Good‐bye to tropical alpine plant giants under warmer climates? Loss of range and genetic diversity in Lobelia rhynchopetalum

Abstract The main aim of this paper is to address consequences of climate warming on loss of habitat and genetic diversity in the enigmatic tropical alpine giant rosette plants using the Ethiopian endemic Lobelia rhynchopetalum as a model. We modeled the habitat suitability of L. rhynchopetalum and assessed how its range is affected under two climate models and four emission scenarios. We used three statistical algorithms calibrated to represent two different complexity levels of the response. We analyzed genetic diversity using amplified fragment length polymorphisms and assessed the impact of the projected range loss. Under all model and scenario combinations and consistent across algorithms and complexity levels, this afro‐alpine flagship species faces massive range reduction. Only 3.4% of its habitat seems to remain suitable on average by 2,080, resulting in loss of 82% (CI 75%–87%) of its genetic diversity. The remaining suitable habitat is projected to be fragmented among and reduced to four mountain peaks, further deteriorating the probability of long‐term sustainability of viable populations. Because of the similar morphological and physiological traits developed through convergent evolution by tropical alpine giant rosette plants in response to diurnal freeze‐thaw cycles, they most likely respond to climate change in a similar way as our study species. We conclude that specialized high‐alpine giant rosette plants, such as L. rhynchopetalum, are likely to face very high risk of extinction following climate warming.

two-to threefolds compared to that recorded during the 20th century (Nogués-Bravo, Araújo, Errea, & Martinez-Rica, 2007). As mountainous areas present steep biotic and abiotic gradients, such a dramatic change is expected to have a pronounced impact on biodiversity, ecosystems, and the services they provide (Price, 2000).
Upward shifts in response to recent climate change are already measurable, although they tend to lag behind the velocity of climate change (Bertrand et al., 2011). Several historical surveys using permanent plots have shown that species with high thermal demands are invading higher zones, whereas species with lower thermal requirements are retreating to even higher altitudes (Keller et al., 2000;Lenoir et al., 2008;Pauli et al., 2012). A study of 26 peaks exceeding 3,000 m in Western Austria and Eastern Switzerland found up to 70% increase in species richness since the beginning of the 20th century due to immigration of lower-elevation species (Grabherr et al., 1994). A comparison of the altitudinal distributions of 171 forest species between 1905 and 2005 showed, on the average, 29 m upward shift of the species' distribution centers per decade (Lenoir et al., 2008). Nearly the same average upward shift rate (27 m per decade) was reported for 125 species endemic to the Himalaya Mountains (Telwala, Brook, Manish, & Pandit, 2013). An analysis of a long-term data set on the upper limits of plants from the beginning of the 20th century for 25 mountains in the European Alps showed an upward shift exceeding 100 m for most species, with an average of about 10 m per decade, and 33 species were recorded as new to the summits (Frei, Bodin, & Walther, 2010). The nival species of Swiss and Austrian peaks were reported to migrate 5 m upwards per decade (Grabherr et al., 1994).
There are also conceptual (Lenoir et al., 2010) and observationbased reports (Lenoir et al., 2008;Pauli et al., 2012) indicating that a considerable proportion of species does not follow this general trend, with many species not moving, or even moving their centers of elevation distribution downslope. Habitat modification, changes in rain and snowfall regimes, and changing competitive interactions among species are thought to explain these downward range shifts (Lenoir et al., 2010). In Mediterranean mountains, species richness has been found to decrease probably due to decreased moisture availability (Pauli et al., 2012). This illustrates that the effects of changing temperatures are not always easy to project, as climate change may have cascading effects on abiotic and biotic factors.
One of the most important features characterizing tropical alpine habitats is the lack of pronounced seasonal variation in temperature.
Instead, the diurnal variation in temperature is extremely high with warm days alternating with very cold nights, when temperatures can drop well below 0°C. Tropical alpine habitats are therefore unique, and they are famous for their conspicuous giant rosette plants, which show intricate adaptations to such extreme diurnal temperature variations (Fetene, Gashaw, Nauke, & Beck, 1998;Halloy, 1983;Hedberg, 1964;Nagy & Grabherr, 2009). Because of prolonged cold stress and snow accumulation in winter, temperate alpine habitats do not sustain plants of such large life forms, while the highest mountains in tropical Africa are renowned for their many, and often vicariant, species of giant groundsels (Dendrosenecio) and giant lobelias (Lobelia; Hedberg, 1964).
