Size-dependent mortality rate profiles

• A conservation law theory of the shape of size distributions in populations is extended.
• Size-specific mortality rate equations of populations for realistic cases are derived.
• Two applications of contrasting characteristics are presented as case studies.
• The extended theory is useful in the stock assessment of small-scale and invertebrate fisheries.

Knowledge of mortality rates is crucial to the understanding of population dynamics in populations of free-living fish and invertebrates in marine and freshwater environments, and consequently to sustainable resource management. There is a well developed theory of population dynamics based on age distributions that allow direct estimation of mortality rates. However, for most cases the aging of individuals is difficult or age distributions are not available for other reasons. The body size distribution is a widely available alternative although the theory underlying the formation of its shape is more complicated than in the case of age distributions. A solid theory of the time evolution of a population structured by any physiological variable has been developed in 1960s and 1970s by adapting the Hamilton–Jacobi formulation of classical mechanics, and equations to estimate the body size-distributed mortality profile have been derived for simple cases. Here I extend those results with regards to the size-distributed mortality profile to complex cases of non-stationary populations, individuals growing according to a generalised growth model and seasonally patterned recruitment pulses. I apply resulting methods to two cases in the marine environment, a benthic crustacean population that was growing during the period of observation and whose individuals grow with negative acceleration, and a sea urchin coastal population that is undergoing a stable cycle of two equilibrium points in population size whose individuals grow with varying acceleration that switches sign along the size range. The extension is very general and substantially widens the applicability of the theory.