This paper introduces a novel methodological approach for deriving length growth transition matrices using B-splines. The methodology, which does not depend on predefined probability structures, enables a more accurate representation of growth transitions and individual variability in fish growth. We demonstrate how metrics derived from a set of matrices, obtained from consecutive years of observations, can be utilized to infer population dynamics of the stock.We apply this methodology to the Barents Sea capelin as a case study. Our results show that growth in length dynamics effectively captures the population dynamics, including both stable and collapse stock states. Consequently, this approach serves as a predictive tool for forecasting trends and provides a quantitative framework for linking growth transition patterns to variations in population states.

This paper presents a modeling framework for simulating capelin spawning migration in the Barents Sea by integrating artificial neural networks (ANNs), a genetic algorithm (GA), an individual-based model (IBM), and environmental variables. ANNs determine movement direction based on inputs like temperature and ocean currents and are trained using a GA whose fitness function dynamically adapts to temperature and proximity to spawning routes. The model successfully reproduces the southeastward migration observed in 2019, aligning with historical spawning sites along northern Norway's eastern coast. Validation against empirical data confirms its accuracy, and results highlight the importance of the adaptive fitness function in such learning-based models. The proposed model is also compared to three alternatives based on passive swimming, gradient detection, and restricted-area search. It outperforms them by accurately replicating migration patterns, with most simulated fish reaching spawning sites. In contrast, the alternative models often failed due to current-induced drift, particularly in the gradient detection and restricted-area search approaches.

Climate-driven changes in the Subarctic will directly impact capelin populations and the ecosystem they inhabit, including their predators, prey, and physical habitats. Consequently, incorporating ecosystem considerations in future capelin fisheries management is crucial. In this study, a multidisciplinary group of experts critically evaluated whether the current capelin stock assessment and management frameworks for the four main capelin stocks in the Barents Sea (BS), Iceland-East Greenland-Jan Mayen (IEGJM), Newfoundland and Labrador shelf (NL) and Alaska (AK) align with the principles of an Ecosystem Approach to Fisheries Management (EAFM). An evidence-based ranking of our knowledge on current capelin dynamics across ecological, economic, and social dimensions was conducted, using expert knowledge supported by literature. This exercise also identified data currently used for assessment and management, which highlighted that the existing capelin assessment frameworks include varying degrees of EAFM elements across stocks, such as considerations of trophic interactions, bottom-up processes, accounting for ecosystem uncertainty, and stakeholder engagement in the advisory process. Nonetheless, there is room for improvement where data and knowledge are lacking. We provide some key tactical (short-term) and strategic (long-term) recommendations from our perspective on what is required to ensure the sustainable management of capelin in the circumpolar region over the coming decades.

Motivated by investigating spatio-temporal patterns of the distribution of continuous variables, we consider describing the conditional distribution function of the response variable incorporating spatio-temporal components given predictors. In many applications, continuous variables are observed only as threshold-categorized data due to measurement constraints. For instance, ecological measurements often categorize sizes into intervals rather than recording exact values due to practical limitations. To recover the conditional distribution function of the underlying continuous variables, we consider a distribution regression employing models for binomial data obtained at each threshold value. However, depending on spatio-temporal conditions and predictors, the distribution function may frequently exhibit boundary values (zero or one), which can occur either structurally or randomly. This makes standard binomial models inadequate, requiring more flexible modeling approaches. To address this issue, we propose a boundary-inflated binomial model incorporating spatio-temporal components. The model is a three-component mixture of the binomial model and two Dirac measures at zero and one. We develop a computationally efficient Bayesian inference algorithm using Pólya-Gamma data augmentation and dynamic Gaussian predictive processes. Extensive simulation experiments demonstrate that our procedure significantly outperforms distribution regression methods based on standard binomial models across various scenarios.

