Size-structured Population Research Articles

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.

Abstract linear partial differential equations related to size-structured population models with diffusion

We study abstract linear partial differential equations in Banach spaces and/or Banach lattices related to size-structured population models with spatial diffusion and their dual problems. We introduce mild solutions through semigroup theory and characteristic method and investigate differentiability of mild solutions. Existence of a unique mild solution is shown. Also, a comparison result is obtained as well as the boundedness of mild solutions is investigated in the Banach lattice setting. Furthermore, we consider the dual problems, and then we introduce weak solutions and establish their uniqueness.

On the uniqueness of a positive stationary state in the dynamics of a population with asymmetric competition

For a nonlinear model of the dynamics of a size-structured (exploited) population with asymmetric form of competition, we prove a uniqueness theorem for a positive stationary solution under sufficiently general assumptions on the parameters of the model.

Global existence of solutions for a nonlinear size-structured population model with distributed delay in the recruitment

In this paper, we study a nonlinear size-structured population model with distributed delay in the recruitment. The delayed problem is reduced into an abstract initial value problem of an ordinary differential equation in a Banach space by using the semigroup techniques. The local existence and uniqueness of solution as well as the continuous dependence on initial conditions are obtained by using the general theory of quasi-linear evolution equations in Banach spaces, while the global existence of solution is obtained by the estimates of the solution and the extension theorem.

Stability patterns for a size-structured population model and its stage-structured counterpart

In this paper we compare a general size-structured population model, where a size-structured consumer feeds upon an unstructured resource, to its simplified stage-structured counterpart in terms of equilibrium stability. Stability of the size-structured model is understood in terms of an equivalent delayed system consisting of a renewal equation for the consumer population birth rate and a delayed differential equation for the resource. Results show that the size- and stage-structured models differ considerably with respect to equilibrium stability, although the two models have completely identical equilibrium solutions. First, when adult consumers are superior foragers to juveniles, the size-structured model is more stable than the stage-structured model while the opposite occurs when juveniles are the superior foragers. Second, relatively large juvenile (adult) mortality tends to stabilise (destabilise) the size-structured model but destabilise (stabilise) the stage-structured model. Third, the stability pattern is sensitive to the adult–offspring size ratio in the size-structured model but much less sensitive in the stage-structured model. Finally, unless the adult–offspring size ratio is sufficiently small, the stage-structured model cannot satisfactorily capture the dynamics of the size-structured model. We conclude that caution must be taken when the stage-structured population model is applied, although it can consistently translate individual life history and stage-specific differences to the population level.

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.

Sensitivity analysis for a nonlinear size-structured population model

Sensitivity analysis for a nonlinear size-structured population model

Four types of interference competition and their impacts on the ecology and evolution of size-structured populations and communities

We investigate how four types of interference competition – which alternatively affect foraging, metabolism, survival, and reproduction – impact the ecology and evolution of size-structured populations. Even though all four types of interference competition reduce population biomass, interference competition at intermediate intensity sometimes significantly increases the abundance of adult individuals and the population׳s reproduction rate. We find that foraging and metabolic interference evolutionarily favor smaller maturation size when interference is weak and larger maturation size when interference is strong. The evolutionary response to survival interference and reproductive interference is always larger maturation size. We also investigate how the four types of interference competition impact the evolutionary dynamics and resultant diversity and trophic structure of size-structured communities. Like other types of trait-mediated competition, all four types of interference competition can induce disruptive selection and thus promote initial diversification. Even though foraging interference and reproductive interference are more potent in promoting initial diversification, they catalyze the formation of diverse communities with complex trophic structure only at high levels of interference intensity. By contrast, survival interference does so already at intermediate levels, while reproductive interference can only support relatively smaller communities with simpler trophic structure. Taken together, our results show how the type and intensity of interference competition jointly affect coexistence patterns in structured population models.

Generalized sensitivity functions for size-structured population models

Abstract Size-structured population models provide a popular means to mathematically describe phenomena such as bacterial aggregation, schooling fish, and planetesimal evolution. For parameter estimation, a generalized sensitivity function (GSF) provides a tool that quantifies the impact of data from specific regions of the experimental domain. This function helps to identify the most relevant data subdomains, which enhances the optimization of experimental design. To our knowledge, GSFs have not been used in the partial differential equation (PDE) realm, so we provide a novel PDE extension of the discrete and continuous ordinary differential equation (ODE) concepts of Thomaseth and Cobelli and Banks et al. respectively. We analyze a GSF in the context of size-structured population models, and specifically analyze the Smoluchowski coagulation equation to determine the most relevant time and volume domains for three, distinct aggregation kernels. Finally, we provide evidence that parameter estimation for the Smoluchowski coagulation equation does not require post-gelation data.

Population drivers of aThalia democraticaswarm: insights from population modelling

The small salp, Thalia democratica, is known globally to form dense swarms, yet the physiological and oceanographic factors that influence the magnitude and occurrence of these swarms are poorly understood. In this study, two numerical models were used to explore T. democratica population dynamics. A Lefkovitch matrix model identified that the survival of juvenile oozoids was the most important population metric promoting population growth and a size-structured population model was developed to further explore the dynamics of T. democratica. The size-structured model tracks cohorts of four life stages and incorporates size-dependent reproduction and mortality. Model outputs of average generation time and mean abundances of each life stage corresponded well to previously reported values. Factors that promote juvenile oozoid abundance were the most sensitive parameters influencing salp abundance. A 10-year time-series simulation using this population model identified that salp abundances in the Tasman Sea are proportionally higher in winter and spring, consistent with previous studies. This model shows that temperature and phytoplankton concentration are sufficient drivers of large-scale patterns of salp abundance across season and latitude. These results demonstrate the importance of future research focusing on identifying the environmental conditions that influence the size at which females reproduce.

