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Abstract
We study a model of bird migration between the summer breeding ground and the winter refuge site. The model involves time lags for the transition time between the patches, and the model parameters are periodic in time as the biological activities related to the migration and reproduction are seasonal. It has been shown in previous studies that the model system exhibits threshold dynamics: either all solutions converge to the trivial solution, or the system has a positive and globally attractive periodic solution. Two issues remain and will be addressed in this paper: how to express the threshold condition in terms of model parameters explicitly (rather than the abstract spectral radius of a certain monodromy operator) and how to describe the temporal pattern of the positive periodic solution. We make an interesting and surprising observation that the delay differential system is completely characterized by a finite-dimensional ordinary differential system and then a finite-dimensional map in the sense that the bird population at the initial time of spring migration determines the future status of the system. As a consequence, we derive the threshold condition, explicitly in terms of the model parameters, for the extinction and persistence of the considered bird species.
Highlights
The functions mws(t) and msw(t) describe the migration rates between the two patches (w, winter; s, summer; ws, from the winter patch to the summer patch; sw, from the summer patch to the winter patch)
The above model represents a simplified version of the patchy model developed in Bourouiba et al (2010) and Gourley et al (2010) to capture the essential dynamic features of bird migration, and to explore the roles of bird migration in the global spread of highly pathogenic H5N1 avian influenza
A major observation is that a solution of system (1.1) is given by an enlarged system of periodic ordinary differential equations (ODEs), with state variables corresponding to the numbers of birds in different intervals of a given year
Summary
We follow Bourouiba et al (2010) and Gourley et al (2010) to consider a single species bird population migrating between a summer breeding patch and a winter refuge patch. The aforementioned studies parametrized the model using satellite tracking records of a dozen Bar-headed geese migrating between the summer breeding site in the northern part of their migration path (e.g. Mongolia) and the wintering grounds (e.g. India), and established the model’s threshold dynamics under very general conditions on the birth rate. They proved that either all solutions converge to the trivial solution, or the system has a positive and globally attractive periodic solution. We will discuss how these assumptions can be relaxed
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