Abstract
In this paper, we study a three-stage tick population model with three development delays. Using the total delay τ as the bifurcation parameter, we conduct local and global Hopf bifurcation analysis. Especially, we examine the onset and termination of Hopf bifurcations of periodic solutions from the unique positive equilibrium. We locate all of the stability switches for the equilibrium and demonstrate that the global Hopf bifurcation branches are bounded. This result implies the system undergoes oscillatory behavior only with bounded delays. The key step in the proof is the exclusion of 2τ-periodic solutions due to the negative Lozinskiĭ measure of the corresponding compound matrix. Numerical simulations are provided to show the complicated patterns of the tick population when the development delays are altered by the temperature and other environmental factors.
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