplified modeling of potato crops started during the 1980s (Fishman et al. 1985; Mackerron and Waister 1985; Johnson et al. 1986), and the first potato model, Sands Model, was developed about five decades ago (Raymundo et al. 2014). In the 1990s, potato crop models were being used to link tuber yield with growth resources such as soil–water and soil-nutrients, using simulation routines (Fig. 1). Potato models vary in their functions, structures, description of involved processes and time scales (Table 2). Some of the well-studied ones are SUBSTOR-Potato, LINTUL-Potato, SOLANUM, APSIMPotato, SPUDSIM, POMOD, SIMPOTATO and Potato Calculator (Fleisher et al. 2017; Raymundo et al. 2014; Saue and Kadaja 2014). In a more recent time, the models were explored for systems analysis by exploring management options, including nitrogen fertilization, irrigation management, and the impacts of climate variability (MacKerron 2008). In many cases, this required an adjustment of the existing models, often crop models other than potato, which led to the development of new potato models (Fig. 2). Hence several potato models can share common data input requirements, routine modules and predictive features A good example of model adaptation is the LINTUL family of models, including LINTUL, LINTUL-FAST, LINTUL-NPOTATO and LINTUL-POTATO (Kooman and Haverkort 1995), which all have been widely applied to potato crop (Spitters 1990; Shibu et al. 2010).

Table 2 Data input requirements or simulation methods for the potato plant growth models
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It is in the 2000s when more potato models were developed and used for a range of applications. Just over a decade ago, potato modeling was first formally recognized as an advanced branch of potato science to traditional husbandry. In more recent years, researchers and scientists have developed quantitative methods, which attempt to account for conceptual models of potato growth. From the models listed in Table 2, 27 models can include a water routine either in a stand-alone runtime or being coupled with other routines. These routines are developed for estimating, predicting and managing water distribution and fluxes, at the soil-atmosphere interface, as a function of various parameters that are used for describing soil and watershed characteristics (Gioia et al. 2011; Manfreda et al. 2005). The commonly required inputs are atmospheric data (e.g., rainfall and temperature) while the model parameterization includes watershed characteristics like the topographic relief, geomorphology, bedrock, soil and vegetation properties (i.e.: models based on physical concepts). The model subroutines of the soil–water balance vary across models, as some of the early models did not include precipitation (Fishman et al. 1985). The main focus of the soil water balance calculation considers soil water dynamics which include evapotranspiration as input data. The water dynamics in the soil profile are usually simulated in one dimension by the tipping bucket approach or the Richards equation (Table 2). The SPUDSIM potato model can simulate soil–water dynamics in two dimensions (Fleisher et al. 2010). AquaCrop is a water-driven model that requires a relatively low number of parameters and input data to simulate the potato yield response to water (Steduto 2003). Wesseling et al. (1989) addressed the SWACROP model to potato which combined SWATRE (Belmans et al. 1983) and CROPR (Feddes 1982) models. Currently, there is a strong and growing overlap and synthesis between traditional potato cropping and potato crop modeling. At the same time, with the increase in the use of computerized techniques there has been a corresponding increase in the demand for quantitative potato information. Multiple sources of information are required for different stages and phases of potato models and at a variety of extents and resolutions.

With decades of development, there are many models available for simulating potato growth (Table 2). There are similarities between these models because similar underlying mechanisms or equations are used to simulate the same processes. The similarity and differences between different models are well reflected in the input data used in the model and the output data the model can produce (Table 2). For example, most models use leaf area index to estimate leaf canopy. This is a very basic method as it does take into account the vertical layout and structure of the leaves, which can affect the efficiency of how plant can use sun light. Several models (e.g., DAISY, DANUBIA and SPUDSIM) have made improvements on this and used various methods to account for the canopy layer structure (Table 2). An example of a widely used model is LINTUL which is suited for ideal growing conditions without any water, nutrient shortages and in absence of pests, diseases and adverse soil conditions. Later versions of LINTUL also made improvements to take into account water-limited (LINTUL2) and nitrogen-limited (LINTUL3) growing conditions. Another widely used yet exhaustive data driven model SUBSTOR-potato considers the growth-stimulating effect of elevated atmospheric CO2 and the impact of temperature on crop growth and development. However, when compared with a wide range of field data, the SUBSTOR-potato model can underestimate the impact of elevated atmospheric CO2 concentrations and also can fail to simulate high temperature effects on potato growth (Raymundo et al. 2017). Although, SUBSTOR has been adapted to take into account temperature and CO2 responses under changing climates (Raymundo et al. 2018). Moreover, not all models can simulate potential yield. From the models listed in Table 2, three models can simulate potential yield, 13 models include a water routine, and 14 models include water and nitrogen routines. The SPUDSIM is the only potato model that simulates soil–water dynamics in two dimensions.

