Ecology

Knowing the distribution of species is crucial for effective conservation action. However, accurate and high-resolution spatial data are only available for a limited number of species1,2. For mammals and birds, the most comprehensive and widely used global distribution dataset is the set of range maps compiled as part of the assessments for the International Union for Conservation of Nature (IUCN) Red List. These represent each species’ distributional limits and tend to minimize omission errors (i.e. false absences) at the expense of commission errors (i.e. false presences)3,4. Therefore, they often contain sizeable areas not regularly occupied by the species.Maps of the Area of Habitat (AOH; previously known as Extent of Suitable Habitat, ESH) complement range maps by indicating potential occupancy within the range, thereby reducing commission errors5. AOH is defined as ‘the habitat available to a species, that is, habitat within its range’5. These models are produced by subtracting areas unsuitable for the species within their range, using information on each species’ associations with habitat and elevation5,6,7,8. Comprehensive sets of AOH maps have been produced in the past for mammals6 and amphibians7, as well as subsets of birds8,9. The percentage of a species’ range covered by the AOH varies depending on the methodology used to associate species to their habitats, and their habitats to land-cover, the coarseness of the range map, the region in which the species is distributed, and the species’ habitat specialization and elevation limits5. For example, Rondinini et al.6 found that, when considering elevation and land cover features for terrestrial mammals, the AOH comprised, on average, 55% of the range. Ficetola et al.7 obtained a similar percentage when analyzing amphibians (55% for forest species, 42% for open habitat species and 61% for habitat generalists). Beresford et al.8 found that AOH covered a mean of 27.6% of the range maps of 157 threatened African bird species. In 2019, Brooks et al.5 proposed a formal definition and standardized methodology to produce AOH, limiting the inputs to habitat preferences, elevation limits, and geographical range.AOH production requires knowledge of which habitat types a species occurs in and their location within the range1. Information on habitat preference is documented for each species assessed in the IUCN Red List10, following the IUCN Habitats Classification Scheme11. However, the IUCN does not define habitat classes in a spatially explicit way, therefore, we used a recently published translation table that associates IUCN Habitat Classification Scheme classes with land cover classes12. Species’ elevation limits were also extracted from the IUCN Red List.We developed AOH maps for 5,481 terrestrial mammal species and 10,651 terrestrial bird species (Fig. 1). For 1,816 bird species defined by BirdLife International as migratory, we developed separate AOH maps, for the resident, breeding, and non-breeding ranges, according to the migratory distribution of the species (Fig. 2). The maps are presented in a regular latitude/longitude grid with an approximate 100 m resolution at the equator. On average, the AOH covers 66 ± 28% of the geographical range for mammals and 64 ± 27% for birds. We used the resulting AOH maps to produce four global species richness layers for: mammals, birds, globally threatened mammals and globally threatened birds13 (Fig. 3).Fig. 1Spatial distribution maps of Tangara abbas. Maps represent (a) the geographic range21, and (b) the Area of Habitat (AOH) of the species. The AOH was produced by subtracting unsuitable habitats from the geographical range. This species’ habitats are forest and terrestrial artificial habitats and has elevation range of 0 – 1600 m.Full size imageFig. 2Spatial distribution maps of Cardellina rubrifrons, divided into resident, breeding and non-breeding areas for this migratory species. Maps represent (a) the geographic range21, and (b) the Area of Habitat (AOH) of the species. The AOH was produced by subtracting unsuitable habitats from the ranges. This species is a forest species with elevation rangelimits of 1500 – 3100 m.Full size imageFig. 3Global species richness maps for (a) terrestrial mammals (considering 5,481species) and (b) terrestrial birds (considering 10,651 species). Calculated by overlaying all species’ AOH per class, resulting inon the number of species at each grid cell, latitude/longitude grid at a resolution of 1°/1008 or approximately 100 m at the equator (EPSG:4326) with the ellipsoid WGS 1984.Full size imageThe AOH maps presented in this paper are more useful for some purposes than global species distribution models, as they reduce and standardize commissions14. They are especially useful for not well-known and wide-range species. However, we note that for well-known species alternative sources may have more accurate distributions15. Moreover, AOHs are affected by the bias and errors of the underlying data, especially relevant errors