On the invertibility of Leslie systems

Abstract

The problem of reconstructing the stocking rate of exploited renewable resources from an output time series is considered. The resource is described by a linear age-structured model of the Leslie type, and the measured output is a linear and positive combination of the number of individuals in each reproductive class. The problem can be formally answered by using a well-known result of system theory: a linear system is invertible if and only if the zeros of its transfer function are all smaller than one in modulus. In the present case, this would imply working with a polynomial of degree n − 1 ( n being the maximum age of the population). By explicitly computing the transfer function, we show that the invertibility of the system can indeed be established by checking if the zeros of a polynomial of degree r − 1 ( r being the number of reproductive age classes) are smaller than one in modulus. Moreover, we investigate two particular measurement schemes and we show that the first one, which is possibly the most commonly used to sample animal populations, gives rise to an invertible system. On the contrary, the second scheme can generate noninvertible systems, although two single conditions for invertibility that we point out in the paper seem to be often satisfied in the case of real populations.