Although many giant rosette plants are only distantly related phylogenetically, they have developed similar distinctive morphological, physiological, and life history adaptations to the diurnal freeze-thaw cycles of tropical alpine habitats (Halloy, 1983). These complex adaptations include the following: (i) retaining old leaves as a shelter, (ii) accumulating water within body parts to minimize thermal shocks, and (iii) having large rosette leaves which fold during the freezing nights through nyctinastic leaf movements to protect the delicate buds. They also share some seemingly paradoxical morphological and physiological traits. In tropical as well as other mountains, the height of plants normally decreases with increasing altitude (Smith, 1980), while giant rosette plants have evolved a noticeable exception to this general pattern as one of their adaptation mechanisms (Fetene et al., 1998;Meinzer, Goldstein, & Rundel, 1985;Smith, 1980;Smith & Young, 1987). The temperature near the ground is very low during night and highly fluctuating throughout the day as compared to the temperature more than 1 m above the ground (Fetene et al., 1998). The main advantage gained by a tall life form is to escape the strong diurnal temperature fluctuation near the ground and to avoid freezing of apical meristems (Fetene et al., 1998). As the freeze-thaw cycles aggravate with increasing altitude, giant rosette plants grow taller at higher altitudes in order to keep the apical meristem farther away from the ground where freezing is more severe. Taller plants have faster growth rate, implying that their growth rate increases with age, representing yet another paradox. In addition, the amount of water stored in their pith and leaves also increases with altitude (Smith & Young, 1987).
In this study, we assess the risk of losing habitat and genetic diversity in the iconic giant rosette plant Lobelia rhynchopetalum Hemsl. due to climate change in the Ethiopian mountains, which represent the largest contiguous high-mountain system of Africa. Under the current climate, this species is widespread and reaches the highest peaks in the country, such as in the Simen and Bale Mountains and in Mount Choke, and it can therefore be expected to lose its range under future warmer climates. Lobelia rhynchopetalum is endemic to the Ethiopian mountains and the only representative of the giant rosette life form in this high-alpine area. While most other plants rarely exceed 0.5 m in height in the alpine zone of Ethiopia, L. rhynchopetalum can grow taller than 10 m (Fetene et al., 1998). It is confined to the alpine zone and thus serves as a conspicuous indicator species of the tropical alpine habitat. In addition, it is a tourist attraction and it has thus not only ecological importance but also recreational and economic value (Geleta & Bryngelsson, 2009).
As L. rhynchopetalum has developed adaptations to the high-alpine tropical climate similar to those of other tropical alpine giant rosette plants through convergent evolution, it can be considered as representative of other giant rosette plants with respect to vulnerability to climate change. Here, we project the current distribution of L. rhynchopetalum and evaluate how its habitat range may respond to different future climate scenarios. We do so by using an ensemble approach using three different statistical methods, and by fitting both a simple and a complex model parameterization (Merow et al., 2014) for each model for building the ensemble. We also assessed its current genetic diversity by genotyping samples from three main mountain systems across the Ethiopian Highlands (Simen, Bale, and Choke) using genome-wide amplified fragment length polymorphism (AFLP) markers. In particular, we estimate to what degree genetic diversity is geographically structured in these archipelago-like African high mountains and predict loss of genetic diversity due to climate-driven range loss.

| Study species and distribution data
Lobelia (Lobeliaceae) comprises ca. 300 mainly tropical/subtropical species, of which 18 occur in Ethiopia (Thulin, 2006). The exclusively alpine L. rhynchopetalum (ca. 3,400-4,500 m) is characterized by a single erect stem growing to 10 m or more in height, by entire, smooth, glossy, lanceolate to oblanceolate rosette leaves up to 80 cm long and 12 cm wide, and by a single, dense, and several meters long terminal inflorescence.
For sampling, we selected three Ethiopian alpine mountain sys-  (Table S1). For the area below the treeline, where L. rhynchopetalum is known to be absent, we generated additional, random pseudo-absence points in a number proportional to the area-weighted observation density made above the treeline. These two data sets were combined and used for model training. In total, we sampled 218 plots (87 presences), and we added 612 pseudo-absences from below the treeline. For model evaluation, we used an independent presenceabsence data set obtained as part of separate field campaigns in the same three mountain systems (for collection of samples for the genetic diversity study, see below). This data set contained 26 presence and seven absence points to which we added 113 pseudo-absence points below the treeline using the same area-based weighting.

