Department of Ecology and Evolutionary Biology, University of Arizona The relationship between environmental productivity and the A unimodal (hump‐ shaped) form of this relationship has long been .. In this process, we considered only those relationships which had a justifiable ecological meaning. "U-shaped relationship" is not a mathematically precise term and there well (or that the jump at the discontinuity has some physical meaning). A humped yield curve is a relatively rare type of yield curve that results when the interest rates on medium-term fixed DEFINITION of Humped Yield Curve Humped yield curves are also known as bell-shaped curves.
Species richness and abundance of the arthropod community increased, with the increase seen across functional feeding groups. Alternative stable states, lags, feedback, and nonlinearity contribute to hysteresis. If, for example, productivity begins increasing from some low value, species richness does not immediately increase because the arrival and establishment of new species take considerable time. The same is true if productivity decreases from some high value.
The system is said to exhibit path dependence; the curves differ depending upon whether the driver i. Low productivity and high productivity states are stable up to a threshold. Returning to our study of ant communities [ 32 ], we find more correlated independent variables Figure 3. Therefore, the intermediate productivity hypothesis can also account for the unimodal species richness-curve of ants on the Fall-line Sandhills.
Thus, available NPP can account for ant species richness as well. In a stepwise multiple regression of species richness on NDVI, spatial heterogeneity, and soil temperature, only spatial heterogeneity emerged from the selection process. This underscores the complexity of most species richness-relationships. Species Richness and Environmental Gradients Extreme environments can often generate unimodal relationships if the gradient is wide enough to be stressful at either extreme [ 1553 ].
Acid-base relationships are often stressful at their extremes.
Grime [ 15 ], for example, described a humpbacked relationship of herbaceous plant species across a range of soil pH 3. Both mean species density mean number of species per m2 and the total number of species per pH bin peaked at a pH of 6.
International Journal of Ecology
Likewise, Graham [ 54 ] showed that species richness of freshwater fishes in New Jersey lakes peaked at a pH of 7. In addition to the stress of extreme pH, productivity varies across a pH gradient. In acidic blackwaters, the entire food web shifts from one based upon primary producers to one based upon detritus. In alkaline marl lakes, phosphorus coprecipitates with calcium carbonate, reducing fertility [ 55 ]. Quadratic regression of species richness on pH for freshwater fishes in 85 New Jersey ponds and lakes, after removing the effect of lake area.
After Graham [ 54 ]. Species richness often peaks in the middle of spatial gradients of elevation and depth [ 56 ]. Nearly half of all elevation gradients, for example, may show a mid-elevation peak [ 57 ]. Organisms showing mid-elevation peaks in species richness include epiphytic ferns [ 58 ], vascular plants and lichens [ 59 ], moths [ 60 ], ants [ 6162 ], birds [ 5863 ], and small mammals [ 64 ]. The reasons for these patterns are varied and difficult to disentangle.
In addition, declining species pool and decreasing intensity of competition with elevation can generate unimodal diversity curves [ 59 ]. Environmental and spatial gradients, such as those associated with soil or aquatic pH, elevation, and depth, can vary in productivity [ 39 ], disturbance, and spatial heterogeneity.
According to Kessler et al. In addition to the difficulty in assigning cause and effect, humpback species richness-curves can arise from geometric constraints on species range boundaries in an environmentally homogeneous area [ 6065 ].
Randomly placing species on a bounded map produces a peak of species richness near the center of the map. This is the mid-domain effect MDEwhich serves as a null model for species richness on environmental gradients.
Before attributing a unimodal species richness-curve to gradients of elevation, depth, pH, productivity, or disturbance, the MDE needs to be addressed but see [ 56 ]. Finally, species richness of different taxonomic and functional groups can peak at different locations on an environmental gradient.
In alpine plant communities of Norway and Finland, for example, species richness of dwarf shrubs peaks at a lower elevation than the species richness of lichens and graminoids, and the species richness of all three of these taxonomic groups peaks at a lower elevation than that of forbs [ 59 ].
