Parameter estimation techniques for interaction and redistribution models: a predator-prey example

Summary The use of parameter estimation techniques for partial differential equations is illustrated using a predatorprey model. Whereas ecologists have often estimated parameters in models, they have not previously been able to do so for models that describe interactions in heterogeneous environments. The techniques we describe for partial differential equations will be generally useful for models of interacting species in spatially complex environments and for models that include the movement of organisms. We demonstrate our methods using field data from a ladybird beetle (Coccinella septempunctata) and aphid (Uroleucon nigrotuberculatum) interaction. Our parameter estimation algorithms can be employed to identify models that explain better than 80% of the observed variance in aphid and ladybird densities. Such parameter estimation techniques can bridge the gap between detail-rich experimental studies and abstract mathematical models. By relating the particular bestfit models identified from our experimental data to other information on Coccinella behavior, we conclude that a term describing local taxis of ladybirds towards prey (aphids in this case) is needed in the model.     

Community stability: effects of limpet removal and reintroduction in a rocky intertidal community

Summary The stability of a high rocky intertidal community was assessed in a controlled field experiment in which the most common consumers, limpets, were temporarily removed. Compared to the unmanipulated plots, the exclusion plots developed greater algal abundance and altered species composition of both algae and barnacles. The community was not perturbed beyond its capacity to recover, since the community structure of the limpet-removal plots converged on the structure of the unmanipulated plots following limpet reintroduction. Different components of the community recovered at different rates, depending on whether or not the species had a size-related escape from the limpets. Algae had no size-related escape from limpets. The difference in algal abundance between manipulated and unmanipulated plots lasted less than six months after limpet reintroduction. Barnacles, however, had a size-related escape from limpets and therefore recovered more slowly. The difference in barnacle species composition between the perturbed and unperturbed plots lasted for 17 months after limpet reintroduction. The length of the limpet removal period (16 or 28 months) did not appear to affect the rate of community recovery.     

Parameterization of mechanistic models from qualitative data using an efficient optimal scaling approach

Quantitative dynamical models facilitate the understanding of biological processes and the prediction of their dynamics. These models usually comprise unknown parameters, which have to be inferred from experimental data. For quantitative experimental data, there are several methods and software tools available. However, for qualitative data the available approaches are limited and computationally demanding. Here, we consider the optimal scaling method which has been developed in statistics for categorical data and has been applied to dynamical systems. This approach turns qualitative variables into quantitative ones, accounting for constraints on their relation. We derive a reduced formulation for the optimization problem defining the optimal scaling. The reduced formulation possesses the same optimal points as the established formulation but requires less degrees of freedom. Parameter estimation for dynamical models of cellular pathways revealed that the reduced formulation improves the robustness and convergence of optimizers. This resulted in substantially reduced computation times. We implemented the proposed approach in the open-source Python Parameter EStimation TOolbox (pyPESTO) to facilitate reuse and extension. The proposed approach enables efficient parameterization of quantitative dynamical models using qualitative data.

     

Effects of avian grazing on the algal community and small invertebrates in the rocky intertidal zone

In this study of a rocky intertidal habitat in northern Japan, feeding by avian consumers had significant effects on algal assemblages and small herbivorous invertebrates. The effects of the birds on algae were different from those of invertebrate grazers such as urchins and gastropods. The abundance of the dominant algal species decreased during the grazing period, increased again after the grazing period, and indirectly affected algal species richness and evenness. Avian grazing also decreased the density of tube-dwelling amphipods on the dominant alga, but did not change the density of mobile and free-living isopods. These results suggest that avian grazers may act as habitat modifiers rather than exploitative competitors for the small herbivorous crustaceans. Avian herbivores consumed only the upper parts of large algal fronds, apparently reducing the amount of suitable microhabitat for the small herbivorous crustaceans, which are subject to a variety of physical or biological stress. Thus, avian herbivores function as ecosystem engineers, regulating community structure in a manner different to invertebrate herbivores in rocky intertidal habitats.     

Scale, environment, and trophic status: the context dependency of community saturation in rocky intertidal communities

Our understanding of the relative influence of different ecological drivers on the number of species in a place remains limited. Assessing the relative influence of local ecological interactions versus regional species pools on local species richness should help bridge this conceptual gap. Plots of local species richness versus regional species pools have been used to address this question, yet after an active quarter-century of research on the relative influence of local interactions versus regional species pools, consensus remains elusive. We propose a conceptual framework that incorporates spatial scale and ecological interaction strength to reconcile current disparities. We then test this framework using a survey of marine rocky intertidal algal and invertebrate communities from the northeast Pacific. We reach two main conclusions. First, these data show that the power of regional species pools to predict local richness disintegrates at small spatial scales coincident with the scale of biological interactions, when studying ecologically interactive groups of species, and in generally more abiotically stressful habitats (e.g., the high intertidal). Second, conclusions of past studies asserting that the regional species pool is the primary driver of local species richness may be artifacts of large spatial scales or ecologically noninteractive groups of species.     

