A Generalized Analysis of the Shepherd Function: Estimating the Degree of Compensation From Short-Term Perturbations in Biomass

The Shepherd function is a versatile tool for modeling the relationship between stock recruitment and spawning stock biomass (which is closely related to fecundity or egg production). The key to its versatility lies in the ‘degree of compensation’ parameter (n), which controls the shape of the functional response (often presented in this form or as the inverse of n). If n = 1, the Shepherd function is equivalent to the Beverton-Holt model; if n > 1, it is equivalent to the Cushing model; if n < 1, it is equivalent to the Ricker model. A key difficulty lies in estimating the value for the degree of compensation. Other than solving the equation for n (which requires knowledge of both the recruitment and spawning stock biomass, as well as the other terms in the equation), Shepherd (J. Cons. int. Explor. Mer., 1982) suggested that the value of n could be estimated based on the physiognomic habitat preferences of the organism in question. For example, pelagic fish may have values of n slightly less than 1, and values greater than 1 only when cannibalism is prevalent. In practice, these solutions either require a great deal of information, or are based on somewhat arbitrary principles. Here we use the technique of Generalized Modeling (Gross and Feudel Phys. Rev. E., 2006) to investigate an alternative method by which n can be estimated. We first build a generalized form of the Shepherd model coupled with density dependent mortality, where we treat each function as unknown. We then normalize the system to an unknown internal equilibrium and derive biologically interpretable relationships for the normalized per-capita recruitment and mortality rates, respectively. This procedure enables us to derive equations governing the derivatives (often called elasticities) of these functions with respect to the normalized equilibrium. Finally, we map the specific Shepherd model to our generalized analysis, and solve for the degree of compensation parameter in terms of the generalized parameters. We suggest that, in some cases, these generalized parameters may be easier to approximate as they exclusively rely on observed system states, rather than unnatural states (e.g. rates at saturation) that cannot be observed directly.