In freshwater fish communities of New Jersey, species richness of native species and sunfishes Centrarchidae peaks at a lower pH than exotic species and minnows Cyprinidae [ 54 ]. Species Richness and Time Succession Margalef [ 1 ] and Horn [ 66 ] suggested that species diversity richness and evenness peaks during the early or middle stages of succession. Research has not, however, always supported this generalization. Three examples demonstrate the point.
A year successional series, from old-field annuals and perennials, to pine, and oak-hickory woodland, on the Georgia Piedmont shows plant species richness increasing and eventually flattening out with no apparent decrease [ 67 ].
A year successional sequence culminating in prairie and oak savanna at the Cedar Creek Natural History Area in Minnesota also shows plant species richness increasing continuously with no hump [ 68 ]. Species richness of ants on a year series of coastal dunes also increases monotonically [ 69 ]. The reasons for such monotonic relationships are difficult to understand; the ratio of productivity to biomass usually increases and then decreases during succession, as does spatial heterogeneity [ 66 ].
Both patterns, if present, should generate unimodal species richness-curves in a long successional series. Species Richness and Narrow Latitudinal Gradients Species richness of most taxonomic and functional groups decreases monotonically from the equator to the poles [ 70 ].
Parasitic wasps of the family Ichneumonidae are an exception. According to Janzen, resource fragmentation can explain the pattern. Alternatively, tropical Lepidoptera may be more distasteful, on average, than temperate Lepidoptera nasty-host hypothesis or predation on parasitoids may be more intense in the tropics.
Sime and Brower [ 72 ] found that the data support the nasty-host hypothesis better than either the resource fragmentation or predation hypotheses. Numbers of ichneumonid species per km2 across a latitudinal gradient in North America. Data, from Janzen [ 71 ], fit with a natural spline. Scale Scale is a necessary component of ecological theory [ 73 ]. The response of dependent variables, for example, can change across multiple spatial and temporal scales [ 7475 ].
The three components of scale are grain, focus, and extent [ 12 ]. The grain is the sampling unit. In our studies of ant communities at Fort Benning, the grain was a 28 m2 area containing 5 pitfall traps whose contents were pooled. The focus is an aggregation of sampling units for which inference is made. At Fort Benning, the focus was a 4 ha site of homogeneous vegetation. The extent is the scale of the entire set of sampling units.
At Fort Benning, the extent was the Fall-line Sandhills section of the 73, ha military base containing all 40 sites. Whittaker [ 76 ] was among the first to elaborate upon species diversity and scale. He defined three kinds of diversity, which varied according to the scale at which diversity was measured.
Alpha diversity is the number of unique species measured within a particular habitat or ecosystem. It is an ecological attribute of a particular place. Beta diversity is the difference in diversity between two or more habitats, expressed as the number of unique species found in each habitat.
Beta diversity measures either compositional heterogeneity or the turnover of species among different habitats [ 7778 ], arising from environmental gradients, as well as distance among populations, dispersal of community members, productivity differences, and other elements of structural heterogeneity [ 7980 ]. Gamma diversity is the total number of species within a larger region composed of many different habitats.
It is a function of alpha diversity and the number of different habitat types present within the larger geographic region. The drivers of species distribution and abundance can be affected simultaneously, and to differing degrees, by factors operating in nested spatial scales. For example, patterns of amphibian richness and abundance differ among headwater streams grouped according to factors measured at multiple spatial scales [ 81 ]. Despite this long recognition, only recently have more papers explicitly dealt with scale effects in examining patterns of species richness [ 82 ].
Patterns of species richness are scale dependent [ 83 ]. The functional relationship between productivity and species richness, for example, is scale dependent [ 12 ]. Part of scale dependence can be attributed to sampling artifacts at local scales [ 84 ] or methodological differences among studies occurring at different scales. But recent efforts have explicitly tested for scale dependence by measuring the same variables across scales.
Other studies have also shown similar scale dependence [ 124186 ]. Proposed mechanisms responsible for scale dependence include different rates of species turnover due to differences in environmental heterogeneity [ 8587 ]. Species Richness-Curves Contingent upon Organism and Environment Ecologists have generated a vast literature as they set out to study unimodal species richness-curves for diverse taxa, at multiple spatial and temporal scales, in various biomes.
