Demographic analysis of gummy shark (Mustelus antarcticus) and school shark (Galeorhinus galeus) off southern Australia by applying a generalized Lotka equation and its dual equation

Abstract

Although Lotka's equation is commonly used for calculating the intrinsic rate of increase with time of a fish population in demographic analysis, its dual equation has never been derived. In this paper, we establish an explicit relationship between the intrinsic rate of increase with time of a fish population and its instantaneous rate of natural mortality from an age-dependent population dynamics model, derive a generalized Lotka equation for calculating the intrinsic rate of increase with time, and derive its dual equation for calculating the intrinsic rate of decrease with age. The virginal intrinsic rate of increase with time of the gummy shark (Mustelus antarcticus) population was calculated as 0.115957·year-1 and its intrinsic rate of decrease with age as -0.312957·year-1. The virginal intrinsic rate of increase with time of the school shark (Galeorhinus galeus) population was calculated as 0.109480·year-1 and its intrinsic rate of decrease with age as -0.216980·year-1. The generalized Lotka equation and its dual equation thus derived imply that both reproductive schedules of a population of animals and its instantaneous rate of total mortality determine its intrinsic rate of increase with time, whereas its reproductive schedules alone determine its intrinsic rate of decrease with age.