Quantifying shrub-shrub competition in drylands using aerial imagery and a novel landscape competition index.

The Jornada Basin Long-Term Ecological Research Site (JRN-LTER, or JRN) is a semiarid grassland-shrubland in southern New Mexico, USA. The role of intraspecific competition in constraining shrub growth and establishment at the JRN and in arid systems, in general, is an important question in dryland studies. Using information on shrub distributions and growth habits at the JRN, we present a novel landscape-scale (c. 1 ha) metric (the 'competition index', CI), which quantifies the potential intensity of competitive interactions. We map and compare the intensity of honey mesquite (Prosopis glandulosa, Torr.) competition spatially and temporally across the JRN-LTER, investigating associations of CI with shrub distribution, density, and soil types. The CI metric shows strong correlation with values of percent cover. Mapping CI across the Jornada Basin shows that high-intensity intraspecific competition is not prevalent, with few locations where intense competition is likely to be limiting further honey mesquite expansion. Comparison of CI among physiographic provinces shows differences in average CI values associated with geomorphology, topography, and soil type, suggesting that edaphic conditions may impose important constraints on honey mesquite and growth. However, declining and negative growth rates with increasing CI suggest that intraspecific competition constrains growth rates when CI increases above c. 0.5.

Male starling floaters preferentially visit nests of males with reduced resource holding potential.

Floaters are sexually mature individuals that are not able to reproduce by defending breeding resources. Floaters often visit active nests, probably to gather public information or to compete for a nesting site. We tested the hypothesis that floaters preferentially prospect nests in which they have a better chance of taking over, and that they do so by assessing the owners' resource holding potential (RHP). We manipulated the flight capacity of male and female breeders in a population of spotless starlings (Sturnus unicolor) by clipping two flight feathers per wing before egg laying, thus increasing their wing-load and likely impairing their condition. We subsequently monitored breeder and floater activity by means of transponder readers during the nestling period. We found that nests owned by wing-clipped males were visited by a greater number of male floaters than control nests. This effect was absent in the case of wing-clipped females. The number of male floaters also increased with increasing nestling age and number of parental visits. The experiment shows that male floaters preferentially prospect nests in which the owner shows a reduced RHP, a strategy that likely allows them to evict weak owners and take over their nests for future reproductive attempts.

Experimental evidence that competition strength scales with ecological similarity: a case study using Anolis lizards.

Interspecific competition is widely considered a powerful process underlying species coexistence and ecological community structure. Although coexistence theory predicts stronger competition between more ecologically similar species, empirical support has largely relied on inferring competition from patterns of species co-occurrence. Coexistence theory also posits that species can only coexist when individuals compete more with conspecifics than with other species, however, few field studies-particularly in reptiles-have simultaneously estimated the strength of both intra- and interspecific competition among co-occurring species. Using an array of 12 experimental plots, we manipulated species presence and population size by plot of three native Anolis lizard species to empirically estimate the strength of competition on one anole species driven by two other species of varying ecological similarity. We observed that the strength of competition-as determined by relative growth rates and gravidity-was highly predictable and correlated to ecological similarity. Interspecific competition was strongest among species of highest ecological similarity, and intraspecific competition-induced by the addition or removal of conspecifics-was consistently the most intense. By employing direct experimental manipulations, our study provides an empirical investigation of the strength of competition as it relates to ecological similarity.

Damming of streams due to the construction of a highway in the Amazon rainforest favors individual trophic specialization in the fish (Bryconops giacopinii).

In Amazonian streams, damming caused by road construction changes the system's hydrological dynamics and biological communities. We tested whether the degree of specialization in fish (Bryconops giacopinii) individuals is higher in pristine stream environments with intact ecological conditions than in streams dammed due to the construction of a highway in the Amazon rainforest. To achieve this, stomach content data and stable isotopes (δ13C and δ15N) in tissues with varying isotopic incorporation rates (liver, muscle, and caudal fin) were used to assess the variation in consumption of different prey over time. The indices within-individual component (WIC)/total niche width (TNW) and individual specialization were employed to compare the degree of individual specialization between pristine and dammed streams. The condition factor and stomach repletion of sampled individuals were used to infer the intensity of intraspecific competition in the investigated streams. The species B. giacopinii, typically considered a trophic generalist, has been shown to be, in fact, a heterogeneous collection of specialist and generalist individuals. Contrary to our expectations, a higher degree of individual specialization was detected in streams dammed by the highway. In dammed streams, where intraspecific competition was more intense, individuals with narrower niches exhibited poorer body conditions than those with broader niches. This suggests that individuals adopting more restricted diets may have lower fitness, indicating that individual specialization may not necessarily be beneficial for individuals. Our results support the notion that intraspecific competition is an important mechanism underlying individual specialization in natural populations. Our results suggest that environmental characteristics (e.g., resource breadth and predictability) and competition for food resources interact in complex ways to determine the degree of individual specialization in natural populations.