Conditional responses of organisms to habitat structure: an example from intertidal mudflats

Habitat structure is often assumed to be a predictor of habitat function. However, habitat structure may be insufficient to predict the functional significance of a habitat if the level of resources in the habitat is a consequence of the interaction between the habitat structure and physical or biological factors. In this study, we investigated whether depressions in tidal flat sediments generated by stout razor clams, Tagelus plebeius, affect the spatial patterns of pit digging by deposit-feeding burrowing crabs, Chasmagnathus granulata. The pits dug by crabs while feeding overlapped with clam siphon holes at a frequency higher than expected at random, and measurements of pit-digging by crabs integrated over several days indicated a higher frequency of feeding in the sediment of depressions. The daily frequency of pit-digging by crabs in depressions was positively related to the organic matter content of their sediments, but was significantly higher than the frequency of pit-digging away from clam siphon holes only after events of high bedload sediment transport, when the organic matter in the sediments of these depressions peaked. This example demonstrates the conditional nature of the relationship between habitat structure and function by illustrating how a physical process—bedload sediment transport—may introduce variation in the function that depressions play as feeding sites for burrowing crabs. Published information suggests that such conditional responses of organisms to habitat structure: (1) occur in a variety of habitats; (2) involve a variety of structures either of biotic or abiotic origin; and (3) are the consequence of either physical or biological controls that vary in importance according to the general mechanism through which habitat structure affect resources. This broad experimental evidence suggests that the accuracy of predictive models linking habitat structure and function can be improved by incorporating a mechanistic perspective that allows recognition of the potential for conditional responses of organisms to habitat structure.     

Merging resource availability with isotope mixing models: the role of neutral interaction assumptions

Background

Bayesian mixing models have allowed for the inclusion of uncertainty and prior information in the analysis of trophic interactions using stable isotopes. Formulating prior distributions is relatively straightforward when incorporating dietary data. However, the use of data that are related, but not directly proportional, to diet (such as prey availability data) is often problematic because such information is not necessarily predictive of diet, and the information required to build a reliable prior distribution for all prey species is often unavailable. Omitting prey availability data impacts the estimation of a predator''s diet and introduces the strong assumption of consumer ultrageneralism (where all prey are consumed in equal proportions), particularly when multiple prey have similar isotope values.

Methodology

We develop a procedure to incorporate prey availability data into Bayesian mixing models conditional on the similarity of isotope values between two prey. If a pair of prey have similar isotope values (resulting in highly uncertain mixing model results), our model increases the weight of availability data in estimating the contribution of prey to a predator''s diet. We test the utility of this method in an intertidal community against independently measured feeding rates.

Conclusions

Our results indicate that our weighting procedure increases the accuracy by which consumer diets can be inferred in situations where multiple prey have similar isotope values. This suggests that the exchange of formalism for predictive power is merited, particularly when the relationship between prey availability and a predator''s diet cannot be assumed for all species in a system.     

Species richness and diversity across rocky intertidal elevation gradients in Helgoland: testing predictions from an environmental stress model

Environmental stress affects species richness and diversity in communities, but the precise form of the relationship is unclear. We tested an environmental stress model (ESM) that predicts a unimodal pattern for total richness and diversity in local communities across the full stress gradient where a regional biota can occur. In 2008, we measured richness and diversity (considering all macrobenthic species) across the entire intertidal range on three rocky shores on Helgoland Island, Germany. Intertidal elevation is known to be positively related to abiotic stress. Since Helgoland is between the northern and southern biogeographic boundaries for the cold-temperate NE Atlantic intertidal biota, it exhibits low stress levels for this biota at low elevations and high stress at high elevations because of long (>6 h) emersion times. Thus, we predicted a unimodal trend for richness and diversity across elevation. On all three shores, richness increased from high to middle elevations, but remained similar between middle and low elevations. Diversity followed the same trend on one shore and different trends (although also non-unimodal) on the other two. Evenness explained the trend differences between richness and diversity. Overall, our study yielded little support for the ESM. Reasons for richness and diversity not decreasing at low elevations may be related to influences of mostly subtidal species, Helgoland’s intertidal range, or sampling resolution. Our study also suggests that the ESM must be developed further to differentiate between richness and diversity. We offer recommendations to improve future ESM research using intertidal systems.     

