Volume 19, Number 1, Spring 2011
GLOBAL DYNAMICS OF A TICK IXODES
YIJUN LOU AND JIANHONG WU
Abstract. Lyme disease remains the world's most frequently recorded vector-borne disease in the temperate zone, with the black-legged tick, Ixodes scapularis Say, as the primary vector in eastern and mid-western United States and Canada. A preliminary to determine the Lyme disease risk is providing detailed information on the tick population. This paper establishes the global dynamics of a tick population model developed to capture dynamical temperature influences on the tick population. It is shown that if the reproduction number for ticks Rtick is less than one, then ticks are doomed to extinction, and it is confirmed that Rtick > 1 implies that ticks can invade into the study region and a positive equilibrium exists. We also use a fluctuation argument to establish the global stability of the positive equilibrium.