The Evolution of Competitive Ability.

AbstractCompetition drives evolutionary change across taxa, but our understanding of how competitive differences among species directs the evolution of interspecific interactions remains incomplete. Verbal models assume that interspecific competition will select for reducing a species' sensitivity to competition with their opponent; however, they do not consider the potential for other demographic components of competitive ability to evolve, specifically, interspecific effects, intraspecific interactions, and intrinsic growth rates. To better understand how competitive ability evolves, we set out to explore how each component has evolved and whether their evolution has been constrained by trade-offs. By setting sympatric and allopatric populations of an annual grass in competition with a dominant invader, we demonstrate (1) that in response to interspecific competition, populations can evolve increased competitive ability through either reduced interspecific or, surprisingly, reduced intraspecific competition; (2) that trade-offs do not always constrain the evolution of competitive ability but rather that parameters may correlate in ways that mutually beget higher competitive ability; and (3) that the evolution of one species can influence the competitive ability of its opponent, a consequence of how competitive ability is defined ecologically. Overall, our results reveal the complexity with which demographic components evolve in response to interspecific competition and the impact past evolution can have on present-day interactions.

The Joint Evolution of Animal Movement and Competition Strategies.

AbstractCompetition typically takes place in a spatial context, but eco-evolutionary models rarely address the joint evolution of movement and competition strategies. Here we investigate a spatially explicit forager-kleptoparasite model where consumers can either forage on a heterogeneous resource landscape or steal resource items from conspecifics (kleptoparasitism). We consider three scenarios: (1) foragers without kleptoparasites, (2) consumers specializing as foragers or as kleptoparasites, and (3) consumers that can switch between foraging and kleptoparasitism depending on local conditions. We model movement strategies as individual-specific combinations of preferences for environmental cues, similar to step-selection coefficients. Using mechanistic, individual-based simulations, we study the joint evolution of movement and competition strategies, and we investigate the implications for the distribution of consumers over this landscape. Movement and competition strategies evolve rapidly and consistently across scenarios, with marked differences among scenarios, leading to differences in resource exploitation patterns. In scenario 1, foragers evolve considerable individual variation in movement strategies, while in scenario 2, movement strategies show a swift divergence between foragers and kleptoparasites. In scenario 3, where individuals' competition strategies are conditional on local cues, movement strategies facilitate kleptoparasitism, and individual consistency in competition strategy also emerges. Even in the absence of kleptoparasitism (scenario 1), the distribution of consumers deviates considerably from predictions of ideal free distribution models because of the intrinsic difficulty of moving effectively on a depleted resource landscape with few reliable cues. Our study emphasizes the advantages of a mechanistic approach when studying competition in a spatial context and suggests how evolutionary modeling can be integrated with current work in animal movement ecology.

Know When You Are Too Many: Density-Dependent Release of Pheromones During Host Colonisation by the European Spruce Bark Beetle, Ips typographus (L.).

Individuals across various animal species communicate their presence to conspecifics. Especially phytophagous and parasitoid insects with their brood developing on limited resources rely on chemical cues, such as host-marking pheromones, to reduce intraspecific competition. Bark beetles are phytophagous insects with some species being economically and ecologically relevant forest pests. Several of them use the volatile compound verbenone to inhibit attraction and reduce intraspecific competition. However, in the Eurasian spruce bark beetle, Ips typographus (L.), temporal emission patterns did so far not quite support the putative function of verbenone as an indicator of densely colonised host trees. More importantly, it is currently unclear how well verbenone emission is actually related to colonisation density and thus intraspecific competition. Here, we inoculated Norway spruce logs with I. typographus at two defined colonisation densities in the greenhouse and measured the emission of verbenone and its precursors α-pinene and verbenol over time. Verbenone emission was 3-7 times greater from colonised logs compared to decaying logs without beetles during the major part of larval development. Furthermore, our data supports the quantitative hypothesis, that the termination of attack on a tree is mediated by a cessation of the release of verbenol and continuous emission of verbenone. The latter is most likely a passively produced host-marking cue reflecting the actual density of conspecifics since per-beetle emission was unaffected by colonisation density. These findings shed new light on the regulation of bark beetle mass aggregations, which are currently causing previously unseen economic damages in temperate forests.

