title: Delay differential systems for tick population dynamics.
abstract: Ticks play a critical role as vectors in the transmission and spread of Lyme disease, an emerging infectious disease which can cause severe illness in humans or animals. To understand the transmission dynamics of Lyme disease and other tick-borne diseases, it is necessary to investigate the population dynamics of ticks. Here, we formulate a system of delay differential equations which models the stage structure of the tick population. Temperature can alter the length of time delays in each developmental stage, and so the time delays can vary geographically (and seasonally which we do not consider). We define the basic reproduction number [Formula: see text] of stage structured tick populations. The tick population is uniformly persistent if [Formula: see text] and dies out if [Formula: see text]. We present sufficient conditions under which the unique positive equilibrium point is globally asymptotically stable. In general, the positive equilibrium can be unstable and the system show oscillatory behavior. These oscillations are primarily due to negative feedback within the tick system, but can be enhanced by the time delays of the different developmental stages.
authors.affiliation: Department of Mathematics and Philosophy, Columbus State University, Columbus, GA, 31907, USA. fan_guihong@columbusstate.edu.
authors.forename: Guihong
authors.initials: G
authors.lastname: Fan
authors.affiliation: School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ, 85287-1804, USA. hthieme@asu.edu.
authors.forename: Horst R
authors.initials: HR
authors.lastname: Thieme
authors.affiliation: Department of Mathematics and Statistics, York University, Toronto, ON, Canada. huaiping@mathstat.yorku.ca.
authors.forename: Huaiping
authors.initials: H
authors.lastname: Zhu
mesh_terms: D000818:Animals; D001096:Arachnid Vectors / Q000254:growth & development / Q000382:microbiology / Q000502:physiology; D050936:Basic Reproduction Number; D005260:Female; D006801:Humans; D008193:Lyme Disease / Q000635:transmission; D008297:Male; D055641:Mathematical Concepts; D008954:Models, Biological; D011157:Population Dynamics; D017282:Tick-Borne Diseases / Q000635:transmission; D013987:Ticks / Q000254:growth & development* / Q000382:microbiology / Q000502:physiology
keywords: Basic reproduction number; Delay differential systems; Global stability; Integral equations; Local stability; Persistence; Stage structure; Tick populations