Retraction Note: Dynamics of a population-hibernation model with genic mutation and impulsive effects in a polluted environment

In this work, we present a population-hibernation system with genic mutation and impulsive effects in a polluted environment. The \((0,0,\widetilde{c_{o}},\widetilde{c_{e}})\) of (1) is globally asymptotically stable. (1) is also proved to be permanent.

In this work, we consider a genic mutational predator–prey system with birth pulse and impulsive cutting on prey population at different moments. All the solutions of the investigated system are proved to be uniformly ultimately bounded. The conditions of the globally asymptotically stable predator-extinction boundary periodic solution of the investigated system are obtained. The permanent conditions of the investigated system are also obtained. Finally, numerical simulations are inserted to illustrate the results. Our results present that the genic mutational rate plays an important role on the permanence of the investigated system. Our results also provide reliable tactic basis for the practical biological economics management.

A predator and prey-hibernation system with mutation and impulsive effects is presented in this work. The globally asymptotically stable solution \((\widetilde{x}, \widetilde{y},0)\) of system (1) exists. System (1) is proved to be permanent. Finally, simulations are presented to explain the results.

Dynamics of a population-hibernation model with genic mutation and impulsive effects in a polluted environment

Retraction Note: Landesman-Lazer type condition for second-order differential equations at resonance with impulsive effects