Relation between mean and variance in catch numbers of fish egg and larva

Abstract

In each of six areas with various horizontal scale from 0.4 to 15.6 nautical miles, many surveys with vertical net hauls are made for sampling fish eggs and larvae. Though values ofC _A are widely spread in each area, the variances ² and the meanm of catch numbers calculated from each survey follow approximately the relation from a Poisson mixture model (PMM).

$$s^2 = m + C_A m^2 $$

. WhereC _A (Kuno's aggegation index) is generally an index of magnitude of variability in catch numbers on the surveyed space (time or time-space). Moreover, this relation is equivalent to the relation between the variance and the mean in a negative binomial distribution model (NBM). However, NBM is a special case in the PMM.

The minimum value ofC _A generally depends on the horizontal scale of each area. Large values ofC _A, however, are not so rare. In order to explain the large values ofC _A, changes ofC _A value are estimated on a simple model area that consists of two sub-areas: in one, some planktonic organisms are abundant; poor in the other. TheC _A values increase with increasing area ratio of the poor sub-area to the whole area. In a widely spread area such as planktonic survey occupied, the change ofC _A value mainly indicates the change of the area ratio as a result.

The Taylor's power law, a famous relation between the variance and the mean in catch numbers, has a conflict with the relation from the PMM. Within a short range of the mean, the relation is nearly a straight line on log scale. The law represented with a strictly straight line was derived mostly from data with short-ranged means, and hence the law is inadequate.