## Environmental Variability

Many of the growth models above are designed for fisheries purposes and therefore, updated on a yearly basis to fit annual sampling programs. This ignores the effects of seasonality on growth and reproduction, which are pronounced in boreal and temporal ecosystems (e.g., because of temperature and light variation) and common even in tropical lakes and oceans (e.g., because of rainy seasons or lunar cycles).

Seasonality in growth arises through direct effects such as temperature limitation on physiological growth rates or

Figure 3 Density-dependent growth. (a) The effect of population density on growth of a freshwater salmonid Coregonus hoyi from Lake Michigan. This population was estimated to have L1 = 53.3 cm, k = 0.21, g = 0.378 cm kg ha~1, average biomass B = 8.8 kg ha~1 (solid line), lowest observed B = 1kg ha~1 and highest observed B = 33 kg ha~1 (upper and lower dashed lines, respectively). (b) Increasing population density B and competition coefficient g decrease the asymptotic length LlB.

productivity cycles in food resources, but also indirectly through establishing fixed points that the annual timing of events has to conform to. For instance, the match-mismatch hypothesis predicts particularly favorable temporal windows for the development of eggs and larvae, which in turn sets constraints for the phenology of migrations, spawning, and thereby also for growth. Such seasonality can be taken into account when modeling fish growth, but it requires shorter time intervals for updating the individual size and explicit modeling of the ecological factors that underlie seasonality.

Typically, physiological rates, such as growth rate, have an optimum for any given environmental variable, and if the level of the environmental variable is above or below this, the physiological rate slows down (Figure 4). This effect can be taken into account when modeling growth. For example, the von Bertalanffy growth equation (eqn [3]) can be modified to include the effect of environmental variability on growth. An environmental variable such as temperature, salinity, or oxygen saturation can be transformed to a coefficient XE:

where Em;n, Emax, and E0pt are the minimum, maximum, and optimum environmental variables, respectively. The growth coefficient k from the von Bertalanffy growth equation can then be calculated by multiplying the growth coefficient in the optimal temperature kopt with the environment coefficient:

Environmental variable

Figure 4 Relationship between an environmental variable E (e.g., temperature) and a physiological rate such as growth rate. Different physiological processes may have different optima, feeding, and digestion might, for example, have higher temperature optimum than metabolic rate or aerobic scope. Adapted from Mallet JP, Charles S, Persat H, and Auger P (1999) Growth modelling in accordance with daily water temperature in European grayling (Thymallus thymallus L.) Canadian Journal of Fisheries and Aquatic Sciences 56: 994-1000, figure 3.

k = koptXE

Environmental variable

Figure 4 Relationship between an environmental variable E (e.g., temperature) and a physiological rate such as growth rate. Different physiological processes may have different optima, feeding, and digestion might, for example, have higher temperature optimum than metabolic rate or aerobic scope. Adapted from Mallet JP, Charles S, Persat H, and Auger P (1999) Growth modelling in accordance with daily water temperature in European grayling (Thymallus thymallus L.) Canadian Journal of Fisheries and Aquatic Sciences 56: 994-1000, figure 3.

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