The anticipated empirical consumption level of a target (prey) species by its predators is an important variable when implementing the constant biomass escapement strategy in fisheries management. For a single predator, the projection of empirical consumption will depend, among others, on projected distribution in space–time of the predator, degree of predator overlap with prey, and considerations of fluctuating environmental variables (e.g., temperature) that may affect the consumption rate. Hence, the projected empirical consumption is usually uncertain, as it involves the integration of uncertain space-time data and high-throughput models and sub-models. The inclusion of such modeling frameworks into an overall assessment model is usually non-trivial. The goal of this paper is to approximate estimates of the empirical consumption rate of capelin by cod in the Barents Sea, using an analytical functional response (FR) model. The empirical consumption rate is estimated by integrating projected space-time (biotic and abiotic) data with a cod stomach evacuation model. We consider the time series of empirical consumption rates as observation data, and use an optimization procedure to estimate parameters of the FR model. We assess the model performance by the degree to which derived uncertainty envelopes encompass the input data, and evaluate its prediction ability using data that were excluded in the parameter estimation process. Our results show that the FR model provides good approximations to estimates of empirical consumption. A major advantage of the FR modeling framework is that it facilitates hypothesis testing, uncertainty quantification, and seamless integration of the FR model into an overall stock assessment model for capelin.

Understanding the relationship between adult fish populations (the "stock") and the number of new fish entering the population (the "recruits") is essential for effective fisheries management. Traditionally, this relationship is represented by a stock-recruitment (SR) function, which is a simplified mathematical model that directly links stock size to recruitment. However, fish populations pass through several life stages, each stage influenced by unique population dynamic factors. Current SR functions often overlook these complexities, assuming that recruitment depends solely on the adult population size. In this study, we use a multi-stage, age-structured discrete-time population dynamic model that accounts for all life stages and the transitions between them. We demonstrate that, in general, a closed-form, univariate SR function may not accurately represent the recruitment process when these life stages are considered. Instead, we identify specific mathematical conditions under which a SR function is equivalent to our multi-stage model. Our findings suggest a re-evaluation of conventional SR models, advocating for multi-stage approaches to support fisheries management decisions.

This paper presents a modeling framework for simulating capelin spawning migration in the Barents Sea. The framework is based on integrating artificial neural networks (ANNs) models, individual-based model (IBM), and environmental variables. The ANNs determine the direction of the fish's movement based on environmental variables such as temperature and ocean currents. The ANNs are trained by an evolutionary algorithm, whose fitness function is dynamically adapted based on the temperature and distance to the spawning route. The proposed model successfully reproduced the southeastward spawning in 2019, capturing the distribution of spawning capelin over the historical spawning sites along the eastern coast of northern Norway. The efficacy of the model is validated by comparing the spatial distributions of modeled and empirical data. Furthermore, the results show that the learning fitness function is crucial in developing such learning-based models. Additionally, three migration models based on three different movement mechanisms—passive swimmers, gradient detection, and restricted-area search—were compared with the proposed approach in terms of their effectiveness in replicating the spawning migration patterns of capelin. The results reveal that our approach outperforms the other models in mimicking the migration pattern. Most simulated spawning stocks managed to reach the spawning sites, unlike in the other models where water currents played a significant role in pushing the fish back from their migration direction towards the coast. The temperature gradient detection model and restricted-area search model were found to be inadequate for accurately simulating capelin spawning migration in the Barents Sea.

This review synthesizes the role of multistability in deterministic food web models of marine ecosystems, focusing on how structural features of ordinary differential equation (ODE) models give rise to multiple stable states. From an initial pool of 178 publications, we systematically selected 35 studies that explicitly report multistability in predator-prey and food web models. These were analyzed according to key structural components-such as functional responses, growth and mortality terms, nonlinearities, and network topologies-to identify the mechanisms underpinning the emergence of bistability, tristability, and higher-order multistability. The reviewed models exhibit a wide range of attractor types, including equilibrium points, limit cycles, and chaotic attractors. Our analysis highlights the influence of non-monotonic functional responses, intraspecific competition, Allee effects, and time-scale separation in generating complex dynamics. Furthermore, we find that bifurcation analysis and numerical simulations are the primary tools used to characterize stability landscapes and attractor transitions. Despite the advances in modeling techniques, the review reveals a significant gap between theory and empirical application. Most models are general in nature and seldom incorporate real-world species, parameterized data, or specific ecosystem structures. We discuss this gap and propose a collaborative modeling approach focused on keystone species and the development of a shared repository of food web models. By categorizing multistable behaviors and clarifying their mechanistic origins, this review contributes towards a more robust theoretical foundation for understanding multistability in marine ecosystems.

