On the stock estimation for some fishery systems

In this work we address the stock estimation problem for two fishery models. We show that a tool from nonlinear control theory called “observer” can be helpful to deal with the resource stock estimation in the field of renewable resource management. It is often difficult or expensive to measure all the state variables characterising the evolution of a given population system, therefore the question arises whether from the observation of certain indicators of the considered system, the whole state of the population system can be recovered or at least estimated. The goal of this paper is to show how some techniques of control theory can be applied for the approximate estimation of the unmeasurable state variables using only the observed data together with the dynamical model describing the evolution of the system. More precisely we shall consider two fishery models and we shall show how to built for each model an auxiliary dynamical system (the observer) that uses the available data (the total of caught fish) and which produces a dynamical estimation of the unmeasurable stock state x(t). Moreover the convergence speed of towards x(t) can be chosen.     

Monitoring in a Lotka-Volterra model

The problem of monitoring arises when in an ecosystem, in particular in a system of several populations, observing some components, we want to recover the state of the whole system as a function of time. Due to the difficulty to construct exactly this state process, we look for an auxiliary system called an observer. This system reproduces this process with a certain approximation. This means that the solution of the observer tends to that of the original system. An important concept for this work is observability. This means that from the observation it is possible to recover uniquely the state process, however, without determining a constructive method to obtain it. If observability holds for the original system, it guarantees the existence of an auxiliary matrix that makes it possible to construct an observer of the system. The considered system of populations is described by the classical Lotka-Volterra model with one predator and two preys and the construction of its observer is illustrated with a numerical example. Finally, it is shown how the observer can be used for the estimation of the level of an abiotic effect on the population system.     

Marine reserves and reproductive biomass: a case study of a heavily targeted reef fish

Recruitment overfishing (the reduction of a spawning stock past a point at which the stock can no longer replenish itself) is a common problem which can lead to a rapid and irreversible fishery collapse. Averting this disaster requires maintaining a sufficient spawning population to buffer stochastic fluctuations in recruitment of heavily harvested stocks. Optimal strategies for managing spawner biomass are well developed for temperate systems, yet remain uncertain for tropical fisheries, where the danger of collapse from recruitment overfishing looms largest. In this study, we explored empirically and through modeling, the role of marine reserves in maximizing spawner biomass of a heavily exploited reef fish, Lethrinus harak around Guam, Micronesia. On average, spawner biomass was 16 times higher inside the reserves compared with adjacent fished sites. Adult density and habitat-specific mean fish size were also significantly greater. We used these data in an age-structured population model to explore the effect of several management scenarios on L. harak demography. Under minimum-size limits, unlimited extraction and all rotational-closure scenarios, the model predicts that preferential mortality of larger and older fish prompt dramatic declines in spawner biomass and the proportion of male fish, as well as considerable declines in total abundance. For rotational closures this occurred because of the mismatch between the scales of recovery and extraction. Our results highlight how alternative management scenarios fall short in comparison to marine reserves in preserving reproductively viable fish populations on coral reefs.     

A statistical light scattering approach to separating fast and slow dynamics

Light scattering is a powerful technique to study the structural and dynamical properties of biomolecular systems or other soft materials such as polymeric solutions and blends or gels. An important application of this technique is the study of the kinetics of formation of supramolecular structures. However, in such cases, the system under study is rapidly changing, and consequently the integration time for each measurement is limited. In order to overcome this difficulty, a statistical approach has been developed based on the analysis of the scattered light intensity distribution (Manno et al. 2006, 2004). Indeed the intensity distribution depends upon the ratio between the integration time of each measurement and the coherence time of scattered radiation. This method has been applied to protein aggregation (Manno et al. 2006) and to sol-gel transition (Manno et al. 2004), to obtain information on the heterogeneity of morphological and dynamical features during such processes. In the present work, we accurately test the validity of this approach by analyzing the statistical properties of the light scattered by a model system: a solution of polystyrene spherical macromolecules of different sizes. Each molecular size is related to a given diffusion coefficient and to a given coherence time of the scattered intensity. The effect of changing the experimental integration time is systematically investigated. A mixture of particles of two different sizes is also analyzed to test the validity and robustness of the model based on the convolution of a gaussian with an exponential distribution. Proceedings of the XVIII Congress of the Italian Society of Pure and Applied Biophysics (SIBPA), Palermo, Sicily, September 2006.     

