FRACTAL CHARACTERISTICS OF LENGTH-WEIGHT RELATIONSHIP IN FISH

In the study of measurable or numerical properties of fish bodies, statisticalmethods are usually employed to establish equations of the correlation among theproperties. For example, the body length (L)-weight (W) relationship has generallybeen described as W=bLα,W= aL+b, W=b1 + b2L+ b3L2, W= beαL or W= beαL,in various literatures. However, only a few authors have defined the biologicalsignificance of the parameters in these equations, but no one has ever considered howto resolve the problem of dimension disharmony when the weight is expressed inrelation to the length. In fact the significant"correlation" may be found statisticallyeven if there is no biological relationship between any two physical measures. But ageneralized applicable expression must be established when the biological significanceand dimension harmony of statistical parameters is considered. In the present paper,the mass of fish body is assumed as being fractally distributed, the length-weightrelationship of fish can be described as W= ρkL3m/D on the basis of the fractalrelationship between surface area and volume in irregular bodies, where and m areconstants, D is the fractal dimension, and k is a factor including D. If b =ρk and α = 3m/D, W= ρkL3m/D can be recorded as W= bLα. It is shown that there areproblems of dimension disharmony in the statistical relationship between the bodylength and weight in fish except the equation W= bLα, and that if the mass of fishbody is a constant the fractal dimension D in the expression W= ρkL3m/D representsthe degree of uniform in the growth speed between length and weight of fish.Moreover, calculations of parameter a within different body length intervals revealedthat from juvenile to senility, the ratio of fish growth between length and weight isinitially varied, then uniformed, and finally back to varied.