This paper presents a new comprehensive growth model which includes numerous historical models as special cases. The new model is derived from a concise biological principle which, unlike earlier theories, relates to growth acceleration. Properties of growth curves, such as asymptotic limits or inflection points, are shown to be incidental in this new context. Possible submodels include not only asymptotic growth (such as von Bertalanffy, Richards, Gompertz, or logistic growth) but also linear, quadratic, or exponential growth. By simple analysis of variance, the observed data can be used directly in deciding which type of model is most appropriate. The new model is cast in terms of parameters which have stable statistical estimates. From this perspective, it is shown how earlier formulations sometimes result in an endless computer search for optimal parameter estimates.Key words: Growth, asymptotic growth, von Bertalanffy, Richards, Gompertz, logistic, length at age, weight at age, nonlinear parameter estimation
It is well known that current equilibrium-based models fall short as predictive descriptions of natural ecosystems, and particularly of fisheries systems that exhibit nonlinear dynamics. For example, model parameters assumed to be fixed constants may actually vary in time, models may fit well to existing data but lack out-of-sample predictive skill, and key driving variables may be misidentified due to transient (mirage) correlations that are common in nonlinear systems. With these frailties, it is somewhat surprising that static equilibrium models continue to be widely used. Here, we examine empirical dynamic modeling (EDM) as an alternative to imposed model equations and that accommodates both nonequilibrium dynamics and nonlinearity. Using time series from nine stocks of sockeye salmon (Oncorhynchus nerka) from the Fraser River system in British Columbia, Canada, we perform, for the the first time to our knowledge, real-data comparison of contemporary fisheries models with equivalent EDM formulations that explicitly use spawning stock and environmental variables to forecast recruitment. We find that EDM models produce more accurate and precise forecasts, and unlike extensions of the classic Ricker spawner-recruit equation, they show significant improvements when environmental factors are included. Our analysis demonstrates the strategic utility of EDM for incorporating environmental influences into fisheries forecasts and, more generally, for providing insight into how environmental factors can operate in forecast models, thus paving the way for equation-free mechanistic forecasting to be applied in management contexts.ecosystem forecasting | fisheries ecology | physical-biological interactions | empirical dynamic modeling | nonlinear dynamics O ne of the fundamental challenges of environmental science is to understand and predict the behavior of complex natural ecosystems. This task can be especially difficult when multiple drivers (e.g., species interactions, environmental influences) interact in a nonlinear state-dependent way to produce dynamics that appear to be erratic and nonstationary (1). In the standard parametric approach, which implicitly assumes that the selected model and its equations are essentially correct, the equations (really just mechanistic hypotheses) can lack the flexibility to describe the nonlinear dynamics that occur in nature. Consequently, these parametric models tend to perform poorly as descriptions of reality, with little explanatory or predictive power (2, 3), and limited usefulness for prediction and management. Parametric Models as HypothesesA common problem when applying the parametric approach to nonlinear systems is that of ephemeral fitting. That is, although population models may assume that demographic parameters such as growth rate or carrying capacity are fixed constants, these quantities are often observed to vary in time or in relation to other variables (e.g., resource availability, changing climate regimes) when tested on actual data (4). This principle is illust...
Juvenile coho salmon (Oncorhynchus kisutch) were raised in six ponds at Rosewall Creek, Vancouver Island, British Columbia, from which releases were made at four times: April 14, May 12, June 10 and July 8, 1975. Prior to each release a portion of the juveniles in each pond were graded into three size groups (small, medium, large) based on size distributions in each pond. The juveniles were nose-tagged according to size group, pond, and release date, and marked by adipose fin removal. A total of 57 groups were released. Returns of adults and precocious males (jacks) to the weir and to the fishery (commercial, sport) were subjected to response surface analysis. Maximum adult returns of 43.5%, to the weir and fishery, were predicted for release of 25.1-g coho juveniles on the 173rd (Julian) day from January 1 (June 22, 1975). A significant interaction was noted between release time and size: maximum returns from early (April 14) releases would be expected from release of 16- to 17-g juveniles. Returns of jacks from the six ponds ranged from 0.0 to 4.65%. Predicted returns of jacks would be maximized from early release of large juveniles (> ~20 g in April). The benefit–cost ratio at the center of the time and size of release surface for maximum adult returns was estimated as 12.2:1. Maximum benefit–cost ratios were calculated for release of smaller juveniles between June 17 and 22, 1975. Benefit–costs associated with releases on the path of joint optimality between the two response centers rose from 12.2:1 (maximum adult returns) to a calculated 16.5:1 for release of 7.5-g juveniles on June 17, 1975. Size and time at juvenile release and success of adult returns are viewed as initial and final aspects of a biological system whose central components as yet are imperfectly understood. Some parts of the system are discussed in relation to possible ways of increasing the efficiency and cost–effectiveness of salmonid hatchery technology, as well as in predicting the success of returns in relation to conditions influencing survival and growth of juvenile migrants from natural stocks.Key words: coho salmon, juvenile salmon, release time and size, response surface analysis and optimization, survival, biomass, benefit–costs
This paper presents a new approach to length–frequency analysis which takes account of biological structure in the mean lengths and standard deviations in length for various age-classes of fish. The new methods help determine biologically meaningful solutions, even when earlier methods lead to an ambiguous set of competing solutions. The structure of the standard deviations turns out to be especially important. For describing the means, new parameters are defined for von Bertalanffy growth which prove to have greater biological meaning and numerical stability than L∞, K, and t0. These new parameters can often be estimated easily from the raw data in cases where the species experiences a slowing of growth with age. This paper also presents χ2 methods which can be used to rank competing solutions, although the results are not definitive. All methods are illustrated using data previously published for pike and abalone. An appendix describes in detail the computer programs required for the analysis.Key words: length–frequency analysis, aging of samples, von Bertalanffy growth, growth, maximum likelihood estimation, nonlinear estimation
This paper presents a general theory for analysis of catch and effort data from a fishery. Almost all previous methods are shown to be special cases, including those of Schaefer, Pella and Tomlinson, Schnute, and Deriso, as well as the stock reduction analysis technique of Kimura and Tagart and Kimura, Balsiger, and Ito. Like that of Deriso, the theory here is based on natural equations for an age structured population. However, instead of a fixed single model, this paper gives a general model that can be tailored to any particular fishery. The problem of determining the appropriate special case is conceptually identical to the model identification problem described by Box and Jenkins in the context of time series analysis, Identification necessarily begins with a suitable class of models. This paper defines such a class, unique to fisheries, complete with mathematical proofs and biological explanations of all important equations.
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