| Predictor variables
Averages of monthly precipitation sum and minimum and maximum monthly temperature at 1 km spatial resolution of both current and future climate using different scenarios were obtained from the global climate data set WorldClim (Hijmans, Cameron, Parra, Jones, & Jarvis, 2005). In order to better characterize the complex mountain topography on the basis of these climate maps, we downscaled the temperature variables to 270-m spatial resolution following the procedure of Zimmermann, Edwards, Moisen, Frescino, and Blackard (2007), and we interpolated the precipitation layers to the same resolution. We computed 19 bioclimatic variables according to Hijmans et al. (2005), resampled the global 90 m resolution SRTM DEM (Jarvis, Reuter, Nelson, & Guevara, 2008) to 270 m resolution, and derived maps of slope angle, slope aspect, and topographic position index (TPI; Jenness, 2006). We used eight categorical classes for aspect (N, NE, E, SE, S, SW, W, NW). To obtain the appropriate scale at which TPI explained the variation in the data set, we produced six different TPI maps (tpi3-tpi8) by setting the neighbor shape to rectangle and by varying the number of neighborhood cells from 3 × 3 (for tpi3) to 8 × 8 (for tpi8) in ArcGIS using Land Facet Corridor Tools (Jenness, Brost, & Beier, 2013) and compared their performance in MaxEnt (Phillips, Anderson, & Schapire, 2006). We used GlobCover (Arino et al., 2012) by clipping it to the study area boundary and resampling it to 270 m using nearest neighbor method to represent a categorical landcover variable. Finally, we overlaid these rasters with the presence and absence points of both plot data sets. For further analyses, we first calculated Pearson's correlations between predictor variables and retained only one of those with correlations ⋝0.7, selecting the one with highest assumed biological importance (Fig.   S1). We finally kept ten variables for model building: slope angle, slope aspect, TPI7, landcover, mean diurnal range of temperature (BIO2), maximum temperature of the warmest month (BIO5), annual temperature range (BIO7), annual precipitation (BIO12), precipitation of the warmest month (BIO13), and precipitation of the coldest quarter (BIO19).
For each model algorithm, we implemented two model versions, one generating simple and one generating complex response shapes. We finally built ensemble maps of current and projected future ranges of the species. The three model algorithms differ in how they relate a response to predictors, and the two model complexity levels differ in the degree to which they fit the response to the data (Merow et al., 2014).
In all three models, we weighted the presences and absences inversely proportional to their fraction, because we had ca. 3× more absences than presences available in both the training and evaluation data set.
For GAM, we used the mgcv package (Wood, 2011). In order to generate shapes of varying complexity, we set the k parameter in the spline smoother to k = 3 for simple and to k = 8 (i.e., the largest possible allowed by the model) for the complex model. For both model complexity settings, a manual predictor variable selection was carried out by stepwise dropping the least significant variable until the two consecutive models revealed a significant difference in the variation they explained at a 95% confidence level in an ANOVA chisquare test.
For GBM, the simple response was generated by setting the tree complexity parameter to two in the "gbm.step" function of the DISMO package (Hijmans, Phillips, Leathwick, & Elith, 2013) and "n.drops" was set to "auto" from the "gbm.simplify" function in the gbm package (Ridgeway, 2013). No interaction among variables is thus allowed in this setting. For the complex response, we set the tree complexity to five and "n.drops" to zero, which allowed for significant (and more

| Genetic analyses
Leaf material of L. rhynchopetalum was obtained from the same three mountain systems (Figure 1a,b) and dried in silica gel. Leaves from five individual plants were collected within a 100 m × 100 m plot to represent a local population. A total of 102 individuals from 21 populations were successfully analyzed for AFLPs (see Appendix S1 for details).
To quantify genetic diversity, we estimated the proportion of polymorphic markers (P), Nei's gene diversity (D), and genetic rarity (Ra) using AFLPdat (Ehrich, 2006). We also counted the number of AFLP markers that were private to each mountain system and the number of markers that were shared among mountain systems. Genetic structure was assessed using principal coordinate analysis (PCoA) based on Dice's similarity coefficient and a Bayesian model-based clustering method. Levels of genetic differentiation were estimated in an analysis of molecular variance (AMOVA; Appendix S1).
