Size-spectrum Model Research Articles

The theoretical foundations for size spectrum models of fish communities

Size spectrum models have emerged from 40 years of basic research on how body size determines individual physiology and structures marine communities. They are based on commonly accepted assumptions and have a low parameter set, making them easy to deploy for strategic ecosystem-oriented impact assessment of fisheries. We describe the fundamental concepts in size-based models about food encounter and the bioenergetics budget of individuals. Within the general framework, three model types have emerged that differ in their degree of complexity: the food-web, the trait-based, and the community models. We demonstrate the differences between the models through examples of their response to fishing and their dynamic behavior. We review implementations of size spectrum models and describe important variations concerning the functional response, whether growth is food-dependent or fixed, and the density dependence imposed on the system. Finally, we discuss challenges and promising directions.

An evaluation of implementing long-term MSY in ecosystem-based fisheries management: Incorporating trophic interaction, bycatch and uncertainty

Maximum sustainable yield (MSY), often defined in relation to a single species considered in isolation, has long been considered a cornerstone in fisheries management. It is difficult to expand the concept of MSY for the purpose of ecosystem-based fisheries management (EBFM). Here we consider MSY for a fishery in a multispecies fish community by addressing issues of trophic interaction and bycatch, along with parameter uncertainty. A size-spectrum model, which fully considers trophic interactions within the fish community, was used for simulating various fisheries scenarios including a single species fishery, a multispecies stow-net fishery, and a multispecies trawl fishery. Population biomasses, yields, and ecological indicators were used to assess the dynamics of the fishery and fish community status. The single species fishery with no bycatch resulted in a low impact on non-target stocks, but led to the collapse of the target stock at low fishing mortality rates. The stow-net and trawl fisheries had broader ecosystem impacts, but allowed target stocks to be fished at much higher fishing mortality rates with higher yields. Ecological indicators were related non-linearly to fishing mortality, possibly diminishing their effectiveness as management tools. Background resource carrying capacity was found to be a greater source of uncertainty than reproductive efficiency. This study demonstrates that bycatch mortality can play an important role in trophic interactions likely via predation release and depensation, and thus influence the resilience of fisheries to fishing pressure. The study indicates that the combination of mixed fisheries and multispecies effects lead to complex fish community dynamics that may present additional challenges for fisheries management.

Assessing uncertainty of a multispecies size-spectrum model resulting from process and observation errors

AbstractEcosystem models, specifically multispecies dynamic models, have been increasingly used to project impacts of fishing activity on the trophodynamics of ecosystems to support ecosystem-based fisheries management. Uncertainty is unavoidable in modelling processes and needs to be recognized and properly quantified before models are utilized. Uncertainty was assessed in this study for a multispecies size-spectrum model that quantifies community structure and ecological characteristics. The uncertainty was assumed to result from errors in fish life-history and metabolic scale parameters, environmental variability, fishing variability, and sampling errors. Given the same level of imprecision, metabolic scale parameters had the dominant influence on the uncertainty of the size spectrum modelling results, followed by life-history parameters. Both types of errors led to “scenario uncertainty”, suggesting the possible existence of alternative states of community structure. Environmental variability, fishing variability, and observation errors resulted in “statistical uncertainty”, implying that such uncertainty can be described adequately in statistical terms. The results derived from such a simulation study can provide guidance for identifying research priorities to help narrow the gap in scientific knowledge and reduce the uncertainty in fisheries management.

Evaluating targets and trade‐offs among fisheries and conservation objectives using a multispecies size spectrum model

Summary Marine environmental management policies seek to ensure that fishing impacts on fished populations and other components of the ecosystem are sustainable, to simultaneously meet objectives for fisheries and conservation. For example, in Europe, targets for (i) biodiversity, (ii) food web structure as indicated by the proportion of large fish and (iii) fishing mortality rates for exploited species that lead to maximum sustainable yield, FMSY, are being proposed to support implementation of the Marine Strategy Framework Directive. Efforts to reconcile any trade‐offs among objectives need to be informed by knowledge on the consequences of alternate management actions. We develop, calibrate and apply a multispecies size spectrum model of the North Sea fish community to assess the response of populations and the community to fishing. The model predicts species' size distributions, abundance, productivity and interactions and therefore provides a single framework for evaluating trade‐offs between population status, community and food web structure, biodiversity and fisheries yield. We show that the model can replicate realistic fish population and community structure and past responses to fishing. We assess whether meeting management targets for exploited North Sea populations (fishing species at FMSY) will be sufficient to meet proposed targets for biodiversity and food web indicators under two management scenarios (status quo and FMSY). The recovery in biodiversity indicators is 60% greater when fishing populations at FMSY than if status quo (2010) fishing rates are maintained. The probability of achieving a food web target was 60% under both scenarios in spite of major community restructuring revealed by other indicators of community size structure. Synthesis and applications. Our model can be applied to evaluate indicator targets and trade‐offs among fisheries and conservation objectives. There is a significant probability that reductions in fishing mortality below FMSY would be needed in Europe if managers make a binding commitment to a proposed large fish indicator target, with concomitant reductions in fisheries yield.