associated with documentation of species’ habitats and elevations, and the translation of habitats into land cover classes, given that habitat is a complex multidimensional concept that is challenging to match to land-cover classes12, and that the current version of the IUCN Habitat Classification Scheme on IUCN’s website is described as a draft version11.The AOH maps have multiple conservation applications5,16,17, such as assessing species’ distributions and extinction risk, improving the accuracy of conservation planning, monitoring habitat loss and fragmentation, and guiding conservation actions. AOH has been proposed as an additional spatial metric to be documented in the Red List5, and is used for the identification of Key Biodiversity Areas18. More

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Surface morphology analysisSEM images were shown in Fig. 1. It showed that the contour of three soils were fairly clear before adsorption. But it became fuzzier and the degree of cementation was increased when phenanthrene was adsorbed on the soils. According to the surface morphology, the silty sand (A) had furrows on the surface before adsorption compared with the fairly smooth without any furrows after adsorption (B). The silts (C) were flaky and the lamellar accumulation decreased (D). The loess (E) had a smooth surface with some flaky and rod like structure, after adsorption (F), the surface of loess increased in clay-like structure.Figure 1SEM micrographs of the three soil samples. (A) Silty sand; (B) Adsorbing 5 h of Silty sand; (C) Silts; (D) Adsorbing 5 h of Silts; (E) Loess; (F) Adsorbing 5 h of Loess.Full size imageAdsorption and desorption experimentsAdsorption and desorption kineticsAdsorption kinetics is one of the most important characteristics governing solute uptake rate and represents adsorption efficiency33. The sorption and desorption kinetics of phenanthrene in three soils were shown in Fig. 2. The results showed that the adsorption processes among all soils were similar. The kinetics of phenanthrene in soils was completed in two steps: a “fast” adsorption and a “slow” adsorption. The adsorption amount increased during 0-18h. It was a rapid reaction from 0 to 200 minutes. From 200 to 600 minutes, the adsorption amount increased slightly into balance. This phenomenon was due to the adsorption of phenanthrene occurred on the surface of soil organic matter. With the increase of time, soil surface adsorption sites were gradually saturated, causing the decrease of adsorption rate until reaching the equilibrium. Phenanthrene was a hydrophobic substance. It was easy to reach the soil surface and adhere to the grain surface. The results were consistent with the study of had also found that the balance time was approximately 18h and the adsorption amount increased with the adsorption reaction time34. Under the same conditions, loess had the highest adsorption capacity, which was mainly due to the highest organic content 18. The maximum phenanthrene sorption capacities ranked as follows: loess > silty sand > silts. As shown in Fig. 2, phenanthrene desorption in soils was relatively quick and its desorption equilibrium time was 3h. To reach an adequate desorption balance while remaining consistent with the adsorption reaction time, the balance time of the adsorption–desorption experiment was set at 18h. Generally, PAHs below 4 cycles could reach the adsorption equilibrium for about 16~24h.Figure 2(a)Adsorption equilibration curves of phenanthrene sorption in soils. (b) Desorption equilibration curves of phenanthrene sorption in soils.Full size imagePseudo-second-order and Elovich models were used to study the phenanthrene adsorption mechanism (Table 3). Phenanthrene sorption kinetics were satisfactorily described by a pseudo-second-order model with coefficients of determination (R2) ranging from 0.99875 to 0.99847, compared with R2 values of 0.26508–0.73901 for the Elovich model. This well-fitting pseudo-second-order model indicated that the rate-limiting step was chemical adsorption, including electronic forces through sharing or exchange of electrons35,36. Moreover, it suggested that sorption was governed by the availability of sorption sites on the soil surfaces instead of by the phenanthrene concentration in solution.Table 3 Constants and coeffients of determination of Pseudo-second-order kinetics and Elovich models of sorption.Full size tableAdsorption and desorption isothermsThe isotherm was used for quantitative analysis of phenanthrene transport from liquid to solid phase and for understanding the nature of interactions between phenanthrene and the soil matrix. The sorption and desorption isotherms of phenanthrene in soils were shown in Fig. 3. The data showed that phenanthrene adsorption and desorption capacities of three soils varied markedly due to their different physicochemical properties. With the increase of phenanthrene concentration, the adsorbed amount increased. At the same temperature, the adsorption capacity of silty sand was