We quantified the expected loss of genetic diversity due to range loss using a similar approach as Alsos et al. (2012). As genetic data were not available from the whole range of the species, in particular not from some of the areas that remained suitable under projected climate change (e.g., Mt Guna, Shewa-Wallo), loss of genetic diversity was estimated by randomly removing a proportion of sampling localities proportional to the area projected to be lost. Genetic diversity in the remaining populations was expressed as the number of remaining markers, as this diversity measure is more sensitive to rapid decrease in population size than Nei's gene diversity (El Mousadi & Petit, 1996).
Assuming that a complete loss of gene diversity ( confidence intervals (CI). As genetic data were not available from the whole range of the species, in particular not from some of the areas that remained suitable under projected climate change (e.g., Mt Guna, Shewa-Wallo), loss of genetic diversity was estimated by randomly removing a proportion of sampling localities proportional to the area projected to be lost. We thus assume that the genetic loss predicted in response to the area loss is proportional to the loss that can be expected for range-wide genetic diversity. To strengthen our assumption, we have also repeated this analysis by constraining the surviving population to different mountains, so that the loss is additionally distributed evenly among mountains.

| Habitat suitability modeling
All the three statistical models generated similarly high AUC values against the independent data set (0.945-0.973; Table S2). GBM performed best at both complexity levels followed by MaxEnt and GAM.
The AUC difference between complexity levels was low. Interestingly, the simple models performed slightly better than the more complex ones when applied to the independent test data set. The predicted distribution patterns obtained from all models and both complexity levels were also quite similar (Fig. S2). However, the binary presence-absence maps produced with different methods for splitting the probability predictions generated quite different results, with the PT producing much smaller suitable habitats and thus contributing a considerable amount of uncertainty to the map products (Figs. S3-S8).
The binary ensemble maps for each method and complexity level S10-S13). The climate model HadCM3 projects higher warming and thus constrains the suitable habitat under future climate scenario even more severely (Table S3).
Overlaying projected future with current habitat suitability maps revealed that no habitat remained suitable with certainty in two of our study areas, Bale Mts and Mt. Choke (Figure 3,  (Figure 3). Another 13.9% of the current range would present uncertain habitat suitability due to differences among climate and statistical models, complexity levels and thresholds (Table 1), which translated to a total of 82.6%-96.6% range loss for L. rhynchopetalum. Currently, L. rhynchopetalum was predicted to occur between 3,300 m and the highest peaks (4,500 m; Fig. S14 (Fig. S15). The PCoA revealed a continuous gradient in genetic variation ranging from Bale via Choke to Simen plants along the first axis, which explained only 8.7% of the variation (Figure 1d). The Bayesian clustering analysis suggested a similar structure with two genetic groups: one contained all plants from Bale and the other contained all plants from Simen, whereas both genetic groups were represented in Choke (Figure 1c, S16). In the AMOVA, most (80%) of the variation was found within populations, but significant albeit moderate variation was also found among mountain systems (12%) and among populations within mountain systems (8%; Table S4).

Removal of 16-20 of the 21 analyzed populations in 200 random
repeats, corresponding to a range loss of 76%-95%, resulted in loss of 37.4%-81.6% of the AFLP markers ( Figure 4). The predicted loss of genetic diversity was largely proportional to the range reduction and decreased rapidly with higher numbers of surviving populations.
As our range modeling projected that L. rhynchopetalum may survive in four different mountain systems, the reduction in genetic diversity was also estimated forcing remaining populations to be on different mountain systems. These results were almost identical to those from the random resampling ( Figure 4; Table S6-S7).

