Size-structured Population Dynamics Model Research Articles

Factors influencing size-structured models’ ability to estimate natural mortality

Natural mortality (M) is crucially important for stock assessment since it strongly influences estimates of spawning stock biomass, MSY and fishing mortality. Variation in M can occur between sexes, maturity stages and temporally. Estimates of M are confounded with those of catchability, recruitment, and growth. Previous studies, which focused on age-structured population dynamics models, suggest that it would be better to estimate M rather than pre-specify it. Size-structured population dynamics models, used for species that are hard to age such as crabs, have unique characteristics such as a size-transition matrix and the possibility of a terminal molt, which could impact the ability to estimate M. Therefore, this study explores the ability of size-structured stock assessment models to estimate M for several scenarios. These scenarios include whether the operating model and estimation method have a terminal molt, whether M is constant, time-varying, or sex- and maturity stage-specific, the quality/sample size of the biomass index and the size-composition data and whether growth is estimated simultaneously with M. The results show that size-structured models can estimate time-varying, sex- and maturity stage-specific M when the estimation method mirrors the operating model. Surprisingly, terminal molt does not affect the ability to estimate M. However, estimating growth simultaneously with M has a negative impact on the ability to estimate M but a positive effect on the quality of the estimates of spawning stock biomass.

Modelling the potential consequences of adaptive closure management in a penaeid trawl fishery

Spatial management of fishing effort can be used to avoid catching undesirable size classes for target species, and improve yield-per-recruit for the exploited stock. Adaptive closure management has been proposed as a means to more effectively utilise spatial management, however these management provisions often lack quantitative evaluation which constrains the information available to inform decisions. We demonstrate the use of a spatially and size structured population dynamics model to evaluate the potential impact of spatial management on a multijurisdictional fishery for a highly migratory species (eastern king prawn, Penaeus [Melicertus] plebejus). Under current conditions in the fishery, the overall effect of closures on harvest was estimated to be comparatively minor, regardless of assumptions about how effort or fisher behavior are affected by spatial management. Alternative assumptions about the movement patterns of eastern king prawn had little influence on the impact of closures on overall harvest. However, when effort was increased to historic levels similar to those observed when the closures were implemented, a much greater impact on overall harvest was observed. The approach taken and simulation outcomes are discussed in the context of spatial management for both eastern king prawn, and penaeid fisheries more broadly.

There is no best method for constructing size-transition matrices for size-structured stock assessments

Abstract Stock assessment methods for many invertebrate stocks, including crab stocks in the Bering Sea of Alaska, rely on size-structured population dynamics models. A key component of these models is the size-transition matrix, which specifies the probability of growing from one size-class to another after a certain period of time. Size-transition matrices can be defined using three parameters, the growth rate (k), asymptotic size (L∞), and variability in the size increment. Most assessments use mark-recapture data to estimate these parameters and assume that all individuals follow the same growth curve, but this can lead to biased estimates of growth parameters. We compared three approaches: the traditional approach, the platoon method, and a numerical integration method that allows k, L∞, or both to vary among individuals, under a variety of scenarios using simulated data based on golden king crabs (Lithodes aequispinus) in the Aleutian Islands region of Alaska. No estimation method performed best for all scenarios. The number of size-classes in the size-transition matrix and how the data are generated heavily dictate performance. However, we recommend the numerical integration method that allows L∞ to vary among individuals and smaller size-class widths.

A model-based approach to incorporate environmental variability into assessment of a commercial fishery: a case study with the American lobster fishery in the Gulf of Maine and Georges Bank

AbstractChanges in bottom-up forcing are fundamental drivers of fish population dynamics. Recent literature has highlighted the need to incorporate the role of dynamic environmental conditions in stock assessments as a key step towards adaptive fishery management. Combining a bioclimate envelope model and a population dynamic model, we propose a model-based approach that can incorporate ecosystem products into single-species stock assessments. The framework was applied to a commercially important American lobster (Homarus americanus) stock in the Northwest Atlantic. The bioclimate envelope model was used to hindcast temporal variability in a lobster recruitment habitat suitability index (HSI) using bottom temperature and salinity. The climate-driven HSI was used to inform the lobster recruitment dynamics within the size-structured population dynamics model. The performance of the assessment model with an environment-explicit recruitment function is evaluated by comparing relevant assessment outputs such as recruitment, annual fishing mortality, and magnitude of retrospective biases. The environmentally-informed assessment model estimated (i) higher recruitment and lower fishing mortality and (ii) reduced retrospective patterns. This analysis indicates that climate-driven changes in lobster habitat suitability contributed to increased lobster recruitment and present potential improvement to population assessment. Our approach is extendable to other stocks that are impacted by similar environmental variability.