Identifying Key Potato Models using CatPCA

With so many models to choose from and each has its strength and weakness, it is difficult for users to decide which model to use for their specific tasks. Besides, the variable range of data inputs required by each model (Table 2) can make the decision complicated. However, as discussed above, there are strong connections between these models. We hypothesized that these connections are reflected in the data summarized in Table 2 and based on the analysis of these data, we will be able to quantify the connections between these models thus identify models that can best represent the different aspects of potato modelling. The method we used for the data analysis was a variant of the classical principal component analysis, the categorical principal component analysis (CatPCA). The CatPCA in the domain of nominal data was applied to identify the most independent ("orthogonal") model. This procedure can simultaneously quantify categorical variables while reducing the dimensionality of the data. Generally, CatPCA assigns numerical quantifiers to each of the categories of qualitative variables, thereby allowing further analysis of the main components of the transformed variables (Meulman 1992; 1998). The numerical values assigned to each class of the original variables are defined by an interactive procedure, which is known as the alternative least squares method, such that the numerical quantifications possess standard metric properties (Maroco 2021). Thus, CatPCA also can be considered as a method to reduce the data size (Meulman 1992; 1998; Meulman et al. 2004). When dealing with a number of potato growth models, the CatPCA approach offers an opportunity to reduce an original set of models into a smaller set of uncorrelated components that represent most of the information found in the original models (Linting and van der Kooij 2012). Reducing the dimensionality could allow for an easy representation and interpretation of the models into a two-dimensional map indicating the most representative ones (Jöreskog and Moustaki 2001). These models are considered "key models" for the evaluation of potential potato management scenarios and predicting potato crop performance.

For the CatPCA of the potato models, we used the data presented in Table 2. The data input (ordinal variables) required for each model were grouped under each model, and then coded numerically on a scale from 1 to up to 9, depending on the level of data input category. For example, leaf canopy having two levels was coded using a scale of 1 to 2, light having three levels was coded using a scale of 1 to 3, and so on. As such, a data matrix of 33 models having 27 categories of data input fields was obtained. Further detail on this methodology can be found in Meulman (1992; 1998) and Meulman et al. (2004). Thus, the numerical values assigned to the original variables presented in Table 2 were subjected to CatPCA. Discretization and ranking of string variables were completed using SPSS v.19 (SPSS Inc., USA). For categorical data analysis using a multivariate principal component approach we followed the methodology described by Linting and van der Kooij (2012). We conducted the CatPCA on the correlation matrix and the selection of the retained factors was based on the explained variance by each principal component (PC or factor) and a plot of their eigenvalues (Meulman 1998). From each of the retained factors, a key variable was identified based on the largest factor loading after checking the communality for the retained factors. This multivariate analysis was performed with SPSS v.19 (SPSS Inc., USA). Such techniques as principal component analysis, factorial analysis, discriminant analysis, structural equation modeling, cluster analysis and others have been used in various research fields of biology, agronomy and medical sciences (Linting et al. 2007; Lattin et al. 2003; Devellis 2012). One of the merits of these techniques is the identification of specific and precise patterns of considerable complexity within a data set, transforming multidimensional information into two- or three-dimensional information (Manly 2005; Hair et al. 2006).

Table 3 displays the eigenvalues for 39 iterations of the analysis. These eigenvalues are useful to guide how many principal components should be retained. Overall, the result of CatPCA was excellent in explaining the dataset variability. The standard PCA solution, i.e., iteration 0, with all models treated as numeric, results in an eigenvalue of 27.70 while the CatPCA begins with an eigenvalue of 29.41 (iteration 1) and increases with each iteration. Eigenvalues were also used to determine the percentage of variance accounted for (a type of effect size) and therefore, larger eigenvalues were preferred over smaller ones. Because the ordinal nature of the models was taken into account rather than simply running a traditional PCA, a better solution was achieved at an eigenvalue of 31.89 at the 39th iteration. For all the principal components (PCs) as well as for each of the PCs, the percentage of variances accounted for was calculated using the eigenvalues. The first PC accounted for 71.43%, the second PC accounted for 13.98% and the third PC accounted for 11.22% of the variance in the optimally scaled matrix. The first three PCs combined accounted for 96.63% of the total variation whereas the first two PCs jointly explained 85.41% of the total variation. Thus, it was sufficient to retain only these three factors.