The role of predation risk in structuring life-history traits of crucian carp (Carassius carassius) in a series of small boreal lakes.

Predation is a major evolutionary force determining life-history traits in prey by direct and indirect mechanisms. This study focuses on life-history trait variation in crucian carp (Carassius carassius), a species well known for developing a deep body as an inducible morphological defence against predation risk. Here, the authors tested variation in growth and reproductive traits in 15 crucian carp populations in lakes along a predation risk gradient represented by increasingly efficient predator communities. Lakes were located in south-eastern Norway and were sampled in summer 2018 and 2019. The authors expected crucian carp to attain higher growth rate, larger size, and later age at maturity with increasing predation risk. In the absence of predators, they expected high adult mortality, early maturity and increased reproductive effort caused by strong intraspecific competition. They found that the life-history traits of crucian carp were clearly related to the presence of piscivores: with increasing predation risk, fish grew in body length and depth and attained larger asymptotic length and size at maturity. This growth was evident at young age, especially in productive lakes with pike, and it suggests that fish quickly outgrew the predation window by reaching a size refuge. Contrary to the authors' predictions, populations had similar age at maturity. High-predation lakes also presented low density of crucian carp. This suggests that fish from predator lakes may experience high levels of resource availability due to reduced intraspecific competition. Predation regulated life-history traits in crucian carp populations, where larger size, higher longevity and size at maturity were observed in lakes with large gaped predators.

Intraspecific competition: A missing link in dermal armour evolution?

Predation is widely regarded as an important selective force in the evolution and maintenance of dermal armour; yet, the basic premise that predation and armour are strongly linked to each other has proven to be difficult to assess. In this concept, I put forward the fighting-advantage hypothesis, the view that aggressive interactions with conspecifics, not predation, might have been a key selective pressure in the evolution of dermal armour. Considering intraspecific competition as a potential explanation could not only reveal previously overlooked aspects of the functional and evolutionary significance of dermal armour, but also advance the emerging field of biomimetics in which such knowledge forms the starting point of technological innovation.

Can attraction to and competition for high-quality habitats shape breeding propensity?

In many animal species, sexually mature individuals may skip breeding opportunities despite a likely negative impact on fitness. In spatio-temporally heterogeneous environments, habitat selection theory predicts that individuals select habitats where fitness prospects are maximized. Individuals are attracted to high-quality habitat patches where they compete for high-quality breeding sites. Since failures in contests to secure a site may prevent individuals from breeding, we hypothesized that attraction to and competition for high-quality habitats could shape breeding propensity. Under this hypothesis, we predicted the two following associations between breeding propensity and two key population features. (1) When mean habitat quality in the population increases in multiple patches such that availability of high-quality sites increases across the population, the resulting decrease in competition should positively affect breeding propensity. (2) When the number of individuals increases in the population, the resulting increase in competitors should negatively affect breeding propensity (negative density dependence). Using long-term data from kittiwakes Rissa tridactyla, we checked the prerequisite of prediction (1), that availability of high-quality sites is positively associated with current mean habitat quality in the population (represented by breeding success). We then applied integrated population modelling to quantify annual fluctuations in population mean breeding success, breeding propensity and number of individuals by breeding status (pre-breeders, breeders, skippers and immigrants), and tested our predictions. Our results showed that breeding propensity acts as an important driver of population growth. As expected, breeding propensity was positively associated with preceding mean habitat quality in the population, and negatively with the number of competitors. These relationships varied depending on breeding status, which likely reflects status dependence in competitive ability. These findings highlight the importance of competition for high-quality breeding sites in shaping breeding propensity. Thereby, we draw attention towards alternative and complementary explanations to more standard considerations regarding the energetic cost of reproduction, and point to possible side effects of habitat selection behaviours on individual life histories and population dynamics.