Ontogenetic shifts in predator diet drive tradeoffs between fisheries yield and strength of predator-prey interactions

In some marine ecosystems, overharvesting marine predators has triggered major changes in trophic structure and ecosystem function. However, harvest levels that are deemed sustainable for one species may still lead to unexpected impacts elsewhere in the ecosystem. For example, by imposing an additional source of mortality, even sustainable harvesting can lead to a reduction in the number of large individuals within a population, and this truncation in size structure is typically more severe when the largest, most valuable size classes are targeted. Often small and large individuals within a species differ in important ways, including in what they consume, so a loss in predator-prey interactions could occur even without changes in overall predator biomass. Here we explore whether a truncation in predator size structure alone can reduce or functionally eliminate linkages between predator and prey. For this outcome to occur, a predator’s diet must change as it grows in size. We examined evidence for changes in diets with size among predators in three large marine ecosystems, and used a size-structured population dynamics model to evaluate the extent to which otherwise sustainable fishing results in disproportionate reductions in predation. Modelling suggests that diet shifts occurring late in life history (diet mid-point >45% of the maximum size) led to losses in predation that were more severe than would have been expected from losses in predator biomass. Further, the form of the fishery selectivity was less important than the degree of reduction in biomass within each size class relative to the timing of diet shifts. Empirical diet information demonstrates that piscivores vary widely in their onset to piscivory, and this may buffer the potential impacts of truncation in size structure. However, over half of the piscivores had diet shifts toward specific fish taxa at sizes at or above that which would lead to disproportionate reductions in prey consumption. Information about when and how diets change with predator size could identify ecosystems where harvest may lead to unexpected losses in predator-prey interactions.

Asymptotic behavior of a hierarchical size-structured population model

We study in this paper a hierarchical size-structured population dynamics model with environment feedback and delayed birth process. We are concerned with the asymptotic behavior, particularly on the effects of hierarchical structure and time lag on the long-time dynamics of the considered system. We formally linearize the system around a steady state and study the linearized system by \begin{document} $C_0-{\rm{semigroup}}$ \end{document} framework and spectral analysis methods. Then we use the analytical results to establish the linearized stability, instability and asynchronous exponential growth conclusions under some conditions. Finally, some examples are presented and simulated to illustrate the obtained results.

Long-Time Behavior of Spatially and Size-Structured Population Dynamics with Delayed Birth Process

This paper is devoted to the study of a spatially and size-structured population dynamics model with delay in the birth process. We investigate some dynamical properties of the equation by using [Formula: see text]-semigroup theory and spectral analysis arguments, through which we obtain some sufficient conditions of asymptotical stability, asynchronous exponential growth at the null equilibrium as well as Hopf bifurcation occurring at the positive steady state of the system. Some examples are presented and simulated to illustrate the obtained results.

Competition for light in forest population dynamics: From computer simulator to mathematical model

In this article we build a mathematical model for forest growth and we compare this model with a computer forest simulator named SORTIE. The main ingredient taken into account in both models is the competition for light between trees. The parameters of the mathematical model are estimated by using SORTIE model, when the parameter values of SORTIE model correspond to the ones previously evaluated for the Great Mountain Forest in USA. We see that the best fit of the parameters of the mathematical model is obtained when the competition for light influences only the growth rate of trees. We construct a size structured population dynamics model with one and two species and with spatial structure.

The effects of applying mis-specified age- and size-structured models

Stock assessments estimate biomass and fishing mortality in absolute terms and relative to reference points. Most of the world’s stock assessments are based on age- or size-structured population dynamics models, with few stock assessments directly accounting for both age and size dynamics. However, the life history parameters of fish and invertebrate populations are often functions of both age and size. An operating model that is based on an age- and size-structured population dynamics model is used to evaluate the performance of assessment methods based on age-, size- and age- and size-structured population dynamics models. Variants of the operating model and the assessment methods, which include ‘platoons’ to better represent individual variation in growth, are considered to explore the impact of this source of uncertainty on the performance of stock assessment methods. Simulation experiments based conceptually on the assessment for Pacific cod in the Eastern Bering Sea are used to explore the consequences of applying assessment methods that are mis-specified in terms of the population dynamics model and the way variation in growth is modelled. The age-structured assessment methods perform poorest, most likely because they mis-represent how mortality, due to fishing, is removed from the population in the operating model, but also because of mis-specification of the growth curve. The assessment methods with platoons can outperform those that ignore platoon structure when this is present, but their performance when there are no platoons is such that, overall, simpler assessment methods based on size-structured, or age- and size-structured population dynamics models, appear best. Conducting assessments using multiple model types and selecting the best model based on the lowest AIC was, however, the best approach overall.