Study on the efficiency in the numerical integration of size-structured population models: Error and computational cost

We describe a procedure which is useful to select an appropriate numerical method in a size-structured population model. We consider four different numerical methods based on finite difference schemes or characteristics curves integration. We compute an analytical approximation in terms of the discretization parameters for the theoretical error principal terms and the computational cost. Thus, we show the efficiency curve that allows to select the best relationship between the discretization parameters for each numerical method. Finally, we obtain the most efficient numerical method for each test.

Stationary Regime of Exploitation of Size-Structured Population with Hierarchical Competition

For a given exploitation intensity of size-structured population with asymmetric competition form we prove the existence of a nontrivial stationary state in the population dynamics. Bibliography: 7 titles.

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.

One model of size-structured population dynamics

A linear model of size-structured population dynamics is considered. It is described by a linear partial differential equation, namely, by the transport equation. A solution in a constructive form is built for this model under a nonlinear global boundary condition, which has a biological interpretation. The model is of interest for biological applications, in particular, in forestry. The results make it possible, for example, to study the qualitative behavior of solutions of the formulated nonstandard boundary value problem with a global boundary condition.

Optimal exploitation of two competing size-structured populations

For a model describing the dynamics of two competing size-structured populations under chosen intensities of their exploitation, the existence and uniqueness of a stationary solution is proved. It is shown that there exist exploitation intensities that maximize a given profit functional on the stationary solution corresponding to them.

Fisheries-induced disruptive selection.

Commercial harvesting is recognized to induce adaptive responses of life-history traits in fish populations, in particular by shifting the age and size at maturation through directional selection. In addition to such evolution of a target stock, the corresponding fishery itself may adapt, in terms of fishing policy, technological progress, fleet dynamics, and adaptive harvest. The aim of this study is to assess how the interplay between natural and artificial selection, in the simplest setting in which a fishery and a target stock coevolve, can lead to disruptive selection, which in turn may cause trait diversification. To this end, we build an eco-evolutionary model for a size-structured population, in which both the stock׳s maturation schedule and the fishery׳s harvest rate are adaptive, while fishing may be subject to a selective policy based on fish size and/or maturity stage. Using numerical bifurcation analysis, we study how the potential for disruptive selection changes with fishing policy, fishing mortality, harvest specialization, life-history tradeoffs associated with early maturation, and other demographic and environmental parameters. We report the following findings. First, fisheries-induced disruptive selection is readily caused by commonly used fishing policies, and occurs even for policies that are not specific for fish size or maturity, provided that the harvest is sufficiently adaptive and large individuals are targeted intensively. Second, disruptive selection is more likely in stocks in which the selective pressure for early maturation is naturally strong, provided life-history tradeoffs are sufficiently consequential. Third, when a fish stock is overexploited, fisheries targeting only large individuals might slightly increase sustainable yield by causing trait diversification (even though the resultant yield always remains lower than the maximum sustainable yield that could be obtained under low fishing mortality, without causing disruptive selection). We discuss the broader implications of our results and highlight how these can be taken into account for designing evolutionarily informed fisheries-management regimes.

Analysis of a Characteristics Scheme for a Quasilinear Hierarchical Size-Structured Model

In this article, we consider a characteristics scheme for solving a hierarchical size-structured population model with nonlinear growth, reproduction and mortality rates. We fully analyze the scheme to establish consistency, stability, and convergence results.

Interference versus exploitative competition in the regulation of size-structured populations.

Competition is a major regulatory factor in population and community dynamics. Its effects can be either direct in interference competition or indirect in exploitative competition. The impact of exploitative competition on population dynamics has been extensively studied from empirical and theoretical points of view, but the consequences of interference competition remain poorly understood. Here we study the effect of different levels of intraspecific interference competition on the dynamics of a size-structured population. We study a physiologically structured population model accounting for direct individual interactions, allowing for a gradient from exploitative competition to interference competition. We parameterize our model with data on experimental populations of the collembolan Folsomia candida. Our model predicts contrasting dynamics, depending on the level of interference competition. With low interference, our model predicts juvenile-driven generation cycles, but interference competition tends to dampen these cycles. With intermediate interference, giant individuals emerge and start dominating the population. Finally, strong interference competition causes a novel kind of adult-driven generation cycles referred to as interference-induced cycles. Our results shed new light on the interpretation of the size-structured dynamics of natural and experimental populations.

Size-dependent diffusion promotes the emergence of spatiotemporal patterns.

Spatiotemporal patterns, indicating the spatiotemporal variability of individual abundance, are a pronounced scenario in ecological interactions. Most of the existing models for spatiotemporal patterns treat species as homogeneous groups of individuals with average characteristics by ignoring intraspecific physiological variations at the individual level. Here we explore the impacts of size variation within species resulting from individual ontogeny, on the emergence of spatiotemporal patterns in a fully size-structured population model. We found that size dependency of animal's diffusivity greatly promotes the formation of spatiotemporal patterns, by creating regular spatiotemporal patterns out of temporal chaos. We also found that size-dependent diffusion can substitute large-amplitude base harmonics with spatiotemporal patterns with lower amplitude oscillations but with enriched harmonics. Finally, we found that the single-generation cycle is more likely to drive spatiotemporal patterns compared to predator-prey cycles, meaning that the mechanism of Hopf bifurcation might be more common than hitherto appreciated since the former cycle is more widespread than the latter in case of interacting populations. Due to the ubiquity of individual ontogeny in natural ecosystems we conclude that diffusion variability within populations is a significant driving force for the emergence of spatiotemporal patterns. Our results offer a perspective on self-organized phenomena, and pave a way to understand such phenomena in systems organized as complex ecological networks.