They have done so using observational studies, experimental approaches, or a combination of the two [ 26 ]. Additionally, species richness is either the number of species or the species density, and species diversity is one of several indices, including the Shannon index and Hurlbert's probability of interspecific encounter.
Likewise, natural disturbances such as waves, fire, and salinity, or anthropogenic stressors such as pollution can be measured along gradients indicating magnitude, frequency, or a combination of the two. Species richness responses can also be a function of productivity, which also has myriad indices or proxies. This diversity in both the dependent and independent variables makes meta-analytic approaches challenging [ 7 ]. Below, we briefly survey this literature to provide highlights across major taxa.
Marine Benthic and Intertidal Organisms The notion of unimodal species richness-curves arose following seminal work in rocky intertidal zones on Pacific and Atlantic coasts of North America. These studies examined predation and herbivory [ 3637 ], wave-generated disturbance of substrate and algal communities [ 88 ], and storm-generated disturbance of coral reefs [ 188990 ]. As in other taxonomic groups and ecological systems, tests of unimodal species richness-curves in marine benthic and rocky intertidal communities have been observational and experimental.
They have also occurred across a range of environmental gradients and spatial scales. Connell's classic intermediate disturbance hypothesis, based on his work with coral reefs, suggested that, under low disturbance, competitive exclusion keeps species richness low because dominant species monopolize resources.
Under high disturbance most species are extirpated, leaving moderately disturbed sites with the highest species richness see also [ 91 ].
Similarly, Sousa's [ 88 ] study of waves, which frequently turn over small boulders and infrequently damage large boulders, showed that intermediate boulder size had the most biologically diverse communities. He concluded that, although communities may be globally stable, at local scales they are, generally, in nonequilibrium states.
A further offshoot of this research is the concept of alternative stable states [ 9293 ]; large disturbances can cause ecosystems to follow unique trajectories towards different equilibria [ 94 ]. Unimodal patterns of species richness are also prevalent in benthic marine communities, with recent studies showing scale dependence and interactions of productivity and disturbance. Species richness of algal communities on the coastline of Sweden was humpbacked along both disturbance waves and NPP biomass gradients [ 95 ].
They used a factorial design to test the effects of disturbance biomass removal and nutrient enrichment and found that the unimodal relationship was damped in the oligotrophic, but not in the eutrophic, bays see also [ 97 ].
The interaction between productivity and disturbance may alter the location of the mode i. These interactions, however, are not always evident; nutrient enrichment in a marine ecosystem on the west coast of Sweden had no effect on unimodal species richness-patterns [ ].
Although our focus in this review is at the local scale, species richness increases from the poles to the equator at a global scale [ 70 ]. They found that unimodal species richness-latitude patterns occurred at both local and regional scales, with a more pronounced relationship at regional scales. This strong effect of regional species pools on local diversity has also been seen for coral reefs [ ] and serpentine plants [ ].
These systems typically have few limiting resources and theory predicts that the best competitors should monopolize resources. Like the global distribution of study sites reported by Whitman for benthic macrofauna, Irigoien et al. Using biomass of these communities as a proxy for productivity, they found unimodal diversity-productivity relationships Shannon index for both phytoplankton and zooplankton.
Contrary to expectation, however, zooplankton, and phytoplankton biomass were uncorrelated. The unimodal relationship held despite other regulatory gradients, such as light and temperature, that differed among sites. Lacustrine species richness is affected by primary productivity and lake area.
They found unimodal species richness-productivity relationships for phytoplankton and zooplankton rotifers, cladocerans, and copepodsbut not for fish. Lake size was a significant covariate for the species richness of fish and phytoplankton. The diversity-productivity relationship is not consistently unimodal within and among trophic levels. Cohen [ ] did not find support for the intermediate disturbance hypothesis; species richness of ostracods declined with increasing sedimentation in an African lake known for high levels of ostracod endemism.