The effect of seasonal harvesting on stage-structured population models

In most models of population dynamics, increases in population due to birth are assumed to be time-independent, but many species reproduce only during a single period of the year. We propose an exploited single-species model with stage structure for the dynamics in a fish population for which births occur in a single pulse once per time period. Since birth pulse populations are often characterized with a discrete time dynamical system determined by its Poincaré map, we explore the consequences of harvest timing to equilibrium population sizes under seasonal dependence and obtain threshold conditions for their stability, and show that the timing of harvesting has a strong impact on the persistence of the fish population, on the volume of mature fish stock and on the maximum annual-sustainable yield. Moreover, our results imply that the population can sustain much higher harvest rates if the mature fish is removed as early in the season (after the birth pulse) as possible. Further, the effects of harvesting effort and harvest timing on the dynamical complexity are also investigated. Bifurcation diagrams are constructed with the birth rate (or harvesting effort or harvest timing) as the bifurcation parameter, and these are observed to display rich structure, including chaotic bands with periodic windows, pitch-fork and tangent bifurcations, non-unique dynamics (meaning that several attractors coexist) and attractor crisis. This suggests that birth pulse, in effect, provides a natural period or cyclicity that makes the dynamical behavior more complex.This work is supported by National Natural Science Foundation of China (10171106)     

A 27-year study of brown trout population dynamics and exploitation in Lake Songsjøen, central Norway

From 1968–1984 (period I), a brown trout Salmo trutta , population in a 70-ha oligotrophic lake in central Norway was exploited using larger mesh gill-nets selectively removing the larger fish. From 1985–1994 (period II), intermediate sized fish were removed using smaller-mesh sizes gill-nets. Fishing mortality and CPUE were correlated positively with effort and numbers of fish >3 years old for period II. The gill-net catchability was correlated negatively with spawner biomass and number of trout >3 years old. The significant positive correlation between natural mortality and stock biomass and spawning stock biomass indicated density-dependent mortality. The significant correlation between spawning stock and recruitment described by the Ricker model, indicated density-dependent recruitment of 1-year-old trout. The fishing regimes in the two periods affected the population dynamics and density differently. Selective removal of smaller fish permitted the larger fish to survive, and was beneficial in reducing fish density and maintaining stocks at low levels, consequently, achieving the expected increase in fish growth rates.     

Investigating the Effect of Recruitment Variability on Length-Based Recruitment Indices for Antarctic Krill Using an Individual-Based Population Dynamics Model

Antarctic krill (Euphausia superba; herein krill) is monitored as part of an on-going fisheries observer program that collects length-frequency data. A krill feedback management programme is currently being developed, and as part of this development, the utility of data-derived indices describing population level processes is being assessed. To date, however, little work has been carried out on the selection of optimum recruitment indices and it has not been possible to assess the performance of length-based recruitment indices across a range of recruitment variability. Neither has there been an assessment of uncertainty in the relationship between an index and the actual level of recruitment. Thus, until now, it has not been possible to take into account recruitment index uncertainty in krill stock management or when investigating relationships between recruitment and environmental drivers. Using length-frequency samples from a simulated population – where recruitment is known – the performance of six potential length-based recruitment indices is assessed, by exploring the index-to-recruitment relationship under increasing levels of recruitment variability (from ±10% to ±100% around a mean annual recruitment). The annual minimum of the proportion of individuals smaller than 40 mm (F40 min, %) was selected because it had the most robust index-to-recruitment relationship across differing levels of recruitment variability. The relationship was curvilinear and best described by a power law. Model uncertainty was described using the 95% prediction intervals, which were used to calculate coverage probabilities and assess model performance. Despite being the optimum recruitment index, the performance of F40 min degraded under high (>50%) recruitment variability. Due to the persistence of cohorts in the population over several years, the inclusion of F40 min values from preceding years in the relationship used to estimate recruitment in a given year improved its accuracy (mean bias reduction of 8.3% when including three F40 min values under a recruitment variability of 60%).     

Statistical interpretation of the interplay between noise and chaos in the stochastic logistic map

A recurring problem in population biology - as well as other stochastic dynamical systems in biology, the physical and social sciences - is the distinction between the ‘true’ dynamics of a system and observational noise: i.e. can we from present data reliably infer e.g. biological mechanisms, or are signals swamped by noise.Here, we approach this problem using the canonical model for simple systems that exhibit complex behaviour, the logistic map. At each time-point noise is added, which allows us to study the long-term behaviour of a system which exhibits both non-linear dynamics and intrinsic noise.We show that the interplay between deterministic non-linear dynamics and simple Gaussian noise results in a perplexingly simple system when viewed statistically.In particular we show that for the case of Gaussian noise it is possible to derive at very reliable approximations for the time until the system has reached an absorbing state. This generic model allows us, for example, to study the life-time of molecular species involved in noisy feedback loops.     

