Complex dynamics in a delay differential equation with two delays in tick growth with diapause

We consider a delay differential equation for tick population with diapause, derived from an age-structured population model, with two time lags due to normal and diapause mediated development. We derive threshold conditions for the global asymptotic stability of biologically important equilibria, a...

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Bibliographic Details
Published in:Journal of Differential Equations 2020-12, Vol.269 (12), p.10937-10963
Main Authors: Shu, Hongying, Xu, Wanxiao, Wang, Xiang-Sheng, Wu, Jianhong
Format: Article
Language:English
Subjects:
Delay dependent parameters
Diapause
Geometric criterion
Global Hopf branch
Tick population dynamics
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Summary:We consider a delay differential equation for tick population with diapause, derived from an age-structured population model, with two time lags due to normal and diapause mediated development. We derive threshold conditions for the global asymptotic stability of biologically important equilibria, and give a general geometric criterion for the appearance of Hopf bifurcations in the delay differential system with delay-dependent parameters. By choosing the normal development time delay as a bifurcation parameter, we analyze the stability switches of the positive equilibrium, and examine the onset and termination of Hopf bifurcations of periodic solutions from the positive equilibrium. Under some technical conditions, we show that global Hopf branches are bounded and connected by a pair of Hopf bifurcation values. This allows us to show that diapause can lead to the occurrence of multiple stability switches, coexistence of two stable limit cycles, among other rich dynamical behaviours.
ISSN:0022-0396
1090-2732
DOI:10.1016/j.jde.2020.07.029