minimum while loess was maximum. This is mainly related to the soil physicochemical properties. At the same initial concentration, the temperature increase from 20 °C to 40 °C showed that the adsorption and desorption capacity decreased with temperature increase. On the one hand, the rise of temperature can increase the phenanthrene solubility in the liquid phase. On the other hand, it could reduce various forces between the soil surface and phenanthrene37.Figure 3(a)20 °C adsorption isotherms for phenanthrene in soils. (b)30 °C adsorption isotherms for phenanthrene in soils. (c)40 °C adsorption isotherms for phenanthrene in soils. (d) 20 °C desorption isotherms for phenanthrene in soils. (e) 30 °C desorption isotherms for phenanthrene in soils. (f) 40 °C desorption isotherms for phenanthrene in soils.Full size imageThe Freundlich isotherm was used mainly for adsorption surfaces with nonuniform energy distribution, and the Langmuir isotherm was used for monolayer adsorption on perfectly smooth and homogeneous surfaces38. The experimental data were fitted with the Langmuir and Freundlich adsorption models, and the isotherm parameters logKF, 1/n, KL, qmax and the coefficient of determination (R2) of phenanthrene in soils were listed in Table 4.Table 4 Isotherm parameters for Phenanthrene sorption in soils.Full size tableAs shown in Table 4, according to the coefficients of determination (R2), all soils were better fitted with the Freundlich model, which assumes that phenanthrene sorption and desorption occurs on a heterogeneous surface with the possibility of sorption being multi-layered39. This phenomenon has also been observed in humic acid and nanometer clay mineral40. It showed that the soil adsorption of organic matter was not only surface adsorption but also the process of soil organic matter distribution41,42,43 reached the equilibrium isotherm fitted well with the Freundlich equation when studying the adsorption behavior of aromatic compounds by solids.Adsorption and desorption thermodynamicsTo clarify the adsorption mechanisms, the thermodynamic parameters mentioned earlier were calculated and presented in Table 5. Generally, the value of Gibbs free energy changeΔG0 indicated the spontaneity of a chemical reaction. Therefore, it could evaluate whether sorption was relate to spontaneous interaction44. Negative values of ΔG0 indicated that the feasibility and spontaneous nature. The research was under the temperature range about 293–313 K. For adsorption process, all soils ΔG0 was  0 and desorption ΔH  1, P  temperature  > phenanthrene concentration  > pH. In the interaction, the phenanthrene concentration and organic matter have a significant effect on the silt adsorption rate. The coefficient of determination of the silt complex correlation is R2 = 0.9464, indicating that the response model has a good fit, and the experimental error is within the acceptable range. Adjusting the complex correlation coefficient R2 = 0.8982 indicates that the regression relationship can explain 89.82% of the change in the dependent variable. Therefore, this The model can be used to analyze and predict the effect of different factors on the adsorption rate of phenanthrene.3D response surface analysisIn response surface optimization, the three-dimensional response surface graph reflects the influence of the interaction of the other two variables on the response value, and the slope of the response surface reflects the significance of the interaction of the two variables on the response value. The more significant the interaction effect is on the response value, when the slope is gentle, the effect is not significant. If the contour map is elliptical, it indicates that the interaction between the two variables is significant, and if the contour map is circular, it is not significant46. In addition, the slope and density of the contour line also reflect the influence of the variable on the response value. The steeper the contour line and the greater the density, the greater the influence of the variable on the response value47.

(1) Loess Fig. 5 is a three-dimensional response surface diagram of the interaction between initial phenanthrene concentration and pH to phenanthrene adsorption on loess. It can be seen from the figure that the slope of the response surface graph is steep, and the contour line is an approximate circle, indicating that the interaction between phenanthrene concentration and pH is not significant for the response value. With the increase of pH, the adsorption rate of phenanthrene on loess showed a slow decline at first to the lowest point at 6, and then gradually increased. When the soil pH was close to 6, with the increase of the initial phenanthrene concentration, the adsorption rate of loess also showed a trend of first decreasing and then increasing. According to the F value, F = 0.337, P = 0.5532  > 0.05, it can be concluded that soil pH and initial phenanthrene concentration of the solution have no significant interaction on the adsorption rate of loess.