Modeling particle aggregation using size class and size spectrum approaches

[1] Particle aggregation plays an important role in many marine biogeochemical processes such as determining the vertical flux of particulate material and trace metal scavenging. Models of particle aggregation vary in complexity and in this paper I compare the behavior of a detailed size spectrum model with that of a series of simple, two size class models with different representations of aggregation, all of which have appeared in the literature. The simplest model uses a first-order representation of aggregation kinetics, while two other models have nonlinear representations of aggregation. The simplest model is unable to reproduce the dynamic or steady state behavior of the size spectrum model. Results from the two nonlinear size class models show better agreement with the behavior of the size spectrum model. I find that the mode of aggregation and the dependence of aggregation on particle size are crucial for understanding the differences between the models.

Taylor's law and body size in exploited marine ecosystems

Taylor's law (TL), which states that variance in population density is related to mean density via a power law, and density-mass allometry, which states that mean density is related to body mass via a power law, are two of the most widely observed patterns in ecology. Combining these two laws predicts that the variance in density is related to body mass via a power law (variance-mass allometry). Marine size spectra are known to exhibit density-mass allometry, but variance-mass allometry has not been investigated. We show that variance and body mass in unexploited size spectrum models are related by a power law, and that this leads to TL with an exponent slightly <2. These simulated relationships are disrupted less by balanced harvesting, in which fishing effort is spread across a wide range of body sizes, than by size-at-entry fishing, in which only fish above a certain size may legally be caught.

Trait diversity promotes stability of community dynamics

The theoretical exploration of how diversity influences stability has traditionally been approached by species-centric methods. Here we offer an alternative approach to the diversity–stability problem by examining the stability and dynamics of size and trait distributions of individuals. The analysis is performed by comparing the properties of two size spectrum models. The first model considers all individuals as belonging to the same “average” species, i.e., without a description of diversity. The second model introduces diversity by further considering individuals by a trait, here asymptotic body size. The dynamic properties of the models are described by a stability analysis of equilibrium solutions and by the non-equilibrium dynamics. We find that the introduction of trait diversity expands the set of parameters for which the equilibrium is stable and, if the community is unstable, makes the oscillations smaller, slower, and more regular. The stabilizing mechanism is the variation in growth rate between individuals with the same body size but different trait values.

Ecological drivers of stability and instability in marine ecosystems

A stability analysis of the steady state of marine ecosystems is described. The study was motivated by the approximate invariance of biomass in logarithmic size intervals, which is widely observed in marine ecosystems. This invariance is recovered as the steady state of dynamic models of size spectra, which, unlike traditional species-based models of food webs, explicitly account for the mass gained by an individual organism when it eats a prey item. Little is known about the ecological conditions affecting the stability of the steady state, and a new method is developed to examine this. The results show that stability is enhanced by: (a) decreasing the mean predator-to-prey mass ratio (PPMR), (b) increasing the diet breadth of predators, (c) increasing the strength of intrinsic mortality relative to predation mortality, (d) increasing the biomass conversion efficiency. When perturbed from steady state, size spectra develop a wave-like shape, with an average wavelength especially sensitive to the mean PPMR. These waves move from small to large body size at an average speed which depends on the rate of growth of organisms. In contrast to traditional food web models, stability is enhanced as connectance (diet breadth) increases and as food chain length is increased by reducing the PPMR.