Table 3 The eigenvalues for each iteration of the CatPCA analysis for the 33 potato models
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When CatPCA was used to explore the relationships between the models, some differences were observed. The scree plot outlines the contribution of each PC and the “elbow” appearing at the third PC suggest that the first three PCs should be retained (Fig. 3). The communalities of the variables for the first three PCs (indicating the proportion of each variable's variance that can be explained by the principal components) are given in Table 4. Most of the variables had high communalities. For those models with mean loadings less than 0.5 on the centroid coordinate dimension (from SWACROP to SOLANUM), their total contributions to the PCs were optimized in the vector dimension (Table 4). The variance of some variables, such as the INFOCROP-POTATO, SUBSTOR-Potato, WOFOST and NPOTATO models, was almost entirely accounted by the first three PC's (communality = 0.95). The component loadings on a vector space dimension for each potato model on each dimension (the first, second and third PC) are given in Table 5. The first PC was dominantly associated with the three models, INFOCROP- POTATO, SUBSTOR-Potato and Potato Calculator with the highest loadings (0.977). The loadings of the other models: WOFOST and LINTUL- NPOTATO (0.975); and Exprt-N-SPASS (0.974) were also very strong on the first PC. Sands-model, Ingram- model and SOLANUM were strongest on the second PC and contributed equally with a loading of 0.857. The third PC was represented by only one model, ROTASK 1.0, with the loading on PC3 of 0.835.

Fig. 3
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Scree plot of the 3-component CatPCA solution. VAF—Variance Accounted For

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Table 4 Communalities of the Principal Components (PCs) of the CatPCA analysis showing the contribution of each potato model to the PCs
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Table 5 Component loadings, showing the coordinates for each potato model on each dimension of the CatPCA analysis
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The fact that these seven models (INFOCROP-POTATO, SUBSTOR-Potato, Potato Calculator, Sands-model, Ingram-model, SOLANUM and ROTASK 1.0) having high loadings indicates their close alignment along the three PC axis. They represent three clusters of models where variation within the clusters were minimized while that between any two cluster groups were maximized. Therefore, these models can be generally considered as key potato models. The component axis of the first PC combines the range of data input required for each model, hence focuses on the wider range of functionalities simulated by each model. The component axis of the second PC incorporates the developmental linkage and output focus of each model while the component axis of the third PC indicates information on climate and geographical origin of these models.

Selecting Key Potato Models for Rain-Fed Potato Production Systems in Atlantic Canada

Comparing the three models clustered around the first PC, the SUBSTOR-Potato model entails a much wider range of data inputs compared to the other six models. Hence offers the exceptional possibility to integrate a variety of information in the model output. Both, INFOCROP-POTATO (Singh et al. 2005) and Potato Calculator (Jamieson et al. 2004) lack the ability to utilize data from ‘soil available water in the root zone’, ‘relative humidity’ and Richards approach for ‘water dynamics’: all essential for rainfed potato production system in Canadian maritime and Atlantic provinces. Besides, INFOCROP-POTATO was originally developed for subtropical conditions while Potato Calculator is more specialized for anticipating future N requirement and scheduling N applications. Considering these, SUBSTOR-Potato clearly stands out as the leading model of the first PC.

For the three models clustered around the second PC (Sands-model, Ingram-model and SOLANUM), the Ingram-model can explain the temperature effects on the fraction of net crop dry matter assimilation partitioned to tubers by growth temperature responses of different potato crop components and simulate cultivar specific tuber yield response (Ingram and McCloud 1984). Sands-model is more suited for pattern of potato tuber bulking in Australian potato cultivation systems (Sands et al. 1979) while SOLANUM is more useful for potato germplasm diversity: analyzing and describing the growth characteristics and yield of different potato germplasm (Condori et al. 2010). Taking these facts into account, Ingram-model appears to be the most representative model in the second PC to be used for rain-fed potato production systems in AC.

For the third PC, the Rotask 1.0 model was the only model identified as the key model according to the CatPCA. The Rotask 1.0 model was originally developed for tillage systems under temperate conditions (Jongschaap 2006). The model can quantitatively evaluate crop rotation strategies, continuous potato cropping and its consequences can be specifically assessed through continuous simulation of agro-ecological processes such as crop growth, water and nutrient fluxes, which are common in temperate potato cropping and tillage systems of Atlantic Canada. Overall, based on the analyses and critical considerations we selected SUBSTOR-Potato (from PC1), Ingram- model (PC2) and Rotask 1.0 model (PC3) as key models for potato crop modeling in Atlantic Canada.

Conclusion

Potato models are important research and management tools. These models simulate potato growth under the effect of water stress, soil, weather, nutrient supply, genotype and many other factors. A vast body of knowledge about potato crop models exists, which has allowed the understanding and prediction of potato plant biology and production, as a function of environment and management goals. There are many appealing directions for potato model use. In this study, a statistical technique, CatPCA, was used to identify key potato models that are relevant for the rain-fed potato production systems in Atlantic Canada. Three models, SUBSTOR-Potato, Ingram- model and Rotask 1.0, were identified based on the CatPCA as well as the underlying characteristics of the models. Most of the reviewed models have received field evaluation, but none of them have been tested under conditions of climate change in AC. The authors see a clear opportunity for future work to evaluate the usefulness of the key models using historical climate data and future climate change scenarios .