Discrete stage-structured tick population dynamical system with diapause and control

A discrete stage-structured tick population dynamical system with diapause is studied, and spraying acaricides as the control strategy is considered in detail. We stratify vector populations in terms of their maturity status as immature and mature subgroups. The immature subgroup is divided into two categories: normal immature and diapause immature. We compute the net reproduction number $ R_0 $ and perform a qualitative analysis. When $ R_0 < 1 $, the global asymptotic stability of tick-free fixed point is well proved by the inherent projection matrix; there exists a unique coexistence fixed point and the conditions for its asymptotic stability are obtained if and only if $ R_0 > 1; $ the model has transcritical bifurcation if $ R_0 = 1. $ Moreover, we calculate the net reproduction numbers of the model with constant spraying acaricides and periodic spraying acaricides, respectively, and compare the effects of the two methods on controlling tick populations.