Figure 6 shows the effects of initial phenanthrene concentration and organic matter on phenanthrene adsorption on Loess under the condition that pH value and temperature are at the central point. It can be seen from the figure that the initial phenanthrene concentration and soil organic matter contour are steep, indicating that their interaction is significant. The range of phenanthrene adsorption rate is 70 ~ 95, and the change of surface is steep. From the Loess error analysis, it can be seen that if f value is 6.05 and P value is 0.0275  More

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Study systemOur focal ecosystem is in Selvíria, state of Mato Grosso do Sul, Brazil ((hbox {20}^{circ }) (22′) (41.86”) S, (hbox {51}^{circ }) (24′) (58.90”) W), on a property owned by the São Paulo State University (UNESP). The location covers 350 ha of pasture composed of liverseed grass (Urochloa decumbens). The native vegetation was removed, pasture areas were implemented, and livestock was introduced in the 1970s, maintaining this configuration during the following 50 years. The climate of this area is categorized as equatorial savanna, with dry periods concentrated mostly during the winter, from April to August. During our sampling period (from November 23th, 1989, to November 19th, 2015), no vermifuges and insecticides that could affect negatively the community of dung beetles associated with cow pads were used1.The native dung beetle community at this site is composed of dwellers and tunnelers. Dwellers comprise the Aphodiinae subfamily, whereas all the tunnelers belong to the Scarabaeinae subfamily31. In total, there were eight species classified as dwellers (Ataenius crenulatus, A. picinus and Atanius aequalis-platensis grouped as one species, Blackburneus furcatus, Genieridium bidens, Labarrus pseudolividus, Nialaphodius nigrita and Trichillum externepunctatum) and ten native tunnelers (Ateuchus nr. puncticollis, A. vividus, Canthidium nr. pinotoides, Dichotomius bos, D. semiaeneus, D. sexdentatus, Ontherus appendiculatus, O. dentatus, O. sulcator). These species were chosen for our study because, as the invasive tunneler D. gazella (also from the Scarabaeinae subfamily), they all co-occur in pasture and exploit the same resource (cow pad)32. The initial establishment of D. gazella caused the loss of most of the native tunnelers from the community, with the invader becoming the overwhelming representative of the functional group, and an initial decrease of abundance for dwellers. Differently from native tunnelers, however, dwellers were able to recover their number a few years after invasion (Fig. 1a, Fig. S1).As reported in1, the abundance of dung beetles was significantly affected by both local minimum temperature and relative humidity. The influence of these two factors is expected, as they determine egg and larval survival and development of dung beetles. For example, because dung beetles are poikilotherms, environmental temperature is key to their development and fecundity33. One of the main dweller species, Labarrus pseudolividus, is widely found in locations with temperature averages ranging between (hbox {12},^{circ }hbox {C}) and (hbox {18},^{circ }hbox {C})34, making it tolerant to colder local temperatures. On the other hand, for D. gazella the lower developmental threshold is (hbox {15.5},^{circ }hbox {C}) (individuals cannot survive below this temperature), and the optimum temperature for population growth is (hbox {28},^{circ }hbox {C})35. For both groups, physiological growth and reproduction rates are maintained even when outside temperatures are close to the lower developmental threshold; dwellers, for example, live inside the dung pile, where temperature is higher and less variable than outside36,37. However, while tunnelers oviposit deep in the soil to protect the eggs, warmer and drier conditions reduce dweller egg viability on dung piles since they are exposed38. Low humidity conditions lead to drier dung and can cause egg and insect dessication. In addition, dwellers from our focal system have Palearctic evolutionary origins39; D. gazella’s natural distribution ranges from central to southern Africa40, presenting high physiological plasticity that allows it to tolerate high temperatures and low relative humidity better than other tunneler species41.Functional-group data collection and community structure characterizationDung beetles were collected once a week in a black-light flight intercept trap42, which guarantees the collection of coprophagic beetles. During all collection periods, climate variables were also collected from a meteorological station located within 2 km of our collecting site. See1 for the complete description of the collection process and database. For our purposes, we retained the species, number of individuals per species, and climate variables for each week sampled (Supplementary Information, SI, Figs. S1–S2).We focused first on the weekly abundance data, which we needed to process in order to avoid spurious results in our analyses stemming from the measurement protocol. Specifically, we filtered out seasonal low values associated with sampling in the coldest periods, when few beetles are captured because the reduced activity in all functional groups restricts their spatio-temporal distribution43. Including such samples would not be representative of the community and could bias the analysis since we are investigating community composition (i.e. proportions, very sensitive to low