Detecting the coevolution of biosequences--an example of RNA interaction prediction

A probabilistic graphical model is proposed in order to detect the coevolution between different sites in biological sequences. The model extends the continuous-time Markov process of sequence substitution for single nucleic or amino acids and imposes general constraints regarding simultaneous changes on the substitution rate matrix. Given a multiple sequence alignment for each molecule of interest and a phylogenetic tree, the model can predict potential interactions within or between nucleic acids and proteins. Initial validation of the model is carried out using tRNA and 16S rRNA sequence data. The model accurately identifies the secondary interactions of tRNA as well as several known tertiary interactions. In addition, results on 16S rRNA data indicate this general and simple coevolutionary model outperforms several other parametric and nonparametric methods in predicting secondary interactions. Furthermore, the majority of the putative predictions exhibit either direct contact or proximity of the nucleotide pairs in the 3-dimensional structure of the Thermus thermophilus ribosomal small subunit. The results on RNA data suggest a general model of coevolution might be applied to other types of interactions between protein, DNA, and RNA molecules.     

Anthropogenic harvesting pressure and changes in life history: insights from a rocky intertidal limpet

The importance of large breeding individuals for maintaining the health of marine fish and invertebrate populations has long been recognized. Unfortunately, decades of human harvesting that preferentially remove larger individuals have led to drastic reductions in body sizes of many of these species. Such size-selective harvesting is particularly worrisome for sequentially hermaphroditic species where the larger size classes are composed primarily of one sex. Whether these species can maintain stable sex ratios under sustained harvesting pressure depends on the level of plasticity of their life-history traits. Here, we show that populations of a marine limpet (Lottia gigantea) can adjust a fundamental aspect of their life history (the timing of sex change) when subjected to size-selective harvesting. As predicted by theoretical models, individuals from harvested populations change sex at smaller sizes and grow at slower rates compared to individuals from protected populations. In addition, the relative size at which the change from male to female occurs remains constant (~0.75; size at sex change/maximum size) across populations, regardless of harvesting pressure. Our results show that population-level demographic and life-history data, in conjunction with existing theory, can be sufficient to predict the responses of sequential hermaphrodites to harvesting pressure. Furthermore, they suggest such species can potentially adapt to size-selective harvesting.     

Mobilifilum chasei: Morphology and ecology of a spirochete from an intertidal stratified microbial mat community

Spirochetes were found in the lower anoxiphototrophic layer of a stratified microbial mat (North Pond, Laguna Figueroa, Baja California, Mexico). Ultrastructural analysis of thin sections of field samples revealed spirochetes approximately 0.25 m in diameter with 10 or more periplasmic flagella, leading to the interpretation that these spirochetes bear 10 flagellar insertions on each end. Morphometric study showed these free-living spirochetes greatly resemble certain symbiotic ones, i.e., Borrelia and certain termite spirochetes, the transverse sections of which are presented here. The ultrastructure of this spirochete also resembles Hollandina and Diplocalyx (spirochetes symbiotic in arthropods) more than it does Spirochaeta, the well known genus of mud-dwelling spirochetes. The new spirochete was detected in mat material cellected both in 1985 and in 1987. Unique morphology (i.e., conspicuous outer coat of inner membrane, large number of periplasmic flagella) and ecology prompt us to name a new free-living spirochete.This article is dedicated to the memory of our dear friend and colleague David G. Chase (1935–1987)     

Effects of intensity and seasonal timing of disturbances on a rocky intertidal benthic community on the southern coast of Korea

The effects of intensity and timing of disturbances on recovery of marine benthic organisms were investigated on a rocky intertidal shore in Gwangyang Bay, Korea. We hypothesized that the recovery pattern of the benthic community structure would be affected by disturbance intensity and season. Twenty-eight permanent plots were set up, with disturbance intensity (cleared plots and sterile plots) and seasonal disturbance (fall 1999 and spring 2000) incorporated into the experimental design. To monitor natural seasonal variation in benthic community abundances, we established seven permanent unmanipulated plots. Turf-forming algae were observed in the unmanipulated plots throughout the experimental period, whereas green algae and invertebrate presence varied with season. In the disturbance-intensity experiment, turf-forming and green algae were dominant in cleared plots. The highest coverage of sessile organisms was observed in sterile plots, which exhibited the highest species richness because of their relatively low macroalgal coverage. Seasonal effects of disturbance were an important factor in the recovery pattern of benthic organisms under high disturbance intensity. Coverage of green algae was higher in sterile spring plots than in sterile fall plots; this result was attributed to low spatial competition, as the disturbances occurred just before green algal blooms. On the other hand, the abundances of barnacles and bivalves were highest on sterile fall plots, as these organisms were suppressed by green algal blooms in other periods. These results indicate that the effects of disturbance intensity on benthic community recovery patterns can be influenced by season of disturbance.     

