Combining sources in stable isotope mixing models: alternative methods

Stable isotope mixing models are often used to quantify source contributions to a mixture. Examples include pollution source identification; trophic web studies; analysis of water sources for soils, plants; or water bodies, and many others. A common problem is having too many sources to allow a unique solution. We discuss two alternative procedures for addressing this problem. One option is a priori to combine sources with similar signatures so the number of sources is small enough to provide a unique solution. Aggregation should be considered only when isotopic signatures of clustered sources are not significantly different, and sources are related so the combined source group has some functional significance. For example, in a food web analysis, lumping several species within a trophic guild allows more interpretable results than lumping disparate food sources, even if they have similar isotopic signatures. One result of combining mixing model sources is increased uncertainty of the combined end-member isotopic signatures and consequently the source contribution estimates; this effect can be quantified using the IsoError model (). As an alternative to lumping sources before a mixing analysis, the IsoSource mixing model () can be used to find all feasible solutions of source contributions consistent with isotopic mass balance. While ranges of feasible contributions for each individual source can often be quite broad, contributions from functionally related groups of sources can be summed a posteriori, producing a range of solutions for the aggregate source that may be considerably narrower. A paleohuman dietary analysis example illustrates this method, which involves a terrestrial meat food source, a combination of three terrestrial plant foods, and a combination of three marine foods. In this case, a posteriori aggregation of sources allowed strong conclusions about temporal shifts in marine versus terrestrial diets that would not have otherwise been discerned.     

Getting to the fat of the matter: models,methods and assumptions for dealing with lipids in stable isotope analyses

Within an organism, lipids are depleted in ¹³C relative to proteins and carbohydrates (more negative δ¹³C), and variation in lipid content among organisms or among tissue types has the potential to introduce considerable bias into stable isotope analyses that use δ¹³C. Despite the potential for introduced error, there is no consensus on the need to account for lipids in stable isotope analyses. Here we address two questions: (1) If and when is it important to account for the effects of variation in lipid content on δ¹³C? (2) If it is important, which method(s) are reliable and robust for dealing with lipid variation? We evaluated the reliability of direct chemical extraction, which physically removes lipids from samples, and mathematical normalization, which uses the carbon-to-nitrogen (C:N) ratio of a sample to normalize δ¹³C after analysis by measuring the lipid content, the C:N ratio, and the effect of lipid content on δ¹³C (Δδ¹³C) of plants and animals with a wide range of lipid contents. For animals, we found strong relationships between C:N and lipid content, between lipid content and Δδ¹³C, and between C:N and Δδ¹³C. For plants, C:N was not a good predictor of lipid content or Δδ¹³C, but we found a strong relationship between carbon content and lipid content, lipid content and Δδ¹³C, and between and carbon content and Δδ¹³C. Our results indicate that lipid extraction or normalization is most important when lipid content is variable among consumers of interest or between consumers and end members, and when differences in δ¹³C between end members is <10–12‰. The vast majority of studies using natural variation in δ¹³C fall within these criteria. Both direct lipid extraction and mathematical normalization reduce biases in δ¹³C, but mathematical normalization simplifies sample preparation and better preserves the integrity of samples for δ¹⁵N analysis.     

Estimating resource preferences of a native bumblebee: the effects of availability and use–availability models on preference estimates

Identifying resource preference is considered essential for developing targeted conservation plans but, for many species, questions remain about the best way to estimate preference. Resource preferences for bees are particularly difficult to determine as the resources they collect, nectar and pollen, are challenging to estimate availability and collection. Resources are traditionally measured at the flower or inflorescence level, but these measures of availability do not correspond to the resources actually used by bees. Additionally, it is unclear as to whether common models including availability are appropriate for bees which may target resources regardless of available quantities. Here we first compare two common hypotheses of resource use – the ‘random use hypothesis’ and the ‘linear preferences hypothesis’ – using three different measures of availability (pollen, flower and inflorescence) – to determine if one measure of availability was better for understanding bee pollen use. Next, the superior model using availability was compared to a novel model of bee pollen use the ‘target use hypothesis’. This model assumes that bees target some resources regardless of how much of each resource is available (but assuming resources are present at a site), and thus models preference without availability data. Of the models including availability, the linear preference model using inflorescence availability best explained the pollen use data. This suggests that bumblebee pollen use is non‐random and that cues to identify and locate resources (i.e. display size and quantity) may be more important than the quantity of the resource available (i.e. pollen availability). Additionally, in most cases the target use model explained the data equal to or better than the other models suggesting bee resource use may be better modeled without measured availability data compared to linear models. These results could be important for expanding resource use analysis of bees that are difficult to quantify availability.     

