Volume 20, Number 4, Winter 2012
DIFFUSIVE SYSTEMS WITH SEASONALITY:
EVENTUALLY STRONGLY
ORDERPRESERVING 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 orderpreserving periodic process, but only some iteration
of the periodic map can be strongly orderpreserving due to
the seasonal onoroff 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 reactiondiusion 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
orderpreserving periodic processes with strictly subhomogeneous nonlinearities to a certain iteration of the associated
periodic map.
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