In empirical predator-prey systems, understanding the inherent dynamics typically comes from analyzing a structural model fitted to observation data. However, determining an appropriate model structure and its parameters is often complex and highly uncertain. A promising alternative is to learn the model structure directly from time series data of both predator and prey. This study explores the capability of a data-driven algorithm, Sparse Identification of Nonlinear Dynamics (SINDy), to accurately capture the dynamics of a predator-prey system. We apply SINDy to derive a Learned Model (LM) from data generated by a Reference Model (RM), whose predator-prey dynamics are well understood. The study compares the dynamics of the LM to the RM using criteria such as equilibrium points, stability, sensitivity, and bifurcation analysis. Our results demonstrate general consistency between the RM and LM dynamics, though notable differences remain. We discuss the implications of these differences in the broader context of using learned models to uncover the inherent drivers of predator-prey dynamics and ecological implications. ### Competing Interest Statement The authors have declared no competing interest.

The stock-recruit relationship is a foundational concept in fisheries science, bridging the connection between parental populations (stock) and progeny (recruits). Traditional approaches describe this relationship using closedform analytical functions, which represent only a restricted subset of the broader class of possibilities. This paper advocates for a novel approach that integrates discrete time modeling with a life-history cycle framework, incorporating distinct stanzas and developmental processes. By breaking down the life cycle into identifiable stages, we capture the step-wise progression of life history traits and the factors influencing recruitment outcomes. Through numerical simulations, we explore the advantages of this approach, including complexity handling, dynamic behavior modeling, and scenario exploration. Our simulation results show that we are able to generate a broad spectrum of stock-recruit relationships (including the traditional ones), which best reflect variability observed in nature. We demonstrate how this framework allows for the identification of critical stages, and integration of various factors that influence recruitment. This holistic approach enhances our comprehension of the intricate interactions shaping stock-recruit relationships and advances our understanding of sustainable population dynamics. Highlights ### Competing Interest Statement The authors have declared no competing interest.

This paper presents a modeling framework for tracking the spawning migration of the capelin, which is a fish species in the Barents Sea. The framework combines an individual-based model (IBM) with artificial neural networks (ANNs). The ANNs determine the direction of the fish's movement based on local environmental information, while a genetic algorithm and fitness function assess the suitability of the proposed directions. The framework's efficacy is demonstrated by comparing the spatial distributions of modeled and empirical potential spawners. The proposed model successfully replicates the southeastward movement of capelin during their spawning migration, accurately capturing the distribution of spawning fish over historical spawning sites along the eastern coast of northern Norway. Furthermore, the paper compares three migration models: passive swimmers, taxis movement based on temperature gradients, and restricted-area search, along with our proposed approach. The results reveal that our approach outperforms the other models in mimicking the migration pattern. Most spawning stocks managed to reach the spawning sites, unlike the other models where water currents played a significant role in pushing the fish away from the coast. The temperature gradient detection model and restricted-area search model are found to be inadequate for accurately simulating capelin spawning migration in the Barents Sea due to complex oceanographic conditions.