Data weighting for tagging data in integrated size-structured models

Increasingly, stock assessments for hard-to-age species such as crabs, prawns, rock lobsters, and abalone are being based on integrated size-structured population dynamics models that are fit to a variety of data sources. These data sources include tagging data to inform growth. Diagnostic statistics and plots have been developed to explore how well integrated population models fit the data types typically used for assessment purposes (index data, size- and age-compositions, and conditional age-at-length data). However, such statistics and plots are not available for tagging data, when these data are used to estimate growth. This paper outlines two diagnostic statistics that can be used to evaluate fits to tagging data, and develops a method based on ‘Francis weighting’ for weighting tagging data in integrated models. For illustration, the methods are applied to Aleutian Islands golden king crab (Lithodes aequispinus) in Alaska, and tiger prawns (Penaeus semisulcatus and P. esculentus) in Australia's Northern Prawn Fishery. Some degree of growth model mis-specification was revealed for P. semisulcatus, and there were conflicts in the data for the tiger prawns. The standard errors for the estimates of mature male biomass for golden king crab were larger when the tagging data were downweighted based on the proposed weighting method. This serves to emphasise that assessments and their interpretations can be impacted by how tagging data are weighted.

Asymptotic analysis of a spatially and size-structured population model with delayed birth process

This paper is devoted to the study of a spatially and size-structured population dynamics model with delayed birth process. Our focus is on the asymptotic behavior of the system, in particular on the effect of the spatial location and the time lag on the long-term dynamics. To this end, within a semigroup framework, we derive the locally asymptotic stability and asynchrony results respectively for the considered population system under some conditions. For our discussion, we use the approaches concerning operator matrices, Hille-Yosida operators, spectral analysis as well as Perron-Frobenius theory. We also do two numerical simulations to illustrate the obtained stability and asynchrony results.

Estimation of size–transition matrices with and without molt probability for Alaska golden king crab using tag–recapture data

Size-structured population dynamics models are used for stock assessments of hard to age invertebrate species, such as crabs, and size–transition matrices play an important role in modeling growth in those models. Crabs grow by molting and then incrementing in size. Therefore, the size–transition matrix should ideally contain sub-models for the probability of molt and the growth increment. Size–transition matrices were estimated in an integrated model setting, which included tagging data. Various models, including those that explicitly model molt probability and that include the molt implicitly in the size–transition model, were applied to data for golden king crab (Lithodes aequispinus) in the eastern Aleutian Islands region. Several diagnostic statistics (e.g., covariance matrix, likelihood, AIC, mean growth increment metric, sensitivity of estimates of mature male biomass, and the fits to the tag recapture, catch-per-unit effort, and length frequency data) were used to investigate the implications of the way growth was modeled. Overall, the fit of the integrated model that included the molt probability sub-model was better than that of the model that did not include this sub-model. However, the trends in key stock assessment outputs did not differ markedly between approaches for modeling growth even though the estimates of the size–transition matrices differed.

Estimating growth within size-structured fishery stock assessments: What is the state of the art and what does the future look like?

Most growth studies have considered the relationship between age and growth. Such relationships are essential to age-structured fishery stock assessments. In contrast, assessments based on stage-structured population dynamics models require information on the probability of animals moving from one stage to each of the other stages at each time-step. Size-structured population dynamics models are a special case of stage-structured population dynamics models in which each class represents a unique set of sizes. These models form the basis of assessments for many valuable, hard to age, marine species, including crabs, abalone, lobsters and prawns. Growth within these models is governed by a size-transition matrix. The values for the parameters of a size-transition matrix can either be estimated externally to the assessment, generally utilizing data from tag-recapture experiments, or the estimation of growth can be integrated within the assessment model. This paper reviews the approaches used to construct size-transition matrices, including the underlying structural formulation, statistical estimation framework, and the consequences of error when specifying these matrices on the ability to estimate population size and manage populations sustainably.

A size-structured population dynamics model of Daphnia

The stability of a size-structured population dynamics model of Daphnia coupled with the dynamics of an unstructured algal food source is investigated for the case where there is also an inflow of newborns from an external source. We determine the steady states and study the stability of the nontrivial steady states. We also identify a demographic-algae parameter that determines a condition for the stability.

