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Volume 20,     Number 4,     Winter 2012

 

DIFFUSIVE SYSTEMS WITH SEASONALITY:
EVENTUALLY STRONGLY
ORDER-PRESERVING PERIODIC PROCESSES
AND RANGE EXPANSION OF TICK
POPULATIONS

Dedicated to the 70th birthday of Professor Herb Freedman

XIAOTIAN WU AND JIANHONG WU

Abstract. Differential systems with periodic coecients arise naturally from population or disease transmission dynamics of species whose ecological or epidemiological activities are highly regulated by seasonality. Such a system may generate an order-preserving periodic process, but only some iteration of the periodic map can be strongly order-preserving due to the seasonal on-or-off biological activities. Hence how to derive qualitative properties of the periodic map from those of its iterations is of practical implication. Here, we consider a special case of such a periodic system arising from the consideration of range expansion of tick populations involved in the Lyme disease spread in Canada. We develop a periodic system of reaction-di usion equations capturing some key stages and stage developments of the tick population as well as the spatial random movements of involved hosts, and we establish the existence of periodic traveling waves and calculate the range expansion speed by applying some existing results of strongly order-preserving periodic processes with strictly subhomogeneous nonlinearities to a certain iteration of the associated periodic map.

 

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