Death rates and time intervals: is there an alternative to the constant natural mortality axiom?

Summary

The ‘constant M axiom’ refers to the often automatic assumption made in fish stock assessment, that the instantaneous natural mortality rate, expressed on an annual basis, is constant throughout the exploited phase of the life history. This uncritical assumption or axiom has come under criticism in recent years, best documented by field determinations of predation rates on juvenile fish coordinated in the North Sea through the International Council for the Exploration of the Sea (ICES). These, and earlier studies, have yielded empirical vectors of predation-related M for juveniles of key species, and in some cases, also for egg and larval stages. To date, perhaps the mathematical function with the widest acceptance for modelling M-at-age in the animal kingdom has been the Weibull distribution. This paper attempts to demonstrate, for a limited sample of highly fecund marine organisms, that a simple, two-parameter reciprocal function of age is a more reasonable, and mathematically parsimonious, description of the form of the life history M-vector between egg and larval stages and the mean parental age (MPA). Mortality rates may increase later in life, especially for semelparous species, and in this case would need to be described by an additional or more complex function, beyond the scope of this paper.

Seasonal and inter-annual variations in M-at-age, though rarely documented, also occur; in the latter case, when associated with density dependence, they give rise to the stock-recruit relationships described in the literature, and lead to annual divergences from mean expectation of death, especially for the earliest life-history stages. Contrary to preconception, however, there seems no overwhelming evidence from egg and larval studies for the consistent action of a critical period: the very high egg and larval M-at-age values predicted by the reciprocal function, though continuous in time, effectively correspond to catastrophic early mortality rates, but provide a mean expectation of death with age that converges smoothly on a constant M value at greater ages.

The use of any steeply declining mortality function of age, in simulating survivorship over a time period T between spawning and maturity of the succeeding generation, reveals a serious difficulty in defining appropriate time units. It is concluded that if the reciprocal mortality function applies, then simulating constant survivors at time T independently of the number of smaller time units (Δt) that T is divided into, is effectively accomplished by using progressively longer time intervals with age t, such that Δt/t is constant for all intervals. This is referred to as ‘proportional time allocation’.

Calculations assuming population stability, and using indicative values for mean lifetime fecundity (MLF) and mean parental age (MPA), are presented for a generalized demersal and a generalized small pelagic fish, employing the reciprocal function with proportional time allocation. Sets of values for the two parameters of the reciprocal function were not rejected, even though non-unique, if they reduced the MLF to two individuals by MPA, and a unique vector of M-at-age presumably could be determined if the ‘constant adult M’ is already known. The new model has the advantage of being compatible with the ‘constant adult M’ hypothesis over periods of roughly 4–5 and 1+ years, respectively, for generalized northern demersals and small pelagics, over the ages just preceding MPA.

The possibility of extension of this model to length-based methods, and its analogies with crustacean moulting models, are discussed.