Abstract: Firstly, we used  the Jury criterion and spectral radius method to study the switching dynamics models of systems with pulse time-delay harvesting and birth, proved the  asymptotic stability of the periodic solutions of the pulse systems and gave the threshold for population extinction. Secondly, we considered the continuous survival of the population and the amount of harvest, discussed the reasonable values of the harvest delay under different circumstances, and verified the theoretical results by numerical simulation.

Key words: pulse time-delay harvesting, pulse birth, Jury criterion, spectral radius,  , asymptotic stability