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Volume 19,     Number 3,     Fall 2011

 

APPROXIMATING PERIODIC PATTERNS AND
DYNAMIC THRESHOLD FOR PATCHY MODEL
OF MIGRATORY BIRDS WITH DELAY
Dedicated to Professor Herb Freedman on his 70th birthday
XIANG-SHENG WANG AND JIANHONG WU

Abstract. Based on finite dimensional reduction methods, we approximate the spatial dynamic model of seasonal migration birds with stopovers by a simple discrete periodic system which captures four seasonal activities: spring migration, summer breeding, autumn migration and winter refuging. The transient functions describing the mapping of bird population from one major patch to another are obtained in terms of model parameters such as the migration rate and mortality rate. Using perturbation techniques, we further derive an explicit asymptotic formula for the persistent/extinct threshold of the bird population. Sensitivity analysis shows that this dynamic threshold is more vulnerable to the change of bird death rates at autumn migratory stopovers than that at spring migratory stopovers. We thus conclude that the effect of repeated epizootic (i.e. H5N1) during autumn season is stronger than spring season. This conclusion supports numerical phenomenon simulated in existing literature.

 

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