sampling). Thus, we considered only samples with a total number of beetles (that is, summing up all groups together) higher than the value of the median of all data, a conservative threshold that retains observations that allow for as much representation of the community as possible. As will become evident in the Results section and Supplementary Information, less conservative choices for the threshold did not alter our main conclusions.Following Mesquita -Filho et al.1, we categorized all sampled species into either dwellers or tunnelers. D. gazella is a tunneler and, as explained above, the native tunneler species experienced massive declines in abundance after its establishment, leaving D. gazella as almost the single representative in the tunneler functional group during the period of observation1. Thus, given the sharp contrast in community composition, we also separated the data into before and after invasion using to that end the 200th week, when D. gazella was first observed at the study site (September 11th, 1993, starting date for what we will call “after invasion”, our focal period henceforth).To describe community functional composition (i.e. system state) through time, we derived a normalized functional group ratio. First, because the abundance of each functional group spanned up to four orders of magnitude, we performed a logarithmic transformation of the number of captured insects from each group i, (log _{10}(N_{i}+K)), following  Yamamura44. Here, we chose (K=1), but the value of K did not alter our results qualitatively. In addition, the original data showed random mismatches in the phenology of each group, which gave the wrong impression of extreme short-term shifts in functional group dominance within the community. To avoid such artifacts, we used nonparametric local regression (LOESS)45 to smooth the dynamics of each group46. For this smoothing, we employed the loess function in the R software 3.6.147 with a smooth parameter equal to 0.25, but other moderate values (or an optimal value calculated with Bayesian inference by the R function optimal_span) did not alter our conclusions. Finally, we extracted back from the smoothed curve the number of beetles within each functional group to calculate the fraction (f_{dwell}) that measures the relative abundance of dwellers:$$begin{aligned} f_{dwell} = frac{N_D}{N_D+N_T} end{aligned}$$
(1)
where (N_D) corresponds to the number of dwellers per week and (N_T) corresponds to the number of native tunnelers (for the period before invasion), or only the number of D. gazella observed per week (after invasion), using their corresponding smoothed curves. Including also native tunnelers after invasion did not alter our conclusions.Climate driverWe devised a single climatic driver variable that merges the weekly measurement of temperature and relative humidity over the years, abiotic factors key to the survival and reproduction of both groups (see above). We first converted minimum temperatures and relative humidity to normalized climate variables using a min-max normalization (a feature scaling that uses the total range of temperatures or relative humidity, respectively, as normalization factor):$$begin{aligned} T = frac{T_{week} – T_{min}}{T_{max}-T_{min}};;,~ ~ ~ ~ ~ ~ RH = frac{RH_{week} – RH_{min}}{RH_{max}-RH_{min}};;, end{aligned}$$
(2)
where T corresponds to the normalized temperature, (T_{week}) is the weekly temperature, and (T_{max}) and (T_{min}) are the absolute maximum and minimum temperatures observed during the whole sampling period, respectively. We used a similar notation for relative humidity, RH. Based on the information above regarding beetle response to climate, the merged climate factor c was defined as the relationship:$$begin{aligned} c = frac{T}{RH};;, end{aligned}$$
(3)
for (RHne 0). That is, higher temperatures and/or drier conditions (expected to favor D. gazella) lead to higher values for c. On the other hand, lower temperatures and/or more humid conditions (expected to favor dwellers) imply lower values for c. Intermediate values of c can represent either moderate or extreme values for both T and RH.Identifying ecological states and quantifying resilienceWith our (f_{dwell}) data as an index of community composition (i.e. system state), we calculated kernel density functions to interpolate a continuous probability distribution of the relative fraction of dwellers in the community, (p_{n}(f_{dwell})) (function density, R software 3.1.647) for a given range of climatic driver c values. We grouped the (f_{dwell}) data using ranges for c of size 0.4, to ensure a significant amount of weekly samples that allowed for the reconstruction of these probability distributions (see Table S1, first column). Note that bins with extreme values showed few data points (see first and last rows in Table S1), and thus were rejected to prevent misleading results due to reduced sampling. Also note that, for the density function, we used the default Gaussian kernel with a smoothing bandwidth adjusted to be (50%) larger than the default value (“adjust” argument set to 1.5). This conservative choice aims to reduce the effect of the different sampling across c bins and to ensure that differences among distributions across c values are not the result of spurious sampling noise.Further, we transformed the kernel density function:$$begin{aligned} V(f_{dwell}) = -ln (p_{n}(f_{dwell})) end{aligned}$$
(4)
