A Method for Determining Mortality Rates Using the Leslie Matrix

A Method for Determining Mortality Rates Using the Leslie Matrix DOUGLAS S. VAUGHAN AND SAUL B. SAILA

Graduate School o/ Oceanography University o/ Rhode Island Kingston, Rhode Island 02881 ABSTRACT

The Leslie matrix algorithm has been utilized to estimate mortality of a year class assuming an equilibrium population for a species. Under this assumption an estimate of the mortality for the 0tn year classof the Atlantic bluefin tuna (Thunnusthynnusthynnus) has been made indicating about five survivorsfrom 10 million eggs in the first year of life. The mortality rates for later year classeswere derived from empirical data.

If the later age-specificmortality rates are determined independently,then the mortality obtain reasonable estimates of early life his- rate for the 0tn year class can be derived. It is necessaryto define the following extory stage mortality rates. In most fishery investigationsit is usual to estimatethe total pressions: let n•.(t) be the number of indimortality rate from the decline in numbers viduals of age x at time t, m• be the fecunof several year classesas they pass through dity (number of young born) per individual a fishery and then to subtract the fishing of age x, l• be the probability that an indimortality rate which has been estimated by vidual survivesto age x, and P:,.be the ageother means, such as marking experiments. specific survival rate or probability that an It usually is unreasonableto extrapolatethis individual reaching age x will survive to age natural mortality rate to the time of birth x+l. In studying the dynamics of natural popu-

lations one of the most difficult

tasks is to

because most fisheries are based on species which are highly fecund. It also seems un-

Based on the above definitions that

reasonable (without prohibitively high cost

it is clear

no(t) = ]• n•,(t) m•

and effort) to make direct estimates of natural mortality rates during the first year of life for many important fish species. This

or the number of individuals in the initial age class is equal to the sum over all age classes note describes an indirect method for estiof their birth products. If mortality is asmating the mortality rate for the 0tn year class sumed invariant over time, based on age-fecunditydata and age-specific n•.(t) = no(t-x) l•, probabilities of survival derived from convenso that, tional fisheries statistics under the assump-

no(t)=]•no(t-x)l•,m.•..

tion of an equilibrium or stable population. LIFE

TABLE

METHODOLOGY

Life tables,developedoriginally for human demography,are used to systematicallypresent age-specificmortality and fecundity. By determining the above parametersthe Leslie matrix algorithm (Leslie 1945; Emlen 1973) allows one to study the stability of the population of a species. Conversely,by assuming an equilibrium population of a speciesand knowing its age-specific fecundity, one can determine the mortality rate for the species.

(1)

Now define a stable age-structuresuch that for all x, R=n•.(t)/n•.(t-1) and further, for any integer y