| Impact of climate change on habitat range
In this study, we predicted that almost all Ethiopian alpine areas are suitable for L. rhynchopetalum under the current climate. This is consistent with the Flora of Ethiopia and Eritrea (Thulin, 2006), where the species is reported from most floristic regions in Ethiopia. Under warmer climates, however, we found that this enigmatic alpine giant will suffer massive reduction in range. Today, the species has its largest proportion of suitable habitat south of the Rift Valley, that is, in the Bale-Arsi massifs. Under all emission scenarios and model algorithms considered, we predicted that no habitat will remain suitable with high certainty in any of the mountain ranges south of the Rift by the end of the 21st century, indicating very high risk of local extinction. Furthermore, no habitat in Mt Choke is projected to remain suitable with high certainty. Under future climates, we found that suitable habitat for L. rhynchopetalum will be confined to only four mountaintops in two areas northwest of the Rift Valley, the Simen Mts and the Shewa-Wallo-Guna massif. Thus, L. rhynchopetalum will lose its range at low latitudes as well as low altitudes.
Such large range reductions are typical of mountain systems that provide little room for upward movement to higher altitudes. In a study of climate change impacts on European mountain plants, similar range losses as we found here were projected for the Pyrenees and the far eastern part of the European Alps, while species in most other systems, such as the main parts of the Alps, the Carpathians, and the Scottish and Scandinavian mountain ranges are projected to be less threatened by climate change (Engler et al., 2011). Also, projected range loss in lowlands is usually much smaller than that projected for mountaintops that barely reach above treeline (Lenoir et al., 2008;Thomas et al., 2004Thomas et al., , 2006Thuiller et al., 2005). Tropical mountains are specifically threatened to such strong range reduction effects due to both the comparably strong fragmentation of the disconnected mountains and the comparably little reach above treelines in most of these mountains, which represent islands in a sea of lowland tropical vegetation.
Our results for L. rhynchopetalum add to a growing body of evidence showing dramatically high rates of range loss and altitudinal shifts in mountain plants (e.g., 27 m in altitude per decade for T A B L E 2 Genetic (AFLP) diversity in Lobelia rhynchopetalum estimated in total per mountain system and as average of population estimates   Telwala et al., 2013). Such high rates can be expected because of the enhanced changes anticipated in climates in mountainous regions (Nogués-Bravo et al., 2007), typically narrow habitat ranges and low genetic diversity that may counteract local adaptation in high-alpine taxa, and steep biotic and abiotic gradients in alpine areas. In L. rhynchopetalum, the habitat predicted to remain suitable with high certainty is very small in spatial extent, only 936.4 km 2 . Furthermore, this small area seems to be fragmented among four mountaintops, which may be too small to sustain viable populations ( Figure 3). Thus, we might lose these high-alpine, charismatic flagship species that render the tropical alpine areas so peculiar.

| Impact of climate change on genetic diversity
Tropical African alpine habitats represent highly fragmented ecosystems being restricted to the mountain peaks that rise above 3,500 m resulting in geographic isolation among populations (Hedberg, 1951).
In L. rhynchopetalum, we found significant albeit moderate geographic structuring in its genetic diversity, corresponding to the three major mountain systems analyzed. This is consistent with the statistically significant variation among populations of L. rhychopetalum from Simen and Bale reported based on sequences of the nuclear ITS region and eight plastid DNA regions (Geleta & Bryngelsson, 2009). In our study, we found considerable numbers of AFLP markers confined to a single mountain system. Only 36.4% of the markers were shared among the three mountain systems, and 23.9% of the total markers were absent from Simen Mts, where we expect the largest patch of suitable habitat to remain under future climates (Fig. S15). Our results clearly show the adverse genetic consequences of habitat loss from the two sampling areas in particular and in other geographic regions in general.
The overall projected habitat loss ranges from 82.6 to 96.6%, with only 3.4% of the habitat being projected to remain suitable with high certainty. Randomly retaining sample populations equivalent in number with the habitat that remains suitable with high certainty and re-computing the genetic diversity indicated an average loss of 82% of the AFLP markers, which is almost as high as the range loss ( Figure 4). However, range loss due to climate change is not random. It mainly occurs at the organisms' lower elevational and latitudinal range edges. Although selection pressures may vary in different parts of the range, we gave equal weight to all loci as typical in studies relying on AFLP markers, of which the majority represents neutral genetic diversity (Alsos et al., 2012). Range loss from different areas is likely to have a similar impact on genetic diversity when genetic diversity is distributed evenly across geographic regions (Alsos et al., 2012). We found similar levels of genetic diversity in L. rhyncopetalum populations from different mountains, and constraining the surviving populations to different mountaintops resulted only in small differences in genetic diversity loss compared to randomly sampling the surviving populations.