Calculating optimal effort and catch trajectories for multiple species modelled using a mix of size-structured, delay-difference and biomass dynamics models

A framework is described whereby effort levels and their associated catches consistent with maximizing the net present value (NPV) of fishery profits over time can be calculated when each harvested species is modelled using a different population dynamics model. Results are presented based on three species ( Penaeus semisulcatus, P. esculentus, and Metapenaeus endeavouri) in Australia's Northern Prawn Fishery and three population dynamics models (size-structured, delay-difference, and biomass dynamics). The results indicate that there is a considerable between-model variation in key model outputs such as the catch predicted for 2010 and the estimated future long-term catches corresponding to maximum economic yield. This variation is comparable with that due to uncertainty about economic parameters when all species are modelled using a size-structured population dynamics model, highlighting the importance of both good population dynamics models and accurate economic parameter inputs. The results also highlight some of the implications (in terms of estimating effort and catch levels which maximize NPV) of not having sufficient data when using population dynamics models to explicitly represent some of the species caught in a multi-species fishery.

Performance of methods for estimating size - transition matrices using tag - recapture data

Management advice for hard-to-age species such as prawns, crabs and rock lobsters are usually based on size-structured population dynamics models. These models require a size–transition matrix that specifies the probabilities of growing from one size-class to the others. Many methods exist to estimate size–transition matrices using tag–recapture data. However, they have not been compared in a systematic way. Eight of these methods are compared using Monte Carlo simulations parameterised using the data for the tiger prawn (Penaeus semisulcatus). Four of the methods are then applied to tag–recapture data for three prawn species in Australia’s Northern Prawn Fishery to highlight the considerable sensitivity of model outputs to the method for estimating the size–transition matrix. The simulations show that not all methods perform equally well and that some methods are extremely poor. The ‘best’ methods, as identified in the simulations, are those that allow for individual variability in the parameters of the growth curve as well as the age-at-release. A method that assumes that l∞ rather than k varies among individuals tends to be more robust to violations of model assumptions.

A semi-empirical growth estimation method for matrix models of endangered species

Matrix models are critical for conservation planning of endangered species or any species with limited data. Sufficient growth data to construct growth-transition matrices required for size-structured population dynamics models may be lacking using traditional methods. We present a simple semi-empirical method for converting limited growth data into estimated transition probabilities required as elements in structured matrix models. Rather than approximating transition probabilities by counting actual transition frequencies between sparsely populated size classes, we assume that a selected function represents the entire data set, we obtain the model parameters by conventional curve fitting, and we construct the matrix model from the assumed model function. To illustrate the method, we use a sparse, scattered sample of growth data from the endangered white abalone. We use the slope and intercept of the von Bertalanffy model function to determine the growth-transition matrix elements, where the paucity and or scatter of the data preclude using the traditional counting method. The method we propose can accommodate both linear and non-linear mappings of size into growth rate, as we demonstrate with a Gaussian function which has been used to model growth of red abalone and red sea urchins. We illustrate how our method can convert confidence intervals from the model function into confidence intervals for the matrix elements. We suggest that this modelling procedure, which is simple to use and is suitable in data poor situations, will be broadly applicable for conservation practitioners in developing quantitative models to evaluate the population viability of endangered species.

Assessing a stock assessment framework for the green sea urchin Strongylocentrotus drobachiensis fishery in Maine, USA

A Bayesian stock assessment framework with a size-structured population dynamics model used to assess the green sea urchin, Strongylocentrotus drobachiensis, fishery in Maine, USA was evaluated, using a simulation approach, for its performance in describing sea urchin population dynamics under different recruitment dynamics and data quality. This study suggests that the current stock assessment model performs well in estimating key sea urchin fishery parameters such as exploitable stock biomass, total stock biomass, natural mortality, and fishing mortality under different simulation scenarios, and can capture the dynamics of the Maine sea urchin population. The recruitment dynamics of the sea urchin are likely to vary greatly with large changes occurring in its ecosystem. The finding that the current assessment framework is able to capture different patterns of recruitment dynamics implies that the current assessment framework will remain effective in future stock assessments of the Maine sea urchin fishery.

Size-structured population dynamics models and their numerical solutions

This paper presents a review of the numerical methods for the solution of the size-structured population balance models. The methods are compared with regards to accuracy, efficiency, generality and mathematical methodology.