In some cases, assembly history can change a unimodal species richness-productivity curve into something else.
As immigration and emigration among metapopulations or metacommunities occur, the sequence of arrival of different species can play a role in the developing community structure. A generation species richness-productivity experiment, with four assembly sequences of freshwater protozoans and rotifers in five levels of nutrient enrichment productivitygenerated unimodal, positive monotonic, u-shaped, and no relationship forms of the curves [ 45 ].
Marine and Lacustrine Vertebrates The emergence of unimodal species richness-curves in lacustrine fish communities is complicated by the effect of lake area on species richness, a phenomenon predicted by the theory of island biogeography  as well as scale-dependent habitat diversity [ ].
As mentioned previously, species richness of native freshwater fish species shows a unimodal relationship on a pH gradient after the effect of lake area is removed [ 54 ]. Species richness is greatest at a pH of 7. Exotic species show a mostly linear relationship, though there is a hint of a decline at the highest pH 9.
The pH of each lake is related to productivity, though the relationship is likely to be complex. Terrestrial Plants According to Tilman and Pacala [ ], plant species richness is a unimodal function of productivity.Examples of Commensalism
Inthese authors could not find a single example of a monotonic relationship between plant species richness and productivity. Since then, the numbers of unimodal relationships have been downgraded considerably.
Of the 15 admissible studies, 7 or Unimodal species richness-disturbance relationships have been just as contentious, even in the tropical rain forests first promoted as examples of the intermediate disturbance hypothesis by Connell [ 18 ]. Even with huge sample sizes, the unimodal relationships were barely obvious, especially in wet and moist forest. The unimodal curve was best developed in dry tropical forest but only explained In the wet forests it only explained an inconsequential 2.
Terrestrial Vertebrates There are few examples of humpbacked species richness-relationships among terrestrial animals, particularly terrestrial vertebrates [ 25 ]. First, the predator or disturbance must reduce densities of competing species, releasing resources. Second, there must be a large species pool available to colonize the resources that have been released.
Third, there must be strong competition among the colonizing species.
The Humpbacked Species Richness-Curve: A Contingent Rule for Community Ecology
This occurs with the invertebrates and attached algae of the rocky intertidal zone studied by Connell [ 18 ], Paine [ 3536 ], and Lubchenco [ 37 ]. Terrestrial predators and herbivores, however, rarely reduce prey densities to the point that resources are released. And when there is a reduction in prey densities, colonization reconstitutes the original set of species from neighboring habitats.
Finally, strong competition is reduced by resource partitioning [ 25 ]. Invertebrates of the rocky intertidal zone compete for space in two dimensions; most terrestrial animals compete for space in three dimensions.
Terrestrial animals also compete for food. Hence, it is easier to partition resources when there are more than two dimensions. Terrestrial Invertebrates The same arguments apply to most terrestrial invertebrates. Ants possess the three criteria necessary for a humpbacked species diversity curve. First, disturbances, such as clear-cuts, substantially reduce their abundance. In Finland, for example, wood ant communities undergo collapse shortly after clear-cutting [ ].
regression - What is a strict definition of U-shaped relationship? - Cross Validated
In northwestern Georgia, we have found pitfall traps nearly devoid of ants following a clear-cut, while those in neighboring reference sites retain a large ant community unpublished data, J. They do not immediately diffuse back into the cleared patch. Second, winged queens are available to establish new colonies following a clear-cut [ ]. Finally, colonizing species of ants compete strongly. Most ant colonies have a regular spatial distribution, indicative of strong competition for space [ ].
In fact, ants show pronounced humpbacked species richness-curves, both for disturbance and productivity as measured by NDVI Figure 3 [ 32]. Humpbacked diversity patterns in local ant communities have been documented along gradients of both stress [ ] and disturbance , with a reduction in diversity linked to exclusion by dominant species see also [ ].
Andersen [ ] actually used the frequencies of dominant ants as surrogates for available productivity, under the assumption that competitively dominant species should prevail in resource-rich environments.