Slower growth of farmed eels stocked into rivers with higher wild eel density.

Farmed anguillid eels are frequently stocked into natural fresh waters to enhance eel resources, but little is known about what happens to these eels or their interactions with wild eels after stocking. A recent study observed a depressed survival and growth rate of farmed Japanese eels when they were reared with wild eels, which indicated that wild eels might interfere with the survival and growth of farmed-and-stocked eels through intraspecific competition. To contribute to improving eel stocking efficiency, the growth of farmed-and-stocked Japanese eels was compared among four rivers with different wild eel densities using mark-and-recapture studies. Based on the 2-year recapture survey after stocking, it was found that the density of the farmed-and-stocked eels was not significantly different among rivers. The daily growth rates of farmed-and-stocked eels in the rivers with lower wild eel density were significantly higher than those of the eels stocked into the rivers with higher wild eel density. The farmed-and-stocked eels moved significantly greater distances downstream than wild eels that showed sedentary behaviour. This and previous studies indicate that significant questions remain about the effectiveness of stocking farmed eels into water bodies where naturally recruited wild eels are present.

Directly quantifying multiple interacting influences on plant competition.

When plants compete what influences that interaction? To answer this we measured belowground competition directly, as the simultaneous capture of soil ammonium and nitrate by co-existing herbaceous perennials, Dactylis glomerata and Plantago lanceolata, under the influence of: species identity; N uptake and biomass of focal and neighbour plants; location (benign lowland versus harsher upland site); N availability (low or high N fertilizer); N ion, ammonium or nitrate production (mineralisation) rate, and competition type (intra- or interspecific), as direct effects or pairwise interactions in linear models. We also measured biomass as an indirect proxy for competition. Only three factors influenced both competitive N uptake and biomass production: focal species identity, N ion and the interaction between N ion and neighbour N uptake. Location had little effect on N uptake but a strong influence on biomass production. N uptake increased linearly with biomass only in isolated plants. Our results support the view that measuring resource capture or biomass production tells you different things about how competitors interact with one another and their environment, and that biomass is a longer-term integrative proxy for the outcomes of multiple separate interactions-such as competition for N-occurring between plants.

Plant intraspecific competition and growth stage alter carbon and nitrogen mineralization in the rhizosphere.

Plant roots interact with rhizosphere microorganisms to accelerate soil organic matter (SOM) mineralization for nutrient acquisition. Root-mediated changes in SOM mineralization largely depend on root-derived carbon (root-C) input and soil nutrient status. Hence, intraspecific competition over plant development and spatiotemporal variability in the root-C input and nutrients uptake may modify SOM mineralization. To investigate the effect of intraspecific competition on SOM mineralization at three growth stages (heading, flowering, and ripening), we grew maize (C4 plant) under three planting densities on a C3 soil and determined in situ soil C- and N-mineralization by 13 C-natural abundance and 15 N-pool dilution approaches. From heading to ripening, soil C- and N-mineralization rates exhibit similar unimodal trends and were tightly coupled. The C-to-N-mineralization ratio (0.6 to 2.6) increased with N availability, indicating that an increase in N-mineralization with N depletion was driven by microorganisms mining N-rich SOM. With the intraspecific competition, plants increased specific root lengths as an efficient strategy to compete for resources. Root morphologic traits rather than root biomass per se were positively related to C- and N-mineralization. Overall, plant phenology and intraspecific competition controlled the intensity and mechanisms of soil C- and N- mineralization by the adaptation of root traits and nutrient mining.

Bifurcations and hydra effects in Bazykin's predator-prey model.

We consider Bazykin's model to address harvesting induced stability exchanges through bifurcation analysis. We examine the existence of hydra effects and analyze the stock pattern under predator harvesting. Prey harvesting cannot produce hydra effects in our model, whereas predator harvesting may cause multiple hydra effects. Our study reveals that type II response function and mutual interference among predators jointly induce multiple hydra effects and bistability. Bifurcations such as single Hopf-bifurcation, multiple Hopf-bifurcations and multiple saddle-node bifurcations appear for increasing harvesting rate on the predators. However, over-exploitation of the predators cannot generate any such bifurcation in our study. In simulations, the maximum sustainable yield (MSY) exists at a globally stable state. When predator is culled under increasing effort, basin of attraction of the equilibrium corresponding to the higher predator stock gets expanded, which alternatively is in favor of stock benefit for predators. The ecological theory developed in this study might be useful to understand conservation policy and fishery management.

