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This paper studies the scaling properties of recruitment fluctuations inrandomly varying environments, for abundant marine species with extremereproductive behavior. Fisheries stock-recruitment data from the North Atlanticdisplay fluctuation scaling, a proportionality between the standard deviationand the average recruitment among stocks. The proportionality covers over fiveorders of magnitude in the range studied. A linear-scaling behavior can be asign of a universal distribution of the normalized data across stocks. In lightof this conjecture, it is demonstrated that the L\'evy-stable model offers abetter effective description of the recruitment distribution than thelog-normal model. Care is devoted to the problem of random sums of randomvariables. Recruitment is calculated by summing random offspring numbers withinfinite variance, where the number of summands (i.e. spawning population size)is also a random process with infinite variance.

Fluctuation scaling in L\'evy-stable recruitment of marine fishes in randomly varying environments