For marine species, several life stages link parents to their progeny (recruits), through a process referred to as recruitment. Current stock recruitment (SR) functions encapsulate this multi-stage relationship in a closed-form mathematical expression, which explicitly relates the biomass of parents, to the number of recruits. This functional relationship (between parents and recruits) is required for management purposes. No study, however, has investigated the conditions that validate the existence of the SR functions when all life history stages are incorporated into a model. In this study, we represent the processes leading to recruitment by a stage-and age-structured discrete-time population dynamic model. We show that in general, a SR function does not correctly represent the parent-progeny relationship. A valid relationship must incorporate information across the complete life cycle over several time periods. For populations with simple life cycle history, a functional relationship may result, which is not necessarily consistent with current SR functions. We present a brief discussion of the relevance of our results to effective management of fisheries.

The schooling behavior of fish can be studied through simulations involving a large number of interacting particles. In such systems, each individual particle is guided by behavior rules, which include aggregation towards a centroid, collision avoidance, and direction alignment. The movement vector of each particle may be expressed as a linear combination of behaviors, with unknown parameters that define a trade-off among several behavioral constraints. A fitness function for collective schooling behavior encompasses all individual particle parameters. For a large number of interacting particles in a complex environment, heuristic methods, such as evolutionary algorithms, are used to optimize the fitness function, ensuring that the resulting decision rule preserves collective behavior. However, these algorithms exhibit slow convergence, making them inefficient in terms of CPU time cost. This paper proposes a CPU-efficient iterative (Cluster, Partition, Refine -- CPR) algorithm for estimating decision rule parameters for a large number of interacting particles. In the first step, we employ the K-Means (unsupervised learning) algorithm to cluster candidate solutions. Then, we partition the search space using Voronoi tessellation over the defined clusters. We assess the quality of each cluster based on the fitness function, with the centroid of their Voronoi cells representing the clusters. Subsequently, we refine the search space by introducing new cells into a number of identified well-fitting Voronoi cells. This process is repeated until convergence. A comparison of the performance of the CPR algorithm with a standard Genetic Algorithm reveals that the former converges faster than the latter. We also demonstrate that the application of the CPR algorithm results in a schooling behavior consistent with empirical observations.

This paper introduces an efficient approach to reduce the computational cost of simulating collective behaviors, such as fish schooling, using Individual-Based Models (IBMs). The proposed technique employs adaptive and dynamic load-balancing domain partitioning, which utilizes unsupervised machine-learning models to cluster a large number of simulated individuals into sub-schools based on their spatial-temporal locations. It also utilizes Voronoi tessellations to construct non-overlapping simulation subdomains. This approach minimizes agent-to-agent communication and balances the load both spatially and temporally, ultimately resulting in reduced computational complexity. Experimental simulations demonstrate that this partitioning approach outperforms the standard regular grid-based domain decomposition, achieving a reduction in computational cost while maintaining spatial and temporal load balance. The approach presented in this paper has the potential to be applied to other collective behavior simulations requiring large-scale simulations with a substantial number of individuals.

The schooling behavior of fish can be studied through simulations involving a large number of interacting particles. In such systems, each individual particle is guided by behavior rules, which include aggregation towards a centroid, collision avoidance, and direction alignment. The movement vector of each particle may be expressed as a linear combination of behaviors, with unknown parameters that define a trade-off among several behavioral constraints. A fitness function for collective schooling behavior encompasses all individual particle parameters. For a large number of interacting particles in a complex environment, heuristic methods, such as evolutionary algorithms, are used to optimize the fitness function, ensuring that the resulting decision rule preserves collective behavior. However, these algorithms exhibit slow convergence, making them inefficient in terms of CPU time cost. This paper proposes a CPU-efficient iterative (Cluster, Partition, Refine – CPR) algorithm for estimating decision rule parameters for a large number of interacting particles. In the first step, we employ the K-Means (unsupervised learning) algorithm to cluster candidate solutions. Then, we partition the search space using Voronoi tessellation over the defined clusters. We assess the quality of each cluster based on the fitness function, with the centroid of their Voronoi cells representing the clusters. Subsequently, we refine the search space by introducing new cells into a number of identified well-fitting Voronoi cells. This process is repeated until convergence. A comparison of the performance of the CPR algorithm with a standard Genetic Algorithm reveals that the former converges faster than the latter. We also demonstrate that the application of the CPR algorithm results in a schooling behavior consistent with empirical observations.

