Estimating a Stochastic Model of Population Dynamics With an Application to Kangaroos

Abstract

In this study, we derive stochastic models of population dynamics and devise a new method of estimating the models. The models allow growth and harvest to be nonlinear functions of stochastic processes and the error terms to be nonlinear and heteroskedastic. Ordinary least-squares estimates would be biased and inefficient and generalized leastsquares estimates cannot be calculated. Therefore, we implement nonlinear maximum likelihood methods to find unbiased and efficient estimates of parameters. The method is applied to the population dynamics of kangaroos in South Australia. Aerial survey data of kangaroo numbers are combined with harvest, effort and rainfall data to estimate the growth and harvest functions and the variances of the stochastic processes which drive the system. Results suggest that growth and harvest should be modeled as functions of stochastic processes and that observations on kangaroo numbers are critical for estimating population dynamics. The results also indicate that the estimation method works well and is a viable alternative to ARIMA and GARCH models, particularly for small data sets.

Monotone methods for semilinear parabolic and elliptic equations on graphs

“…And the key point of this approach is to construct a suitable iterative sequence. However, the habitat of some species is on an island in the sea, like the Kangaroo in Australia [13]. There are many reaction-diffusion equations with initial-boundary values used to describe the spread of species, see [6] and references therein.…”

Inferencia e indicadores de viabilidade para modelos estocasticos de crescimento de populações

“…Este estudo consiste na escolha de um modelo adequado, que se ajuste aos conjuntos de observações do número de indivíduos de uma população animal ao longo do tempo, e na inferência dos parâmetros dos modelos. A modelagem é uma ferramenta importante para se compreender como as incertezas ambientais afetam o crescimento de uma população, vide por exemplo, Boyce, 1992 ;Ludwig, 1996 ;Hertzler et al,1997e Holmes, 2001. Tem-se como objetivo calcular indicadores da viabilidade da população na forma da probabilidade da população atingir um determinado nível e do tempo esperado para a população atingir esse nível.…”