Allometric Growth Allometric equations describe the scaling relationship between two

Chapter 5, Problem 41

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Allometric Growth Allometric equations describe the scaling relationship between two measurements, such as tree height versus tree diameter or skull length versus backbone length. These equations are often of the form Y = bXa (5.8) where b is some positive constant and a is a constant that can be positive, negative, or zero. (a) Assume that X and Y are body measurements (and therefore positive) and that their relationship is described by an allometric equation of the form (5.8). For what values of a is Y an increasing function of X, but one such that the ratio Y/X decreases with increasing X? Is Y concave up or concave down in this case? (b) In vertebrates, we typically find [skull length] [body length]a for some a (0, 1). Use your answer in (a) to explain what this means for skull length versus body length in juveniles versus adults; that is, at which developmental stage do vertebrates have larger skulls relative to their body length?