The impact of repeat spawning of males on effective number of breeders in hatchery operations

In conservation hatcheries, typically a great emphasis is placed on maximizing the effective number of breeders through spawning an appropriate number of spawners, spawning equal numbers of males and females when possible, and by mating either in a single-pair or factorial fashion. A commonly encountered problem, especially in small operations with anadromous salmonids, is a shortage of males at some point during the season, so that some males need to be spawned multiple times. This has an impact on the genetically effective number of breeders, and the commonly used equations for effective number are not applicable. In this paper I derive a relatively simple equation for effective number of breeders when some males are multiply spawned but reproductive contribution within spawning classes are equal: Nb,m≈Ntot−1/μt∑i=1k−1∑j=i+1kNiNj(i−j)2(∑i=1kiNi)2+1, where i is the number of times a male is spawned, Ni the number of males spawned i times, and μt the mean gametic contribution. I also derive an equation that allows relaxation of the assumption of equal contribution within classes. I then develop a number of simple and accurate approximations for the effective number of breeders. The equations can be applied to any situation in which there are discrete spawning classes in which the average reproductive contribution of any one class can be expressed as a multiple of that of another class.