In a compilation of 307 studies of plant species, Nybom (2004) found an average AFLP-based genetic diversity of 0.23 ± 0.08, with highest diversity in long-lived perennials (0.25), outcrossers (0.27), and late successional taxa (0.30), and lowest diversity in annuals (0.13), selfers (0.12), and early successional taxa (0.17). We found very little total gene diversity in L. rhynchopetalum (0.12), even if this species is a long-lived, perennial and outcrosser (Hedberg, 1964). Very little genetic diversity has also been found in other afroalpine species, possibly due to strong fragmentation and bottlenecks during climatically unfavorable periods Masao et al., 2013;Wondimu et al., 2014), rendering them particularly vulnerable to further climate change. A meta-analysis of the impact of habitat fragmentation revealed a substantial decrease in expected heterozygosity, number of alleles, and percentage of polymorphic loci without substantial variation among organisms with different life history traits (Vranckx, Jacquemyn, Muys, & Honnay, 2012). For L. rhynchopetalum, it seems therefore likely that further fragmentation into very small areas under future climates will accelerate decrease in its genetic diversity and possibility for survival. Species which are expected to lose 80% of their populations within 100 years are proposed to be listed as critically endangered by the IUCN (IUCN, 2001). Thus, inferred from habitat loss alone, climate change will seriously threaten the survival of L. rhynchopetalum. In addition, mountain species are expected to suffer more from climate-induced range reductions (Frei et al., 2010;Lenoir et al., 2008;Telwala et al., 2013;Thuiller et al., 2005) and the associated loss in genetic diversity (Alsos et al., 2012) than species from lower and warmer regions.

| Implications from model calibration and sampling design
We found that the different statistical models and the different complexity levels for calibrating the models revealed very similar results. This is unanticipated, as previous studies have shown that the choice of statistical models generates high uncertainty in projected future ranges of suitable habitats (Buisson, Thuiller, Casajus, Lek, & Grenouillet, 2010). In addition, our tests against independent data reveal that the more complex parameterization schemes did not improve, but rather deteriorate the model accuracies. This is in line with the results of Randin et al. (2006), who also found the simpler GLM models to outperform GAM models when transferred between Eastern Austria and Western Switzerland. Merow et al. (2014) suggest using simple modeling schemes for data sets that are small and lack sound statistical designs. We believe that our data were well sampled and larger than minimal and thus would allow for more complex model calibration. We restricted our study area to highlands (>2,700 m), thus resulting in a relatively small area. L. rhynchopetalum is currently ubiquitous across Ethiopian alpine areas (Fetene et al., 1998;Thulin, 2006) and seems to be in equilibrium with its niche requirements. The sampling design of presence and absence observations covers the most important ecological gradients, which are latitude, altitudes, and aspect. This may also have led to few false absences and little noise in the data set which in turn potentially minimized the differences in performance among models and model complexity levels. This may show how sampling strategy may be more important than selections of model algorithms and model settings, especially for species that appear to be in equilibrium with their niche requirements.
Although the different statistical models and the two different complexity levels for calibrating these models produced very similar results under current climate, the different cut thresholds to produce binary presence-absence predictions added considerable uncertainties to the predictions. Notably, the PT was quite different and mapped a much smaller area as suitable than the other two methods. Using the prevalence of the data used for model building as a threshold is not a common practice but it considered as one of the most robust approaches (Liu, Berry, Dawson, & Pearson, 2005). Although it is recommended to be used for data sets in which the presence-absence data are systematically collected (Halvorsen, 2013;Liu et al., 2005), it is still considered useful in other data sets (Liu et al., 2005). The effect of this model step for analyzing projected gain and loss in suitable habitat is not well studied to date and requires further exploration.
Relying on different algorithms for modeling and combining uncertainty sources in an ensemble map is becoming a common practice (Buisson et al., 2010). Here, we produced multiple maps not only by combining different algorithms of different statistical properties but also models of different complexity levels and their binary maps produced using different threshold criteria, which means we have further enhanced the sources of uncertainties. Increasing uncertainty sources by fitting models of different complexity levels and using more than one threshold criterion per model has not been common practice so far. Although it adds more uncertainty sources, the consensus in predicting suitable habitat among these multiple maps gives more confidence for management and conservation practices, especially in delineating priority areas for conservation purposes and reintroductions of endangered species. It also enhances reliability of model results.