Recruitment prediction for Pacific herring (Clupea pallasi) on the west coast of Vancouver Island, Canada

The accurate prediction of recruitment to the fishery is a very important tool within the management structure of any fish stock being exploited. In the case of the Pacific herring, Clupea pallasi, fishery in Canada, a forecast of the abundance of each herring stock is particularly important for formulating an annual catch quota. The sustainable management of the fishery and the resource is based in part on accurate recruitment forecasting because Pacific herring are short-lived and so the recruitment contributes a significant part of the total spawning run targeted by the fishery each year. Several factors are believed be important in determining the success of recruitment besides spawners biomass. Since herrings are “r” strategists, conditions related to the egg, the planktonic, or even the juvenile stage might determine the future level of recruitment. Recently a formula that defines conditions for a semi-quantitative level of recruitment forecast was elaborated using genetic algorithms and current study attempts to improve on this model. Using salinity in two quarterly periods during the planktonic and pre-recruit stages, temperature and spawning biomass for the west coast of Vancouver Island stock, classification rules that define recruitment in 3 different levels (low, medium and high) were developed with a genetic algorithm, setting low and high boundaries for each condition. A 75% success in classifying recruitment was obtained. The model was shown to be particularly effective at predicting when the recruitment would be low, which could be important from the perspective of the Precautionary Approach and the sustainable management of this stock.     

Food web dynamics affect Northeast Arctic cod recruitment

Proper management of ecosystems requires an understanding of both the species interactions as well as the effect of climate variation. However, a common problem is that the available time-series are of different lengths. Here, we present a general approach for studying the dynamic structure of such interactions. Specifically, we analyse the recruitment of the world's largest cod stock, the Northeast Arctic cod. Studies based on data starting in the 1970-1980s indicate that this stock is affected by temperature through a variety of pathways. However, the value of such studies is somewhat limited by the fact that they are based on a quite specific ecological and climatic situation. Recently, this stock has consisted of fairly young fish and the spawning stock has consisted of relatively few age groups. In this study, we develop a model for the effect of capelin (the cod's main prey) and herring on cod recruitment since 1973. Based on this model, we analyse data on cod, herring and temperature going back to 1921 and find that food-web effects explain a significant part of the cod recruitment variation back to around 1950.     

The Dynamics of a Fish Stock Exploited in Two Fishing Zones

This work presents a specific stock-effort dynamical model. The stocks correspond to two populations of fish moving and growing between two fishery zones. They are harvested by two different fleets. The effort represents the number of fishing boats of the two fleets that operate in the two fishing zones. The bioeconomical model is a set of four ODE's governing the fishing efforts and the stocks in the two fishing areas. Furthermore, the migration of the fish between the two patches is assumed to be faster than the growth of the harvested stock. The displacement of the fleets is also faster than the variation in the number of fishing boats resulting from the investment of the fishing income. So, there are two time scales: a fast one corresponding to the migration between the two patches, and a slow time scale corresponding to growth. We use aggregation methods that allow us to reduce the dimension of the model and to obtain an aggregated model for the total fishing effort and fish stock of the two fishing zones. The mathematical analysis of the model is shown. Under some conditions, we obtain a stable equilibrium, which is a desired situation, as it leads to a sustainable harvesting equilibrium, keeping the stock at exploitable densities.     

Shifting dynamic forces in fish stock fluctuations triggered by age truncation?

Accumulating evidence shows that environmental fluctuations and exploitation jointly affect marine fish populations, and understanding their interaction is a key issue for fisheries ecology. In particular, it has been proposed that age truncation induced by fisheries exploitation may increase the population's sensitivity to climate. In this study, we use unique long‐term abundance data for the Northeast Arctic stock of cod (Gadus morhua) and the Norwegian Spring‐Spawning stock of herring (Clupea harengus), which we analyze using techniques based on age‐structured population matrices. After identifying time periods with different age distributions in the spawning stock, we use linear models to quantify the relative effect of exploitation and temperature on the population growth rates. For the two populations, age truncation was found to be associated with an increasing importance of temperature and a relatively decreasing importance of exploitation, while the population growth rate became increasingly sensitive to recruitment variations. The results suggested that the removal of older age classes reduced the buffering capacity of the population, thereby making the population growth rate more dependent on recruitment than adult survival and increasing the effect of environmental fluctuations. Age structure appeared as a key characteristic that can affect the response of fish stocks to climate variations and its consequences may be of key importance for conservation and management.     

