Extinction, persistence and growing in a degenerate logistic model with impulses

This paper deals with an impulsive degenerate logistic model, where pulses are introduced for modeling interventions or disturbances, and the degenerate logistic term may describe refugees or protections zones for a species. First, the principal eigenvalue depending on impulse rate, which is regarded as a threshold value, is introduced and characterized. Second, the asymptotic behavior of the population is fully investigated and sufficient conditions for the species to be extinct, persist, or grow unlimitedly are given. Our results extend those of the well-understood logistic and Malthusian models. Finally, numerical simulations emphasize our theoretical results, highlighting that a medium impulse rate is more favorable for species to persist, a small rate results in extinction, and a large rate leads the species to unlimited growth.