Introduction

The more than 3 m long ocean sunfish Mola mola develops from an egg that is about a millimeter in diameter. The anadromous brown trout Salmo trutta often hatches in a small stream in the midst of a forest, migrates after one or more years into the sea for foraging, and eventually returns as an adult to the natal stream for spawning. As fish grow through their life, they often occupy different habitats, are threatened by different predators, and rely on different food resources. Understanding how these complexities shape lifetime patterns of animal growth is a large and active field of research. In this article, we examine one component of this research by reviewing the most common mathematical models that have been used to describe and interpret the growth of fish. These models have been developed for several purposes, including identifying and comprehending the causes of individual variability, to predict consequences for fisheries yield, and to improve production in fish farming.

Experimental studies and descriptive field-based research have provided valuable insight that has increased our knowledge of fish growth in both the laboratory and the wild. Modeling fish growth is a worthy endeavor because it allows us to better understand the mechanisms underlying how and why a fish grows; it provides a means of calculating parameters that can be readily compared across populations, and we can use these growth models as parts of larger, more comprehensive models to study trophic interactions or ecosystem dynamics. However, modeling has the most to offer when used in conjunction with experiments and field work. In a cyclic manner, models can generate hypotheses for experiments, and experiments can test model assumptions and predictions.

This article only considers models for individual fish growth. Models are also used for describing the growth of an entire fish stock in biomass or abundance, for example, logistic models with exponential population growth limited by a carrying capacity such as in Lotka-Volterra models (see Fishery Models and Growth Models). Such population models underlie general concepts in fisheries science (e.g., the maximum sustainable yield) and are still used in the assessment and management of fish stocks. They do not, however, contain information about individual size, which is a drawback since survival probability and reproductive rates often change as an individual grows. As a consequence, the distribution of individual body sizes in the population will influence population dynamics and the population's overall growth rate. These effects become particularly strong in long-lived species. More recent approaches therefore combine individual growth models of the type described in this article with size-specific rates for survival and reproduction to model population dynamics. With information about size distribution or abundance, one can thereafter scale predictions for individual growth trajectories up to the whole population. This process can result in more accurate estimates of population abundance and biomass, and also offer advantages for research, for example, in population dynamics, ecological interactions, and life-history theory (see Life-History Patterns). Scaling up can be done using cohorts that consist of identical fish with average parameter values. In reality, however, a parameter value such as mean growth rate emerges from individuals that have different physiological growth capacities and experience different environmental conditions. Both these types of variances can be included by modeling the population as a collection of individuals. The individual-based approach is more computer intensive, and offers advantages where individual differences and environmental variation are important components of population dynamics.

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