Effects of resource availability on tolerance of herbivory: a review and assessment of three opposing models

Although it is widely acknowledged that a plant's tolerance of herbivore damage depends on resource availability in the plant's environment, there is no consensus on whether higher resource levels lead to greater or to lower tolerance. The prevailing model, the compensatory continuum hypothesis (CCH), predicts that tolerance of herbivory should be greater in high-resource or low-competition conditions. The main rival hypothesis, the growth rate model (GRM), makes the opposite prediction: tolerance of herbivory should be greater in more stressful conditions. The tolerance predictions of a recently introduced model, the limiting resource model (LRM), are more flexible and depend on the type of resource and herbivore under consideration. We reviewed 48 studies (from 40 published articles) of plant tolerance of leaf damage in conditions differing in levels of light, inorganic nutrients, water stress, or competition. The results of 31%, 48%, and 95% of the studies were consistent with the predictions of the CCH, GRM, and LRM, respectively. Thus, by considering which resource is primarily affected by herbivory and which resource is limiting a plant's fitness, the LRM offers a substantial advance in predicting how tolerance will be affected by environmental differences in resource availability.     

Improving Bayesian isotope mixing models: a response to Jackson et al. (2009)

We recently described a Bayesian framework for stable isotope mixing models and provided a software tool, MixSIR, for conducting such analyses (Ecol. Lett., 2008; 11 :470). Jackson et al. (Ecol. Lett., 2009; 12:E1) criticized the performance of our software based on tests using simulated data. However, their simulation data were flawed, rendering claims of erroneous behaviour inaccurate. A re‐evaluation of the MixSIR source code did, however, uncover two minor coding errors, which we have fixed. When data are correctly simulated according to eqns  (1)–(4) in Jackson et al. (2009) , MixSIR consistently and accurately estimated the proportional contribution of prey to a predator diet, and was surprisingly robust to additional unquantified error. Jackson et al. (2009) also suggested we use a Dirichlet prior on the source proportion parameters, which we agree with. Finally, Jackson et al. (2009) propose adding additional error parameters to our mixing model framework. We caution that such increases in model complexity should be evaluated based on data support.     

Feeding of tropical freshwater fishes: seasonality in resource availability and resource use

Summary Food resources in the environment and in the diets of small fish inhabiting two water bodies in a tropical savanna were studied during both wet and dry seasons. During the wet season (high water, abundant food) most fish species in both habitats fed predominantly on vegetation-dwelling invertebrates. Most fish species switched to alternative foods (algae and detritus) following the drastic decline in invertebrate food available towards the end of the dry season. In one habitat, this change in diet was accompanied by an increase in the volume of food intake. In the second habitat, only two larger species foraged intensively, while smaller species showed low food intake or almost ceased feeding. These differences may be explained by the high risk of predation for small fish in the second habitat. Dietary overlaps among fish species were high at the end of the dry season and moderate in the wet season. However, critical analysis of such factors as food abundance, the size and number of shared prey, and diet breadth showed that all significant overlaps were ecologically unimportant i.e. there was only weak competition for food.     

Cross-feeding niches among commensal leaf bacteria are shaped by the interaction of strain-level diversity and resource availability

Leaf microbiomes play crucial roles in plant health, making it important to understand the origins and functional relevance of their diversity. High strain-level leaf bacterial genetic diversity is known to be relevant for interactions with hosts, but little is known about its relevance for interactions with the multitude of diverse co-colonizing microorganisms. In leaves, nutrients like amino acids are major regulators of microbial growth and activity. Using metabolomics of leaf apoplast fluid, we found that different species of the plant genus Flaveria considerably differ in the concentrations of high-cost amino acids. We investigated how these differences affect bacterial community diversity and assembly by enriching leaf bacteria in vitro with only sucrose or sucrose + amino acids as possible carbon sources. Enrichments from F. robusta were dominated by Pantoea sp. and Pseudomonas sp., regardless of carbon source. The latter was unable to grow on sucrose alone but persisted in the sucrose-only enrichment thanks to exchange of diverse metabolites from Pantoea sp. Individual Pseudomonas strains in the enrichments had high genetic similarity but still displayed clear niche partitioning, enabling distinct strains to cross-feed in parallel. Pantoea strains were also closely related, but individuals enriched from F. trinervia fed Pseudomonas more poorly than those from F. robusta. This can be explained in part by the plant environment, since some cross-feeding interactions were selected for, when experimentally evolved in a poor (sucrose-only) environment but selected against in a rich (sucrose + amino acids) one. Together, our work shows that leaf bacterial diversity is functionally relevant in cross-feeding interactions and strongly suggests that the leaf resource environment can shape these interactions and thereby indirectly drive bacterial diversity.Subject terms: Microbiome, Plant ecology, Biodiversity, Microbial ecology, Bacterial genomics     

