Spatiotemporal Distributions of Migratory Birds: Patchy Models with Delay

Abstract

We derive and analyze a mathematical model for the spatiotemporal distribution of a migratory bird species. The birds have specific sites for breeding and winter feeding, and usually several stopover sites along the migration route, and therefore a patch model is the natural choice. However, we also model the journeys of the birds along the flyways, and this is achieved using a continuous space model of reaction-advection type. In this way proper account is taken of flight times and in-flight mortalities which may vary from sector to sector, and this information is featured in the ordinary differential equations for the populations on the patches through the values of the time delays and the model coefficients. The seasonality of the phenomenon is accommodated by having periodic migration and birth rates. The central result of the paper is a very general theorem on the threshold dynamics, obtained using recent results on discrete monotone dynamical systems, for birth functions which are subhomogeneous. For such functions, depending on the spectral radius of a certain operator, either there is a globally attracting periodic solution, or the bird population becomes extinct. Evaluation of the spectral radius is difficult, so we also present, for the particular case of just one stopover site on the migration route, a verifiable sufficient condition for extinction or survival in the form of an attractive periodic solution. This threshold is illustrated numerically using data from the U.S. Geological Survey on the bar-headed goose and its migration to India from its main breeding sites around Lake Qinghai and Mongolia.