We consider modeling and prediction of Capelin distribution in the Barents sea based on zero-inflated count observation data that vary continuously over a specified survey region. The model is a mixture of two components; a one-point distribution at the origin and a Poisson distribution with spatio-temporal intensity, where both intensity and mixing proportions are modeled by some auxiliary variables and unobserved spatio-temporal effects. The spatio-temporal effects are modeled by a dynamic linear model combined with the predictive Gaussian process. We develop an efficient posterior computational algorithm for the model using a data augmentation strategy. The performance of the proposed model is demonstrated through simulation studies, and an application to the number of Capelin caught in the Barents sea from 2014 to 2019.

Because of its responsiveness to changes in the marine environment, it has been suggested by Rose in 2005 that the capelin, a small pelagic fish that is key to the ecology and fisheries of the North Atlantic, could be seen as a "canary in the coalmine" to detect signals of changes in the Arctic and sub-Arctic Ocean. We describe the historical data that make possible a quantitative assessment of the geographical shift capelin migration-paths and spawning grounds undergo, with increasing temperature, and the time it takes to make these shifts long-lasting. Then we introduce recent data that make these quantitative measurements more accurate and predictive. Direct measurements made in the fall expeditions of Iceland's Marine and Freshwater Research Institute along the East Coast of Greenland, and the Copernicus database of the European Union, are used to examine the evolution of the returning Atlantic water (from Svalbard) that is forming a warmer and saltier boundary current under the colder and fresher East Greenland polar current. The returning Atlantic water has a temperature range (1 to 4 degrees Centigrade) suitable for feeding migrations of the capelin. This current is reaching further north along the coast of North East Greenland and we use simulated data from Copernicus to monitor this evolution. We calibrate the Copernicus data with the direct measurements made by the Marine and Freshwater Research Institute, in Iceland. A trend emerges, both in the direct measurements and in Copernicus data, showing that the returning Atlantic water boundary current may reach Greenland's major Northeastern glacier streams, draining the bulk of the Greenland Glacier in the relatively near future We use the capelin data to predict when this may happen.

It is assumed that maturation in the Barents Sea capelin is length-dependent, and that fish of at least 14 cm will potentially spawn. Current assessment and management models for the stock are based on this assumption of constant maturity at length (MaL). Using data from scientific surveys, this paper examines the validity of the constant MaL assumption, and contrasts it with maturation based on examination of fish gonads. Our analyses, based on time series of 16 years, show that MaL-based estimates of the proportion of maturing stock usually exceed gonad-based estimates. The difference varies consistently with time, and stock-size. We discuss the consequence of our results in the context of uncertainty associated with the current harvest rule.

Identifying spawning sites of fish often involves extensive egg and larval sampling surveys over potential spawning sites, or by backward-tracking advected larvae to their source. Due to the vastness of the Barents Sea capelin spawning areas, back-tracking methods have limited application. Egg and larval surveys that provide information about spawning sites have also been discontinued in recent years. This paper aims at using alternative data sources to egg and larval distribution information, to infer potential spawning regions of the Barents Sea capelin during the period 1994–2020. We use the K-Means clustering technique to cluster historical spawning sites into spawning regions, and the Self Organizing Map (SOM) algorithm to define observed data clusters, which we assign to specific regions. The observation data consists of survey data sets from capelin pre-spawning and post-spawning periods during winter and spring respectively, as well as data from the Norwegian Directorate of Fisheries Electronic Reporting System database (ERS). Our method was efficient in reproducing capelin spawning regions and approximate time windows for commencement of spawning. The results showed that spawning occurred mainly over the eastern part of historical spawning areas during the whole period. A westward extension of the preferred spawning areas occurred in several years regardless of the rising Barents Sea water temperatures, especially during the second half of the period. The ERS data can be used to identify the arrival times and migration fronts of pre-spawning capelin along the coast.