The role of sex ratios and resource availability on the pre-mating behavior of a monogamous fish

Convict cichlid fish (Archocentrus nigrofasciatus) are one of the few monogamous species where the female appears more colorful than the male. We examined whether this sexual dimorphism was reflected in a female-biased actual sex ratio (ASR) and whether convict cichlid females would exhibit behavioral traits typified in monogamous males when they possessed the extravagant colors and/or morphology. Our field observations revealed that females are marginally, but not significantly, more abundant than males but they did initiate more intersexual social interactions (i.e. approach behavior) than males. They also showed more intrasexual aggression. Furthermore, males more commonly chased females rather than vice versa. Our laboratory experiment also indicated that differences in intrasexual aggression were not related to differences in the ASR but did appear with the addition of a breeding site. Thus, while intrasexual competition was more frequently observed between females in the field it was probably related to the availability of breeding sites rather than our estimates of the ASR.     

Parameterization and prediction of community interaction models using stable-state assumptions and computational techniques: an example from the high rocky intertidal

Many ecological communities exist in a stable state where, if undisturbed, no net change will occur in the populations or in the interactions between the component parts of the system. In this paper we present computational methods (evolutionary algorithms and random searches) to parameterize mathematical models that describe communities in stable states. The initial parameterization of the model requires only "best guess" estimates for parameters and can therefore be used in data-poor situations. The technique locates the stable state that occurs with minimum deviation from these parameters. Alternative stable states in which the community may exist after a disturbance event can also be assessed using this technique, even though the number of alternative states may be large. Using available but incomplete data from an intertidal grazer/biofilm community, we created a prediction of the dynamics of both a pre- and post-disturbance community. Using limited data, we then predicted the most likely post-disturbance community, which proved to be a good match to experimental data, indicating the usefulness of this technique as a predictive tool.     

Investigating the interaction between ungulate grazing and resource effects on Vaccinium myrtillus populations with integral projection models

Dense ungulate populations in forest accompanied by high grazing intensities have the potential to affect plant population dynamics, and such herbivory effects on populations are hypothesised to differ along environmental gradients. We investigated red deer grazing and resource interaction effects on the performance and dynamics of the functionally important boreal shrub Vaccinium myrtillus using integral projection models (IPMs). We sampled data from 900 V. myrtillus ramets in 30 plots in two consecutive years across the boreo-nemoral pine forest on the island Svanøy, western Norway. The plots spanned two environmental gradients: a red deer grazing intensity gradient (assessed by Cervus elaphus faecal pellets), and a relative resource gradient (DCA-ordination of species composition). The use of IPMs enabled projections of population growth rate (λ) using continuous plant size instead of forcing stage division upon the demographic data. We used the environmental gradients as continuous variables to explain the dynamics of V. myrtillus populations and found that both increasing grazing intensity and resource levels negatively affected λ of the V. myrtillus populations. Interestingly, these factors interacted: the negative effects of grazing were strongest in the resource-rich vegetation, and higher resource levels reduced λ more strongly than at low resource levels when grazing intensities became higher. Populations with λ > 1 were projected if the grazing intensity was less than or equal to the mean grazing intensity on the island, and indicated that V. myrtillus is relatively tolerant of grazing. Variance decomposing showed that the decrease of λ along the grazing gradient, both at low and high resource levels, was largely caused by reductions in plant growth. The use of IPMs together with important environmental gradients offered novel possibilities to study the synthesised effect of different factors on plant population dynamics. Here, we show that the population response of an abundant boreal shrub to ungulate grazing depends on resource level.     