Dispersal reduces interspecific competitiveness by spreading locally harmful traits.

Just as intraorganismal selection can produce "selfish" elements that lower individual fitness, selection at the organismal level can favour traits that reduce the fitness of conspecifics and potentially impact population survival. Because dispersal can affect how these traits are distributed within species, it may determine whether their negative consequences are restricted locally or spread throughout the species' range. We present an individual-based simulation model that explores the interaction between dispersal rate and traits that increase individual fecundity at the expense of conspecific fitness. We first modelled dispersal as a trait that varied within species and then fixed the within-species dispersal rates and modelled competition between species that differed only in dispersal rate. Reproductive isolation allowed species differences in dispersal rates to become associated with traits moulded by intraspecific competition, but this association did not occur when dispersal variation was distributed within species due to recombination between the dispersal and competition loci. Alleles that reduced the fitness of conspecifics were maintained at lower frequencies in low-dispersal species, resulting in a competitive advantage over high-dispersing species. Although high-dispersal species initially outcompeted low-dispersal species owing to enhanced colonization opportunities, low-dispersal species ultimately showed greater representation across a range of ecological and genetic scenarios. This process may shift the makeup of communities over time towards a greater representation of low-dispersal species.

Stony coral populations are more sensitive to changes in vital rates in disturbed environments.

Reef-building corals, like many long-lived organisms, experience environmental change as a combination of separate but concurrent processes, some of which are gradual yet long-lasting, while others are more acute but short-lived. For corals, some chronic environmental stressors, such as rising temperature and ocean acidification, are thought to induce gradual changes in colonies' vital rates. Meanwhile, other environmental changes, such as the intensification of tropical cyclones, change the disturbance regime that corals experience. Here, we use a physiologically structured population model to explore how chronic environmental stressors that impact the vital rates of individual coral colonies interact with the intensity and magnitude of disturbance to affect coral population dynamics and cover. We find that, when disturbances are relatively benign, intraspecific density dependence driven by space competition partially buffers coral populations against gradual changes in vital rates. However, the impact of chronic stressors is amplified in more highly disturbed environments, because disturbance weakens the buffering effect of space competition. We also show that coral cover is more sensitive to changes in colony growth and mortality than to external recruitment, at least in open populations, and that space competition and size structure mediate the extent and pace of coral population recovery following a large-scale mortality event. Understanding the complex interplay among chronic environmental stressors, mass-mortality events, and population size structure sharpens our ability to manage and to restore coral-reef ecosystems in an increasingly disturbed future.

Local prey shortages drive foraging costs and breeding success in a declining seabird, the Atlantic puffin.

As more and more species face anthropogenic threats, understanding the causes of population declines in vulnerable taxa is essential. However, long-term datasets, ideal to identify lasting or indirect effects on fitness measures such as those caused by environmental factors, are not always available. Here we use a single year but multi-population approach on populations with contrasting demographic trends to identify possible drivers and mechanisms of seabird population changes in the north-east Atlantic, using the Atlantic puffin, a declining species, as a model system. We combine miniature GPS trackers with camera traps and DNA metabarcoding techniques on four populations across the puffins' main breeding range to provide the most comprehensive study of the species' foraging ecology to date. We find that puffins use a dual foraging tactic combining short and long foraging trips in all four populations, but declining populations in southern Iceland and north-west Norway have much greater foraging ranges, which require more (costly) flight, as well as lower chick-provisioning frequencies, and a more diverse but likely less energy-dense diet, than stable populations in northern Iceland and Wales. Together, our findings suggest that the poor productivity of declining puffin populations in the north-east Atlantic is driven by breeding adults being forced to forage far from the colony, presumably because of low prey availability near colonies, possibly amplified by intraspecific competition. Our results provide valuable information for the conservation of this and other important North-Atlantic species and highlight the potential of multi-population approaches to answer important questions about the ecological drivers of population trends.

Spatial segregation in a sexually dimorphic central place forager: Competitive exclusion or niche divergence?

