Density-Dependent Growth and Biological Reference Points for Fisheries Management

In many stock assessments, stock status is determined using biological reference points that are based on spawning per recruit (SPR) or yield per recruit (YPR) calculations. One default assumption for calculating reference points based on SPR or YPR is that the life history parameters of the stock are in steady state and do not change through time as stock biomass increases or decreases due to intraspecific competition and associated density dependence in growth. The assumption that life history parameters are time-invariant, when in fact they are influenced by changes in stock density, introduces approximation errors into biological reference point calculations based on either SPR or YPR.

We show how approximation errors in biological reference points and associated equilibrium biomasses, recruitments, and yields can be quantified when weight at age depends on adult stock biomass. We begin by describing SPR and YPR calculations in a standard discrete time formulation. A first order approximation that describes changes in weight at age as a function of adult biomass is developed and applied to define density-dependent spawning per recruit SPR_D and yield per recruit YPR_D.  We show that assuming growth is independent of stock density, when in reality, it is density dependent leads to the overestimation of SPR and YPR, which in turn, leads to the overestimation of equilibrium biomass, recruitment, and yield at a fixed fishing mortality (F) and selectivity pattern. Although expected fishery yield at a given F will decrease under density-dependent growth, the value of fishing mortality to maximize yield, F_MSY, will generally increase. We illustrate our ideas with a numerical example for Georges Bank haddock (Melanogrammus aeglefinus) and discuss the practical implications for fishery management.