The role of niche breadth, resource availability and range position on the life history of butterflies

We analysed the relationship between three life history characteristics (mobility, length of flight period and body size) and niche breadth (larval host plant specificity and adult habitat breadth), resource availability (distribution and abundance of host plants) and range position (distance between the northernmost distribution record and southernmost point of Finland) of the butterfly fauna of Finland. The data is based on literature and questionnaires. Often in across species studies phylogeny may create spurious relationships between life-history and ecological variables. We took the phylogenetic relatedness of butterfly species into account by analysing the data with phylogenetically independent contrasts (CAIC method). Butterfly mobility was positively related to the niche breadth, resource availability and range position. The length of the flight period was negatively related to the range position, indicating that the species at the northern edge of their distribution range have shorter flight period than species which are further way from the range edge. After controlling for the phylogenetic relatedness we found no significant correlations between body size and niche breadth, resource availability or range position. We suggest that the relationship between the length of the flight period and range position may arise as a consequence of lower hatching asynchrony in edge species as a result of lower environmental variance in larval growth conditions. Our results on the mobility suggest that there is selection pressure towards lower migration rate in species that have restricted niche breadth, low resource availability and in species that are on the northern edge of their geographical distribution range. In such species, selection against mobile individuals is likely to result from the decreased probability of finding another habitat patch suitable for egg laying.     

Network epidemic models with two levels of mixing

The study of epidemics on social networks has attracted considerable attention recently. In this paper, we consider a stochastic SIR (susceptible-->infective-->removed) model for the spread of an epidemic on a finite network, having an arbitrary but specified degree distribution, in which individuals also make casual contacts, i.e. with people chosen uniformly from the population. The behaviour of the model as the network size tends to infinity is investigated. In particular, the basic reproduction number R(0), that governs whether or not an epidemic with few initial infectives can become established is determined, as are the probability that an epidemic becomes established and the proportion of the population who are ultimately infected by such an epidemic. For the case when the infectious period is constant and all individuals in the network have the same degree, the asymptotic variance and a central limit theorem for the size of an epidemic that becomes established are obtained. Letting the rate at which individuals make casual contacts decrease to zero yields, heuristically, corresponding results for the model without casual contacts, i.e. for the standard SIR network epidemic model. A deterministic model that approximates the spread of an epidemic that becomes established in a large population is also derived. The theory is illustrated by numerical studies, which demonstrate that the asymptotic approximations work well, even for only moderately sized networks, and that the degree distribution and the inclusion of casual contacts can each have a major impact on the outcome of an epidemic.     

Mean reproductive traits of fungal assemblages are correlated with resource availability

Organisms have evolved a fascinating variety of strategies and organs for successful reproduction. Fruit bodies are the reproductive organ of fungi and vary considerably in size and shape among species. Our understanding of the mechanisms underlying the differences in fruit body size among species is still limited. Fruit bodies of saprotrophic fungi are smaller than those of mutualistic ectomycorrhizal fungi. If differences in fruit body size are determined by carbon acquisition, then mean reproductive traits of saprotrophic and ectomycorrhizal fungi assemblages should vary differently along gradients of resource availability as carbon acquisition seems more unpredictable and costly for saprotrophs than for ectomycorrhizal fungi. Here, we used 48 local inventories of fungal fruit bodies (plot size: 0.02 ha each) sampled along a gradient of resource availability (growing stock) across 3 years in the Bavarian Forest National Park in Germany to investigate regional and local factors that might influence the distribution of species with different reproductive traits, particularly fruit body size. As predicted, mean fruit body size of local assemblages of saprotrophic fungi was smaller than expected from the distribution of traits of the regional species pool across central and northern Europe, whereas that of ectomycorrhizal fungi did not differ from random expectation. Furthermore and also as expected, mean fruit body size of assemblages of saprotrophic fungi was significantly smaller than for assemblages of ectomycorrhizal species. However, mean fruit body sizes of not only saprotrophic species but also ectomycorrhizal species increased with resource availability, and the mean number of fruit bodies of both assemblages decreased. Our results indicate that the differences in carbon acquisition between saprotrophs and ectomycorrhizal species lead to differences in basic reproductive strategies, with implications for the breadth of their distribution. However, the differences in resource acquisition cannot explain detailed species distribution patterns at a finer, local scale based on their reproductive traits.     

The mechanistic basis of discrete-time population models: The role of resource partitioning and spatial aggregation

The purpose of this paper is to present a unified view to understand mechanistic basis of various discrete-time population models from the viewpoints of resource partitioning and spatial aggregation of individuals. A first-principles derivation is presented of a new population model which incorporates both scramble and contest competition using a site-based framework in which individuals are distributed over discrete resource sites. The derived model has parameters relating to the way of resource partitioning and the degree of spatial aggregation of individuals, respectively. The model becomes various population models in various limits in these parameters. This model thus provides a unified view to understand how various population models are interrelated. The dependence of the stability of the model on the parameters is also examined.