Population dynamics and potential of fisheries stock enhancement: practical theory for assessment and policy analysis

The population dynamics of fisheries stock enhancement, and its potential for generating benefits over and above those obtainable from optimal exploitation of wild stocks alone are poorly understood and highly controversial. I review pertinent knowledge of fish population biology, and extend the dynamic pool theory of fishing to stock enhancement by unpacking recruitment, incorporating regulation in the recruited stock, and accounting for biological differences between wild and hatchery fish. I then analyse the dynamics of stock enhancement and its potential role in fisheries management, using the candidate stock of North Sea sole as an example and considering economic as well as biological criteria. Enhancement through release of recruits or advanced juveniles is predicted to increase total yield and stock abundance, but reduce abundance of the naturally recruited stock component through compensatory responses or overfishing. Economic feasibility of enhancement is subject to strong constraints, including trade-offs between the costs of fishing and hatchery releases. Costs of hatchery fish strongly influence optimal policy, which may range from no enhancement at high cost to high levels of stocking and fishing effort at low cost. Release of genetically maladapted fish reduces the effectiveness of enhancement, and is most detrimental overall if fitness of hatchery fish is only moderately compromised. As a temporary measure for the rebuilding of depleted stocks, enhancement cannot substitute for effort limitation, and is advantageous as an auxiliary measure only if the population has been reduced to a very low proportion of its unexploited biomass. Quantitative analysis of population dynamics is central to the responsible use of stock enhancement in fisheries management, and the necessary tools are available.     

Isolation and characterization of nuclear microsatellite loci in the anadromous marine fish Morone saxatilis

Molecular genetic studies of population structure and gene flow are commonly employed in fish stock assessment and breeding population delineation. Genetically structured breeding populations may have different effective population sizes, distinct reproductive rates and variable susceptibility to harvesting or breeding habitat degradation. Nine microsatellite loci were isolated for Morone saxatilis (Moronidae), an anadromous fish inhabiting the mid‐Atlantic. Microsatellite loci were isolated with a subtractive hybridization method and will be used to estimate population structure. The loci averaged 8.5 alleles each. Seven loci in the Choptank population and two loci in the Potomac population deviated from Hardy–Weinberg expected frequencies of heterozygotes.     

Stocking of captive-bred fish can cause long-term population decline and gene pool replacement: predictions from a population dynamics model incorporating density-dependent mortality

Releasing captive-bred fish into natural environments (stocking) is common in fisheries worldwide. Although stocking is believed to have a positive effect on fish abundance over the short term, little is known about the long-term consequences of recurrent stocking and its influence on natural populations. In fact, there are growing concerns that genetically maladapted captive-bred fish can eventually reduce the abundance of natural population. In this study, we develop a simple model to quantitatively investigate the condition under which recurrent stocking has long-term effects on the natural population. Using a population dynamics model that takes into account a density-dependent recruitment, a gene responsible for the fitness difference between wild and captive-bred fish, and hybridization between them, we show that there is little or no contribution of recurrent stocking to the stock enhancement without a replacement of the wild gene pool by the captive-bred gene pool. The model further predicted that stocking of an intermediate level causes a reduction, rather than enhancement, of population size over the long term. The population decline due to stocking was attributed to the fitness disadvantage of captive-bred fish and strong overcompensation at recruitment stage. These results suggest that it would be difficult to simultaneously attain population size recovery and conservation of the local gene pool when captive-bred fish have fitness disadvantage in the wild, although caution is needed when applying the predictions from the simplified model to a specific species or population.     