Air bubble entrainment by breaking waves and estimation of the related statistical quantities

Based on the experiments, the energy loss from wave breaking εed, the bubble plume penetration depth zb, the rate of air entrainment Q(z), and the turbulent kinetic energy dissipation rate εT(z) are estimated in this study. As a consequence, a simple bubble size spectrum model N(a,z) is proposed and the rate of energy dissipation due to air bubble entrainment and bubble fragment Hinze scale and bubble size spectrum are investigated under different sea states. It is indicated that the rate of energy dissipation due to air bubble entrainment ranges from 01 to 06 in low and moderate sea states, but may be negligible in the high sea state. Bubble fragment Hinze scale downshifts to the lower-frequency band as wind speed increases. Bubble size spectrum not only depends on the rate of air entrainment, the breaking rate, and the turbulent kinetic energy dissipation rate, but also strongly on the wind and sea state.

Towards end-to-end models for investigating the effects of climate and fishing in marine ecosystems

End-to-end models that represent ecosystem components from primary producers to top predators, linked through trophic interactions and affected by the abiotic environment, are expected to provide valuable tools for assessing the effects of climate change and fishing on ecosystem dynamics. Here, we review the main process-based approaches used for marine ecosystem modelling, focusing on the extent of the food web modelled, the forcing factors considered, the trophic processes represented, as well as the potential use and further development of the models. We consider models of a subset of the food web, models which represent the first attempts to couple low and high trophic levels, integrated models of the whole ecosystem, and size spectrum models. Comparisons within and among these groups of models highlight the preferential use of functional groups at low trophic levels and species at higher trophic levels and the different ways in which the models account for abiotic processes. The model comparisons also highlight the importance of choosing an appropriate spatial dimension for representing organism dynamics. Many of the reviewed models could be extended by adding components and by ensuring that the full life cycles of species components are represented, but end-to-end models should provide full coverage of ecosystem components, the integration of physical and biological processes at different scales and two-way interactions between ecosystem components. We suggest that this is best achieved by coupling models, but there are very few existing cases where the coupling supports true two-way interaction. The advantages of coupling models are that the extent of discretization and representation can be targeted to the part of the food web being considered, making their development time- and cost-effective. Processes such as predation can be coupled to allow the propagation of forcing factors effects up and down the food web. However, there needs to be a stronger focus on enabling two-way interaction, carefully selecting the key functional groups and species, reconciling different time and space scales and the methods of converting between energy, nutrients and mass.

Assessing the status of shallow lakes using an additive model of biomass size spectra

Abstract Planktonic biomass size spectra were used to summarize the ecological quality of six shallow lakes sampled in spring, early summer and late summer. A simple additive model fitted to the data was used to assess the applicability of the size spectrum theory to shallow lake ecosystems. The additive model replicated the hierarchical pattern of biomass predicted by the predator–prey theory of aquatic production, and was a more appropriate model for predicting biomass size spectra than the frequently used linear regression. Lakes with varying ecological quality were a significant source of variation in the additive model, and further research into using size spectra to monitor ecological quality in shallow lakes is warranted. Specifically, the production of size spectra from a wider range of sites is needed to provide greater statistical validation. The use of size spectra can provide an attractive and cost‐effective way for classifying lake ecosystems because it circumvents the need for difficult taxonomic description. Copyright © 2007 John Wiley &amp; Sons, Ltd.

Using zooplankton biomass size spectra to assess ecological change in a well-studied freshwater lake ecosystem: Oneida Lake, New York

We explored the sensitivity of three descriptors of zooplankton size spectra (slope, periodic, and Pareto II models) to environmental changes in Oneida Lake, New York, and then used documented environmental changes to model the responses of zooplankton biomass using a general linear model. Using multiple regressions, we identified significant ecological events in Oneida Lake that could affect zooplankton biomass before actual model testing and assessed the three size spectrum models based on their sensitivity to these known variables. The intercept of the slope (linear regression) model was responsive to changes, but the slope was not. The periodic (quadratic) model showed no sensitivity in detecting ecological change. The Pareto II model (probability distribution function) demonstrated the most sensitivity to all ecological variables but was complex to model and there was no direct relationship between its parameters and biological events. The general linear model regression approach proved relatively sensitive to environmental change and had the added benefit of providing a graphical means of biologically assessing differences between years. In general, zooplankton biomass size spectra were responsive to changes in conditions in the Oneida Lake environment, and we believe that size-based approaches have potential as a biotic index in freshwater lake ecosystems.

