The von Bertalanffy Growth Model

The classic among growth models for fish was derived by Ludwig von Bertalanffy and adapted to use in fisheries by Ray Beverton and Sidney Holt. The von Bertalanffy growth model incorporates indeterminate growth and fits well with observed data, both for individual growth trajectories and for population averages (Figure 1). Its mechanistic derivation assumes that the processes of ana-bolism and catabolism have different exponents in their scaling relationships. Anabolism, or the acquisition of resources, is assumed to be proportional to body surface and thus scales with W2/3, whereas the catabolic costs of activity and maintenance are assumed proportional to body mass and scale as W1. Denoting the proportionality coefficient for anabolism and catabolism a and c, respectively, the growth rate in mass can be expressed as dW ~dt

where the exponent for anabolism, mb = 2/3, and the exponent for catabolism, m2 = 1, are as indicated above. The result is that as the individual grows larger, more and more of the available energy will be used for maintenance and growth will slow down and eventually stop. This asymptotic body size is denoted W1 and for weight and length, respectively.

For calculating the length L at a given time t, the above equation can be rearranged to obtain

or, for calculating the weight W at time t,

where k is the growth coefficient and t0 is the hypothetical age at which L (and thus W) equals zero, and b is the exponent of the age-length relationship W = aLb. In Figure 1, the effects of different asymptotic lengths and growth coefficients on the resulting lifetime growth trajectories are illustrated. From these graphs it is easy to see the asymptotic nature of the von Bertalanffy growth curve: it approaches the asymptotic length Lm with a declining growth rate defined by k.

Critique of the von Bertalanffy model

Even though widely used and despite its simplicity and elegance, the von Bertalanffy growth model has drawbacks. First, the exponents that were used in the original derivation have later been shown to be wrong: both catabolism and anabolism scale with exponents in the range 0.7-0.8 in most fish species studied. Second, the mechanism that was used to explain the asymptotic body size in the von Bertalanffy derivation was that all the available energy was used for maintenance, leaving the fish with only enough energy to feed and digest but not to grow (or any other activity for that matter). By adopting the perspective of life-history theory, which dictates that

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Figure 1 The von Bertalanffy growth model. (a) Examples of von Bertalanffy growth curves for different values of asymptotic length (black lines: L„ = 50 cm; gray lines: L1 = 25 cm) and growth parameter k (solid lines: k=0.2, dotted lines: k = 0.1). (b) Length-at-age for three different smallmouth bass Micropterus dolomieu individuals from Lake Opeongo, Canada. Filled symbols represent the observed age at first spawning and fits represent individual von Bertalanffy growth curves. (c) Observations of individual length-at-age across a population of smallmouth bass M. dolomieu from Lake Opeongo, Canada. The line is the fitted population-level von Bertalanffy growth curve.

Figure 1 The von Bertalanffy growth model. (a) Examples of von Bertalanffy growth curves for different values of asymptotic length (black lines: L„ = 50 cm; gray lines: L1 = 25 cm) and growth parameter k (solid lines: k=0.2, dotted lines: k = 0.1). (b) Length-at-age for three different smallmouth bass Micropterus dolomieu individuals from Lake Opeongo, Canada. Filled symbols represent the observed age at first spawning and fits represent individual von Bertalanffy growth curves. (c) Observations of individual length-at-age across a population of smallmouth bass M. dolomieu from Lake Opeongo, Canada. The line is the fitted population-level von Bertalanffy growth curve.

the body is a tool for efficient reproduction, we can see that the fish should stop growing when it is most efficient at acquiring energy that can be used to produce offspring, that is, not when the difference between anabolism and catabolism is zero but when the difference is at its maximum. Third, the von Bertalanffy curve fits well for adult growth but represents juvenile growth less accurately. In the juvenile phase, individuals devote all available energy into growing somatic tissues - muscles, bones, and the like - and do not expend energy for producing reproductive material. Empirically, many studies suggest that length growth is linear prior to sexual maturation, and that growth decelerates when energy is used for gonad development and reproduction. This expected change in growth rate at maturation is not included in the von Bertalanffy growth curve.

In summary, although the von Bertalanffy growth model fits well with observations for fish after maturation, it does less well in describing immature fish growth, and the mechanisms that are underlying it have turned out to be false. Two newer models address these drawbacks and they will be explained in the following two sections: first, Derek Roff included the costs of maturation and derived a mechanistic growth model that is more consistent with fish physiology and empirical observations of gonad maturation; later, Nigel Lester and colleagues developed a mechanistic model similar to Roffs, resulting in a model that, during the postmaturation phase, is mathematically very similar to the von Bertalanffy model but with parameters that are biologically more meaningful.

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