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An impulsive predator-prey model with communicable disease in the prey species only. (English) Zbl 1162.92043
Summary: A system of impulsive differential equations describing predator-prey dynamics with impulsive effect is proposed and analyzed with the assumption that a transmissible disease is spreading among the prey species only. At first, the “semi-trivial” periodic solution $\left(S\left(t\right),0,0\right)$ is given. After that, the existence of an “infection-free” periodic solution $\left(S\left(t\right),0,P\left(t\right)\right)$ and the “predator-free” periodic solution have been obtained via bifurcations. Finally, the method of coincidence degree has been used to derive a set of sufficient conditions for the existence of at least one strictly positive periodic solution. Numerical simulations and a brief discussion conclude the paper.
##### MSC:
 92D40 Ecology 34A37 Differential equations with impulses 92D30 Epidemiology 34C25 Periodic solutions of ODE