The parabolic pattern of animal growth: determination of equation parameters and their temperature dependencies



    1. Israel Oceanographic and Limnological Research, The Yigal Allon Kinneret Limnological Laboratory, PO Box 345, Tiberias, 14102. Israel
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1. Parabolic (power) growth is characteristic of many aquatic poikilothermic animals for certain stages of their development. The parabolic pattern describing growth in weight (or length) under constant ambient conditions can be expressed in the following general form:


where Y is growth rate (or specific growth rate), X is animal size, and Ω and τ are coefficients. The constancy of ambient conditions is of cardinal importance in determining τ. The problem of maintaining a constant level of nutrition can be reliably solved only by the presence of food in excess of demand. Data satisfying these requirements have demonstrated that τ does not depend on factors such as temperature, and can be assumed to be independent of ambient conditions. In the growth rate-weight equation, τ ranges between 0.5 and 0.85 for animals representing a variety of taxonomic groups.

2. The coefficient Ω. is affected by ambient conditions (e. g. temperature, amount of food). Its value reflects the ‘level’ of the growth rate-size relationship under given conditions. For a specific time period, Ω can be computed from the following formula:


where X1 and X2 are the animal sizes (weights, lengths) at time t1 and t2, the beginning and end of the time period. The calculated value of Ω corresponds to the average intensity of the ambient factor (F) affecting the growth during the period between the two observations. If the values of the Ω are calculated for wide range of the factor, the relationship between the Ω. and F, Ω=f(F), can be determined. The function can be then incorporated into the parabolic equation of growth, as


3. Dependence of the development rate (1/D, where D is time interval needed to complete a given stage) on temperature (T), and dependence of Ω on T, are both described by sigmoid-shape curves. The broad intermediate part of these curves, a range to which animals are adapted in nature, can be approximated by straight line functions. For two groups, pan-size sockeye salmon (Oncorhynchus nerka) and different species of chironomid larvae, it was shown that an equation combining parabolic growth and linear temperature patterns describes accurately the variability observed in growth rates under experimental and natural conditions.