Fish biomass size spectra in Chesapeake Bay

Biomass size spectra of pelagic fish were modeled to describe community structure, estimate potential fish production, and delineate trophic relationships in Chesapeake Bay. Spectra were constructed from midwater trawl collections each year in April, June–August, and October 1995–2000. The size spectra were bimodal: the first spectral dome corresponded to small zooplanktivorous fish, primarily bay anchovyAnchoa mitchilli; the second dome consisted of larger fish from several feeding guilds that are supported by multiple prey-predator linkages. Annual production estimates of pelagic fish, derived from a mean production to biomass ratio, varied nearly three-fold, ranging from 162 × 109 kcal (125 × 103 tons) in 1996 to 457 × 109 kcal (352 × 103 tons) in 2000. Seasonally, the biomass level and mean individual sizes of fish in the first dome increased from April to October, while the biomass level of the second dome was relatively stable. Regionally, biomass levels in the second dome were higher than biomasses in the first dome for the upper and lower Bay, but were minimal in the middle Bay where seasonal and episodic hypoxia occurs. To test a benthic-pelagic coupling hypothesis that could explain the higher biomass in the second domes for the lower and upper Bay, a cyclic size-spectrum model was fit that included only species in the zooplanktivorous-piscivorous fish guilds. The mean, normalized slope equaled −1, indicating that zooplanktivorous fish may support piscivore production, but that a benthic-pelagic linkage is required to fully support fish production in the second dome. Interannual variability in slopes and intercepts of modeled size spectra was related to salinity, recruitment level of bay anchovy, and the primary axis of a correspondence analysis (salinity effect) on fish community structure. The spectral slope and intercept of normalized spectra were lowest in 1996, a near-record wet year. Results suggest that fish size spectra can be developed as useful indicators of ecosystem state and response to perturbations, especially if prey-predator relationships are explicitly represented.

A continuous model of biomass size spectra governed by predation and the effects of fishing on them

A new time-dependent continuous model of biomass size spectra is developed. In this model, predation is the single process governing the energy flow in the ecosystem, as it causes both growth and mortality. The ratio of predator to prey is assumed to be distributed: predators may feed on a range of prey sizes. Under these assumptions, it is shown that linear size spectra are stationary solutions of the model. Exploited fish communities are simulated by adding fishing mortality to the model: it is found that realistic fishing should affect the curvature and stability of the size spectrum rather than its slope.

Linking size-based and trophic analyses of benthic community structure

Models of biomass size spectra assume that organisms with higher body mass feed at higher trophic levels, but explicit empirical tests of this pattern are rare. We used nitrogen stable iso- topes (δ 15 N) as an index of the trophic level in a benthic fish and invertebrate size-spectrum, and demonstrated that body mass was positively and significantly related to trophic level. This pattern was consistent with the expectation that predator-prey relationships led to powerful size-based trophic structuring in marine communities and ecosystems. Further analysis of intra- and interspecific relationships between body mass and trophic level in the community showed that increases in trophic level across the size spectrum were predominantly a consequence of intra-specific increases in trophic level with body mass and not a consequence of larger species (species with greater maximum body mass) feeding at higher trophic levels. We confirmed the absence of strong inter-specific rela- tionships between maximum body mass and trophic level with cross-species and phylogenetic com- parative approaches. Size-based models are easier and cheaper to parameterise than most food-web models. Subject to the persistence of relationships between body mass and trophic level in space and time, our results suggest that size spectra could be parameterised with body mass-trophic level rela- tionships and used to describe the trophic structure of some marine communities and ecosystems.

A simple analytical model of aerosol properties with account for hygroscopic growth: 2. Scattering and absorption coefficients

A new microphysical model of aerosol size spectra accounting for hygroscopic growth is used to calculate the aerosol scattering and absorption coefficients of solar and longwave radiation. Dependence of the extinction coefficient is explored theoretically, starting from the Junge [1963] power spectra of the dry aerosol and the Kohler equation [Khvorostyanov and Curry, this issue]. It is shown that the general dependence of the aerosol scattering and absorption coefficients on λ and H can be presented in the form σ∼λ−γ(1 − H)R. This model explains both the empirical angstrom wavelength law and Kasten [1969] Hanel [1976] humidity law and identifies a functional relationship between them. The empirical coefficients of these laws are related to the aerosol microstructure and to each other by a very simple relationship. An explanation is offered for the spectral behavior of the scattering coefficient and its variation with relative humidity for various types of atmospheric hazes. It is shown that at very high humidities a dense haze with visibility less than 1 km may form under subsaturated conditions and without activated droplets. Application of this model to the longwave spectrum allows us to explain quantitatively the negative temperature dependence of atmospheric absorption in the atmospheric window without depending on the presence of water vapor dimers. Recommendations for aerosol optical and microphysical measurements are given. This simple model can be used as a tool in remote sensing for evaluation of aerosol scattering and absorption optical thicknesses and for parameterization of aerosol optical properties in general circulation and cloud models.