This (V(f_{dwell})) function, called potential (e.g.48), shows by design well-defined minima for the most frequently observed values of (f_{dwell}) (i.e. configurations most frequently observed for the community, which conform the modes of the probability distribution) in a given group of data. At these points, the potential exhibits a change of trend from decreasing to increasing, and therefore its derivative shows a change of sign. Eq. (4), thus, provides a simple criterion to identify possible system states, which is a reason why potentials have been used extensively across disciplines49,50,51. Nonetheless, because the position of extrema is invariant under the transformation, using probability distributions instead would not alter our conclusions.Representing the potential obtained from all the (f_{dwell}) system states associated with a same range of climatic driver c values allowed us to identify stable community configurations associated with a specific climate. The comparison of the potentials obtained for different c ranges enabled the description of how the community changed in response to climatic variation. The location of the minima revealed which states were stable for a given value of the climatic driver; the presence of two minima, then, flagged the existence of bistability (i.e. two different community compositions possible for the same c value).These minima are materialized as wells in the potential’s landscape, which provides an easy way to understand the concept of stability: the dynamics of the system for the given value of the driver will eventually “fall” into a well (either a state dominated by dwellers or a state dominated by tunnelers), with the shape of the well (e.g. its depth) determining how difficult it is for the system to “escape” that state. Therefore, the area inside a well provides quantification of the tendency of a system to stay in that specific state, i.e. the resilience of the associated ecological state or how strong a perturbation has to be to move the system from such an ecological state to another2,3,50,51,52,53. Thus, in addition to number and location of wells, measuring their associated area allowed us to further characterize the resilience of the community. To this end, we first set a visualization window common to all potentials. Specifically, we plotted the potentials within a range for the vertical variable (the potential, V) given by ([-1.5,1.5]); the horizontal variable (fraction of dwellers, (f_{dwell})) is by definition bounded between 0 and 1. For potentials that showed one single well, the area of the well was measured as the area above the potential curve within this visualization window. For potentials that showed two wells (bistability), we measured the value of the potential at the local maximum separating the two wells, and established that value as the upper (horizontal) line closing the area of each well. To ensure all cases were comparable and eliminate any arbitrariness of the choices above, we expressed resilience as a relative area; in other words, we further normalized the well area by the total area across wells for that potential, which means that any single-well case will show a resilience (or relative area) of 1, and the resilience of the two wells when there is bistability adds up to 1.Figure 1Left: Community composition by functional group for all weeks of observation1. Green represents dwellers, blue represents tunnelers, and orange represents the invader D. gazella. Right: Sketch of responses of the community composition to the climatic driver (i.e. phase diagram) expected from the physiological and behavioral characteristics of the functional groups in the community as described in text: linear (red), or non-linear but monotonic without (blue) or with (brown) hysteresis.Full size imageIdentifying ecological transitionsMeasuring a state variable, (f_{dwell}), and a driver, c (order and control parameter, respectively, in the jargon of regime shift theory), allowed us to study how their observed behavior over time materializes in a driver-state relationship (the so-called phase diagram) defining the possible shifts in dominance (i.e. regime shifts) that the community may undergo as climate changes12. The non-monotonic temporal behavior of the components of the order parameter (i.e. dwellers and tunneler availability) and the components of the control parameter (i.e. temperature and relative humidity) makes it difficult to predict the shape of the phase diagram, and therefore whether we can expect alternative stable states in the focal example. For such cases, the dominance of the dung beetle community could (1) shift in a linear fashion toward the functional group favored by climatic conditions; (2) shift between functional groups in non-linear threshold response to climatic conditions without hysteresis; or (3) shift between functional groups in non-linear threshold response to climatic conditions with hysteresis –and thus showing bistability (see Fig. 1b, or12). Other possibilities, e.g. a non-linear shift between functional groups where one group is favored at intermediate climatic conditions12 are discarded as the invader is better suited for warmer and drier conditions. To evaluate which of these possibilities occurred, we represented (f_{dwell}) as a function of c, as well as the location of the minima shown by the potentials above. In addition to the emerging shape of this relationship, this plot can reveal the presence of alternative stable states if two or more different points occur for the same value of the control parameter, c. More