Sexual competition is increasingly recognized as an important selective pressure driving species distributions. However, few studies have investigated the relative importance of interpopulation versus intrapopulation competition in relation to habitat availability and selection. To explain spatial segregation between sexes that often occurs in non-territorial and central place foragers, such as seabirds, two hypotheses are commonly used. The 'competitive exclusion' hypothesis states that dominant individuals should exclude subordinate individuals through direct competition, whereas the 'niche divergence' hypothesis states that segregation occurs due to past competition and habitat specialization. We tested these hypotheses in two populations of an extreme wide-ranging and sexually dimorphic seabird, investigating the relative role of intrapopulation and interpopulation competition in influencing sex-specific distribution and habitat preferences. Using GPS loggers, we tracked 192 wandering albatrosses Diomedea exulans during four consecutive years (2016-2019), from two neighbouring populations in the Southern Ocean (Prince Edward and Crozet archipelagos). We simulated pseudo-tracks to create a null spatial distribution and used Kernel Density Estimates (KDE) and Resource Selection Functions (RSF) to distinguish the relative importance of within- versus between-population competition. Kernel Density Estimates showed that only intrapopulation sexual segregation was significant for each monitoring year, and that tracks between the two colonies resulted in greater overlap than expected from the null distribution, especially for the females. RSF confirmed these results and highlighted key at-sea foraging areas, even if the estimated of at-sea densities were extremely low. These differences in selected areas between sites and sexes were, however, associated with high interannual variability in habitat preferences, with no clear specific preferences per site and sex. Our results suggest that even with low at-sea population densities, historic intrapopulation competition in wide-ranging seabirds may have led to sexual dimorphism and niche specialization, favouring the 'niche divergence' hypothesis. In this study, we provide a protocol to study competition within as well as between populations of central place foragers. This is relevant for understanding their distribution patterns and population regulation, which could potentially improve management of threatened populations.

Limited evidence that larger acorns buffer Quercus rubra seedlings from density-dependent biotic stressors.

PREMISE: Plant performance and functional traits vary considerably within species, particularly in response to environmental variation. Plant responses may reflect life-history trade-offs, such as between resource acquisition and resource conservation. Larger seeds may buffer young plants from the negative effects of environmental variation, such as limitations in nutrients or water. However, whether seed size plays a similar role in how plants respond to variation in their biotic environment, including competition and soil microbial communities, remains poorly understood. METHODS: We used a greenhouse experiment to test the interactive effects of intraspecific competition, the origin of the soil microbial community, and seed size on performance and functional traits in Quercus rubra L. seedlings. RESULTS: Intraspecific variation in seedling traits weakly aligned with a resource acquisition-conservation trade-off. Across the wide range of initial acorn mass, competition decreased seedling biomass by about 35%. Competition directly decreased the root mass ratio and indirectly increased specific leaf area and specific root length, via the negative effects on total biomass. In contrast, soil microbial communities had minor effects on seedlings, and we found no differences between plants receiving soil originating from a conspecific adult and plants receiving soil originating from a heterospecific adult. CONCLUSIONS: Competition is a more important driver of intraspecific variation in young Quercus rubra seedling performance and traits, both directly and by delaying ontogenetic development, than soil microbial communities. Seed size is an important predictor of seedling biomass, but a larger seed does not necessarily buffer seedlings from the effects of competition.

Resonance in Physiologically Structured Population Models.

Ecologists have long sought to understand how the dynamics of natural populations are affected by the environmental variation those populations experience. A transfer function is a useful tool for this purpose, as it uses linearization theory to show how the frequency spectrum of the fluctuations in a population's abundance relates to the frequency spectrum of environmental variation. Here, we show how to derive and to compute the transfer function for a continuous-time model of a population that is structured by a continuous individual-level state variable such as size. To illustrate, we derive, compute, and analyze the transfer function for a size-structured population model of stony corals with open recruitment, parameterized for a common Indo-Pacific coral species complex. This analysis identifies a sharp multi-decade resonance driven by space competition between existing coral colonies and incoming recruits. The resonant frequency is most strongly determined by the rate at which colonies grow, and the potential for resonant oscillations is greatest when colony growth is only weakly density-dependent. While these resonant oscillations are unlikely to be a predominant dynamical feature of degraded reefs, they suggest dynamical possibilities for marine invertebrates in more pristine waters. The size-structured model that we analyze is a leading example of a broader class of physiologically structured population models, and the methods we present should apply to a wide variety of models in this class.