Biological interactions and their role in community structure in the rocky intertidal of Helgoland (German Bight,North Sea)

Over 3 successive seasonal cycles (April 1986 to October 1988), field experiments were established within 3 intertidal levels in the sheltered rocky intertidal of Helgoland (North Sea, German Bight). Competitors for space (Mytilus edulis, macroalgae), herbivores (Littorina spp.) and predators (Carcinus maenas) were either excluded from areas (0.25 m²) covered by undisturbed communities or enclosed at natural densities on areas that were cleared before of animals and plants. All the experimental fields (each 0.25 m²) were covered by cages with 4 mm gauze at the sides and a plexiglas top. The results of the experiments in the upper intertidal (occupied byLittorina spp. andEnteromorpha) showed that a natural density of herbivores could not prevent algal settlement and had only little influence on algal growth. Instead abiotic factors (storms, algae washed ashore) decreased the stock of the green algae. Experiments in the mid intertidal, dominated byMytilus (50% cover),Fucus spp. (20%) and grazingL. littorea (100 ind. m^?2) showed that community structure was directly changed both by grazing periwinkles and by competition for space between mussels and macroalgae. WheneverLittorina was excluded, the canopy ofFucus spp. increased continuously and reached total cover within two years. In addition to the increase ofFucus spp., the rock surface and the mussel shells were overgrown byUlva pseudocurvata, which covered the experimental fields during parts of the summer in the absence of herbivores. As soon as perennial species (fucoids) covered most of the experimental areas, the seasonal growth ofUlva decreased drastically. Presence and growth of macroalgae were also controlled by serious competition for space with mussels. EstablishedMytilus prevented the growth of all perennial and ephemeral algae on the rocks. However, the shells of the mussels provided free space for a new settlement ofFucus andUlva. In the lower intertidal (dominated by total algal cover ofF. serratus, herbivores such asL. littorea andL. mariae, and increasing number of predators such asCarcinus), the feeding activity of herbivores can neither prevent the settlement of the fucoid sporelings nor reduce the growth of macroalgae.F. serratus achieved a total canopy on the rock within one year. Doubled density of herbivores prevented the settlement ofFucus and most of the undercover algae. Predation byCarcinus onLittorina spp. had little influence on the herbivore community patterns. However, the crabs supported the establishment of macroalgae by excluding the mussels from the lower intertidal. In summary, the community organization and maintenance in the mid and lower intertidal is influenced to a high degree by biological interactions. Whereas both the relatively important herbivory byL. littorea and competition for space between mussels and macroalgae dominate in the mid intertidal, predation reaches its highest relative degree of importance for community structure in the lower intertidal.     

Intervertebral reaction force prediction using an enhanced assembly of OpenSim models

OpenSim offers a valuable approach to investigating otherwise difficult to assess yet important biomechanical parameters such as joint reaction forces. Although the range of available models in the public repository is continually increasing, there currently exists no OpenSim model for the computation of intervertebral joint reactions during flexion and lifting tasks. The current work combines and improves elements of existing models to develop an enhanced model of the upper body and lumbar spine. Models of the upper body with extremities, neck and head were combined with an improved version of a lumbar spine from the model repository. Translational motion was enabled for each lumbar vertebrae with six controllable degrees of freedom. Motion segment stiffness was implemented at lumbar levels and mass properties were assigned throughout the model. Moreover, body coordinate frames of the spine were modified to allow straightforward variation of sagittal alignment and to simplify interpretation of results. Evaluation of model predictions for level L1–L2, L3–L4 and L4–L5 in various postures of forward flexion and moderate lifting (8 kg) revealed an agreement within 10% to experimental studies and model-based computational analyses. However, in an extended posture or during lifting of heavier loads (20 kg), computed joint reactions differed substantially from reported in vivo measures using instrumented implants. We conclude that agreement between the model and available experimental data was good in view of limitations of both the model and the validation datasets. The presented model is useful in that it permits computation of realistic lumbar spine joint reaction forces during flexion and moderate lifting tasks. The model and corresponding documentation are now available in the online OpenSim repository.     

