Packard (Packard, 2012) restudied Huxley's measurements of Uca pugnax and presented not only a two-parameter power function but also a three-parameter power function; this three-parameter model ‘is better than the two-parameter model for describing the observations’. The two-parameter model has a biological meaning, as explained by Huxley (Huxley, 1924); the three-parameter model has no biological meaning.
I have shown (Geraert, 2004) that there is a constant change in the relationship (and not a constant relationship) between a small amount of growth of body part y compared with that of body part x; mathematically speaking, the ‘second difference’ is constant. This second difference is the growth rate and is present in the quadratic factor of a quadratic equation; the other factors in the equation have no biological meaning but are necessary to position the quadratic curve in a diagram.
In my study (Geraert, 2004), growth is followed from the new-born stage to the adult; in Huxley's study on the fiddler crab, a comparison is made among adult males; as these adults show a very large variation in the development of the claw, Huxley (Huxley, 1924; Huxley, 1932) interpreted this also as ‘growth’. An attempt is made to see whether a quadratic parabola can also be used to describe variation in adults, called here ‘comparative’ growth.
Diagram representing the measurements obtained by Huxley (modified from Huxley, 1932) of the males of the fiddler crab (Uca pugnax); the dashed line is the calculated quadratic parabola.
Diagram representing the measurements obtained by Huxley (modified from Huxley, 1932) of the males of the fiddler crab (Uca pugnax); the dashed line is the calculated quadratic parabola.
In the case of Uca pugnax, a quadratic curve describes the comparative growth in males in a satisfactory way; moreover, it also has a biological meaning. Power curves with two parameters and three parameters (Packard, 2012) describe the phenomenon as well, but the three-parameter curve has no biological meaning. In my study (Geraert, 2004), the quadratic curve describes real growth in such an impressive way that fare-reaching conclusions could be made; it seems not necessary to introduce similar conclusions for comparative growth.
In a study on comparative growth in adults, if a curved line is obtained it is interesting to evaluate whether a quadratic curve is appropriate to describe the phenomenon; if the match is good enough, the ‘growth’ rate, indicated by the quadratic factor, could help us to make assumptions about larger and/or smaller values. The term allometry can be continued, although it can be mathematically formulated by a quadratic curve instead of a power curve.