A simple analytical model of aerosol properties with account for hygroscopic growth: 1. Equilibrium size spectra and cloud condensation nuclei activity spectra

A microphysical model of aerosol size spectra that includes hygroscopic growth is presented. In the presence of atmospheric humidity, an aerosol size spectrum of the Junge [1952] type transforms into a superposition of two equilibrium power law spectra with indices determined by the original Junge index and the fraction of soluble material contained in the aerosol. The aerosol size distributions are presented as explicit functions of relative humidity H for three different humidity regimes: (1) moderate humidity, (2) high humidity but subsaturation, and (3) interstitial cloud aerosol. On the basis of this model of aerosol hygroscopic growth, the Twomey [1959] cloud condensation nuclei (CCN) power law is then derived so that its coefficients are related to the aerosol microstructure and humidity dependence under subsaturation. This simple aerosol model can be used in cloud and climate models that do not treat explicitly CCN activation and for evaluation of aerosol optical properties. An application for aerosol optical properties is considered by Khvorostyanov and Curry [this issue].

Application of a Biomass Size Spectrum Model to Demersal Fish Data from the Scotian Shelf

The distribution of biomass density over organism body size follows an orderly pattern in aquatic ecosystems. A recent theoretical model of such aquatic biomass size spectra predicts an overall primary slope across all trophic groupings in an ecosystem, attributable to physiological causes, and a set of secondary slopes of the biomass size spectrum, thought to be related to predator-prey interactions. Inter-annual variability in demersal fish biomass size spectra from 1970–1991 on the Scotian Shelf was assessed using fits of secondary slope derived from this model. The model was able to pick out 1976 as an anomalous year, when squid abundance on the Scotian Shelf was very high. Similarly, it was shown that the fitted slope of the curve was stable after 1980, a fact that may reflect fisheries management practices implemented in the late 1970s. How the fitted slopes to dome shaped biomass size spectra reflect the food resources available on the Scotian Shelf is discussed.

Models of aquatic biomass size spectra and the common structure of their solutions

Mathematical models of aquatic biomass body-size spectra are usually interpreted as single-valued functions of the body weight of organisms. Where these spectra are recognized as dependent on predator-prey energy transfers, it is shown that solutions of the models consist of a smoothly varying or linear part plus a neglected but not negligible part that is periodic in the logarithm of body weight. The models can also be interpreted as yielding multiple spectra, in which the spectrum for a given trophic position is replicated at regular intervals over the range of body weight. In either interpretation the models support empirical data exhibiting significant quasi-periodic departures from a smoothly varying or straight line, without implying a priori knowledge of underlying mechanisms. Under some conditions the biomass size spectrum in an ecosystem describable by a multiple spectra ecosystem model may be indistinguishable from one characterized by a single-valued spectrum.

A Review of Methods for Prediction of Potential Fish Production with Application to the Great Lakes and Lake Winnipeg

Methods for estimating fish production in aquatic ecosystems range from simple empirically derived estimators, such as morphoedaphic indices, to complex ecosystem simulation models. As first-order estimators, the former are attractive to managers because they are simple and relatively inexpensive to apply and interpret. Application of the latter group has been limited because many of the data inputs are difficult and expensive to obtain. Between these extremes are several models, such as the biomass–size spectrum model, that provide useful information for moderate expenditures of time and effort. Existing and new methods are reviewed in the light of production theory and several are applied to Great Lakes and Lake Winnipeg data. Eight empirical models derived from limnological variables were selected from the literature and used to estimate potential fish yield for the Great Lakes and Lake Winnipeg. The models predicted a fairly narrow range of potential yields, but when compared with historic yields, none was consistent for all lakes. The best overall empirically derived estimator of potential yield in the Great Lakes was the morphoedaphic index. Potential fish production estimated from invertebrate production with Borgmann's biomass – size spectrum model was considerably greater than historic yields or the yield estimates from the empirical models. In a third approach, we calculated life history parameters for "small" and "large" fish in the Great Lakes and combined these with Borgmann's production model, empirical information on population production/biomass ratios from the literature, and classical population dynamics theory to estimate potential production and optimum sustained yield for each group. Historic sustained yield, as a percentage of optimum sustained yield, varied from a low of 6 for "small" Lake Ontario fish to 100 for "large" Lake Erie fish.