Irreversible prey diapause as an optimal strategy of a physiologically extended Lotka–Volterra model

We propose an optimal control framework to describe intra-seasonal predator–prey interactions, which are characterized by a continuous-time dynamical model comprising predator and prey density, as well as the energy budget of the prey over the length of a season. The model includes a time-dependent decision variable for the prey, representing the portion of the prey population in time that is active, as opposed to diapausing (a state of physiological rest). The predator follows autonomous dynamics and accordingly it remains active during the season. The proposed model is a generalization of the classical Lotka–Volterra predator–prey model towards non-autonomous dynamics that furthermore includes the effect of an energy variable. The model has been inspired by a specific biological system of predatory mites (Acari: Phytoseiidae) and prey mites (so-called fruit-tree red spider mites) (Acari: Tetranychidae) that feed on leaves of apple trees—its parameters have been instantiated based on laboratory and field studies. The goal of the work is to understand the decisions of the prey mites to enter diapause (a state of physiological rest) given the dynamics of the predatory mites: this is achieved by solving an optimization problem hinging on the maximization of the prey population contribution to the next season. The main features of the optimal strategy for the prey are shown to be that (1) once in diapause, the prey does not become active again within the same season and hence diapause is an irreversible process; (2) for the vast majority of parameter space, the portion of prey individuals entering diapause within the season does not decrease in time; (3) with an increased number of predators, the optimal population strategy for the prey is to start diapause earlier and to enter diapause more gradually. This optimal population strategy will be studied for its ESS properties in a sequel to the work presented in this article.     

Stock estimation, environmental monitoring and equilibrium control of a fish population with reserve area

For sustainable exploitation of renewable resources, the separation of a reserve area is a natural idea. In particular, in fishery management of such systems, dynamic modelling, monitoring and control has gained major attention in recent years. In this paper, based on the known dynamic model of a fish population with reserve area, the methodology of mathematical systems theory and optimal control is applied. In most cases, the control variable is fishing effort in the unreserved area. Working with illustrative data, first a deterministic stock estimation is proposed using an observer design method. A similar approach is also applied to the estimation of the effect of an unknown environmental change. Then it is shown how the system can be steered to equilibrium in given time, using fishing effort as an open-loop control. Furthermore, a corresponding optimal control problem is also solved, maximizing the harvested biomass while controlling the system into equilibrium. Finally, a closed-loop control model is applied to asymptotically control the system into a desired equilibrium, intervening this time in the reserve area.     

The stabilizability of a controlled system describing the dynamics of a fishery

This work presents two stock-effort dynamical models describing the evolution of a fish population growing and moving between two fishing zones, on which it is harvested by a fishing fleet, distributed on the two zones. The first model corresponds to the case of constant displacement rates of the fishing effort, and the second one to fish stock-dependent displacement rates. In equations of the fishing efforts, a control function is introduced as the proportion of the revenue to be invested, for each fleet. The stabilizability analysis of the aggregated model, in the neighborhood of the equilibrium point, enables the determination of a Lyapunov function, which ensures the existence of a stabilizing discontinuous feedback for this model. This enables us to control the system and to lead, in an uniform way, any solution of this system towards this desired equilibrium point.     

Forecasting future recruitment success for Atlantic cod in the warming and acidifying Barents Sea

Productivity of marine fish stocks is known to be affected by environmental and ecological drivers, and global climate change is anticipated to alter recruitment success of many stocks. While the direct effects of environmental drivers on fish early life stage survival can be quantified experimentally, indirect effects in marine ecosystems and the role of adaptation are still highly uncertain. We developed an integrative model for the effects of ocean warming and acidification on the early life stages of Atlantic cod in the Barents Sea, termed SCREI (Simulator of Cod Recruitment under Environmental Influences). Experimental results on temperature and CO₂ effects on egg fertilization, egg and larval survival and development times are incorporated. Calibration using empirical time series of egg production, temperature, food and predator abundance reproduces age‐0 recruitment over three decades. We project trajectories of recruitment success under different scenarios and quantify confidence limits based on variation in experiments. A publicly accessible web version of the SCREI model can be run under www.oceanchange.uni-bremen.de/ ;SCREI. Severe reductions in average age‐0 recruitment success of Barents Sea cod are projected under uncompensated warming and acidification toward the middle to end of this century. Although high population stochasticity was found, considerable rates of evolutionary adaptation to acidification and shifts in organismal thermal windows would be needed to buffer impacts on recruitment. While increases in food availability may mitigate short‐term impacts, an increase in egg production achieved by stock management could provide more long‐term safety for cod recruitment success. The SCREI model provides a novel integration of multiple driver effects in different life stages and enables an estimation of uncertainty associated with interindividual and ecological variation. The model thus helps to advance toward an improved empirical foundation for quantifying climate change impacts on marine fish recruitment, relevant for ecosystem‐based assessments of marine systems under climate change.