03/18/2009 09:00 AM
Applied Mathematics & Statistics
The relationship between the biomass of reproductively mature individuals (spawning stock) and the resulting offspring added to the population (recruitment), the stock recruitment relationship, is a fundamental and challenging problem in all of population biology. The steepness of this relationship is the fraction of unfished recruitment obtained when the spawning stock biomass is 20% of its unfished level. Since its introduction about 20 years ago, steepness has become widely used in fishery management, where it is usually treated as a statistical quantity. Here, we investigate the reproductive biology of steepness, using both unstructured (biomass) and age-structured models. We show that if one has sufficient information to construct a density independent population model (maximum per capita productivity and natural mortality for the unstructured case or maximum per capita productivity, natural mortality and schedules of size and maturity at age for the structured model) then one can construct a point estimate for steepness. Thus, minimal information about the demography of a cohort leads to inferences about steepness, which cannot be chosen arbitrarily. If one assumes that individual survival fluctuates within populations, then it is possible to construct a prior distribution for steepness from this same minimal demographic information. We develop the ideas for both compensatory (Beverton-Holt) and over-compensatory (Ricker) stock-recruitment relationships. We illustrate our ideas with an example concerning bluefin tuna.