A two-species ratio-dependent predator-prey model with time delay and impulse is investigated. By using the continuation theorem of coincidence degree theory, the existence of a positive periodic solution for this system is established.
An SIR epidemic model with saturated treatment function and nonlinear pulse vaccination is studied. The existence and stability of the disease-free periodic solution are investigated. The sufficient conditions for the persistence of the disease are obtained. The existence of the transcritical and flip bifurcations is considered by means of the bifurcation theory. The stability of epidemic periodic solutions is discussed. Furthermore, some numerical simulations are given to illustrate our results.
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