Humpback species richness-curves are difficult to find in terrestrial invertebrates other than ants. This is surprising, especially for herbivorous insects, because one might expect herbivore richness to track plant species richness.
Humpbacked species richness-curves are well documented in terrestrial plants, and experimental work by Siemann et al.
Tree species diversity also influences the diversity of herbivorous insects [ ]. Logically, then, one would expect a unimodal plant richness curve to generate a unimodal arthropod richness curve. But the relationship between plant and arthropod species richness is weak [ ], and patterns evident in field experiments may be masked by noise in real communities.
Oddly enough, tree species diversity on Barro Colorado Island, Panama, does not influence the diversity of ants and mites of the leaf litter [ ]. Does arthropod diversity fit the intermediate disturbance hypothesis? Some studies of arthropods, especially ants, seem to support it [, ], whereas others do not [ — ] or do so for only a few taxa [ ]. Thus, arthropods other than ants may, or may not, exhibit intermediate disturbance effects.
Microorganisms Do microorganisms show unimodal species richness-curves? According to Smith [ 46 ], a literature survey of microbial diversity-productivity relationships from 70 natural, experimental, and engineered aquatic ecosystems reveals mostly unimodal and monotonic patterns both positive and negative.
The unimodal curves accounted for In both cases, monotonic patterns outnumbered unimodal ones. Mathematical and Statistical Considerations 5. Polynomial Regression and Data Smoothing One can fit unimodal functions most simply with a quadratic polynomial. In fact, polynomial regression is especially useful for distinguishing among monotonic linearunimodal quadraticand multimodal cubic, quartic, etc. Nonlinear response models Huisman-Olff-Fresco or HOF models  have also been used to model species richness-curves [ 21 ].
For all of these models, however, care must be taken that the mode lies within the range of the data [ 7 ]. Alternatively, a variety of data smoothing techniques are available, such as moving averages and locally weighted least squares, or LOESS [ ] see Smith [ 46 ] for an example. Despite not generating an equation, or providing a statistical test, data smoothing has the advantage of approximating any curve. Restricting the range of the independent variable can have dramatic effects on the outcome of a polynomial regression [ 39 ].
Sampling below the mode of a unimodal curve generates a relationship that is monotonically increasing, whereas sampling above the mode generates a relationship that is monotonically decreasing. Finally, a very restricted range near the mode may generate no relationship. How widespread is this problem? Range restriction has shaped the conclusions of our own research. When we began studying species richness of ants at Fort Benning, we selected nine sites in three disturbance classes light, moderate, and heavy.
Species richness seemed to decrease monotonically with increasing disturbance [ ]. But when we expanded the study to 40 sites across a much wider range of habitats and disturbance categories bare ground with widely scattered trees and grasses to deciduous oak-hickory forest we found the relationship was actually unimodal [ 32 ].
Many of the positive and negative monotonic relationships described in the literature may represent one-half of a unimodal relationship. It is unlikely that the converse is true. Consequently, it is important to sample across the widest possible range of habitats to get an accurate appraisal of unimodal relationships. A U-shape means convexity for an application along these lines, see Van Landeghem's "A test for the convexity of human well-being over the life cycle: Longitudinal evidence from a year panel".
A U-shape is a function with weighted average derivative negative until a point, and weighted average derivative positive after that point see Uri Simonsohn's Two-Lines: A U-shape is a function with exactly one turning point.
This corresponds to a function that is quasi-convex but not monotone. One complication that comes up is what if the turning point is close to the ends of the range of the x variable? Should we still consider such a function a U-shape? In my opinion, such a discussion should be had when you define what a U-shape means to you for your application, and when you specify your null hypothesis.
Arbitrary decisions are necessary with this proposed framework. The important thing is to be open about them and check how sensitive results are to changes and to challenge others to do the same. In addition to stating the null hypothesis, as always you should state the assumptions you rely on.
For example, a common assumption is that the regression function is either U-shaped on monotone. The Appropriate Test for a U-Shaped Relationship", where they propose an improvement on the vanilla OLS quadratic test by testing that the derivative of a specified functional form is negative at the beginning of the range, and positive at the end.