Evolutionary dynamics as a component of stage-structured matrix models: an example using Trillium grandiflorum

Evolution by natural selection improves fitness and may therefore influence population trajectories. Demographic matrix models are often employed in conservation studies to project population dynamics, but such analyses have not incorporated evolutionary dynamics. We project evolutionarily informed population trajectories for a population of the perennial plant Trillium grandiflorum, which is declining due to high levels of herbivory by white-tailed deer. Individuals with later flowering times are less often consumed, so there is selection on this trait. We first incorporated selection analyses into a deterministic matrix model in three ways (corresponding to different methods that have been used for analyzing evolution in structured populations). Because it is not clear which of these methods works best for stage-structured models, we compared each with a more realistic, individual-based model. Deterministic models using fitness averaged over the phenotypic distribution gave trajectories that were similar to those of the individual-based model, whereas the deterministic model using fitness at the mean phenotype gave a much faster rate of evolution than that which was observed. This illustrates that subtle differences in the way in which one splices evolution into demographic models can have a large effect on expected outcomes. This study demonstrates that, by combining demographic and selection analyses, one can gauge the potential relevance of evolution to population dynamics and persistence.     

On the prediction of transport properties of ionic liquid using 1-n-butylmethylpyridinium tetrafluoroborate as an example

The Green–Kubo and Einstein–Helfand approaches are examined for calculating diffusion coefficient, electronic conductivity and shear viscosity of ionic liquid using 1-n-butylmethylpyridinium tetrafluoroborate [C₄PY][BF₄] as an example. Both methods suffer numerical errors accumulated at long simulation time, resulting divergences in the integrated autocorrelation time (IAT) and nonlinearity in the mean square displacement (MSD). The numerical errors can be reduced using smaller time step in the simulation. By identifying a converged plateau in IAT and a linear segment in MSD both approaches yield consistent predictions. Using a validated force field, the predicted diffusion coefficient and electrical conductivity agree reasonably well with the experimental data. However, the shear viscosity is significantly underestimated. Analysis of the simulation data indicates that a much slow relaxation in the pressure tensor must be considered, which is unfortunately infeasible due to the accumulated numerical error. Alternatively, the non-equilibrium periodic perturbation method shows promising improvement in the prediction.     

Mechanisms of species coexistence: a field test of theoretical models using intertidal snails

Competitor coexistence is often facilitated by spatial segregation. Traditionally, spatial segregation is predicted to occur when species differ in the habitat in which they are either superior at competing for resources or less susceptible to predation. However, predictions from a behavioural model demonstrate that spatial segregation and coexistence can also occur in the absence of such interspecific trade‐offs in competitive ability and vulnerability to predation. Unlike other models of competitor coexistence this model predicts that when species rank both habitat productivity and ‘riskinesses’ similarly, but differ slightly in their habitat‐specific vulnerabilities to predators, they will tend to segregate across habitats, with the species experiencing the higher ratio of mortality risk across the habitats occurring primarily in the safer habitat. Here, we investigate the hypothesis that intraspecific trade‐offs between resource availability and mortality risk can lead to spatial segregation of competing species by (1) documenting the spatial (i.e. intertidal) distribution of two marine snails, Littorina sitkana and L. subrotundata and (2) performing field experiments to quantify growth and mortality rates of each species at ‘low’ and ‘high’ intertidal heights. Our results indicate that both species agree on the rankings of habitat riskiness and productivity, experiencing higher predation and higher growth in low‐ than in high‐intertidal habitats. However, L. sitkana and L. subrotundata experienced differences in their habitat‐specific mortality risks and growth rates. Despite both species being similarly at risk of predation in high‐intertidal habitats (where mortality was lower), L. subrotundata was subject to significantly higher mortality than L. sitkana at the low‐intertidal height. In contrast, growth rate differences between habitats were greater for L. sitkana than for L. subrotundata. Whereas both species grew at the same rate at the high‐intertidal level (where growth was lower), L. sitkana individuals grew more rapidly than L. subrotundata snails at the low‐intertidal level. As predicted by the behavioural model, the species that experienced the higher ratio of mortality across habitats (i.e. L. subrotundata) occurred exclusively in the safer, high‐intertidal habitat. Taken together, these results provide support for the hypothesis that spatial segregation, and potentially competitor coexistence, can occur in the absence of interspecific trade‐offs in resource acquisition ability or vulnerability to predation.