Bifurcation Analysis of a Generalized Impulsive Kolmogorov Model With Applications to Pest and Disease Control

Abstract

State-dependent impulsive dynamical systems have been widely applied to model and qualitatively analyze discontinuous and state-dependent managements of pests and various infectious diseases. Rich dynamical behavior and bifurcation phenomena of the state-dependent impulsive models by extending special ODE systems have been studied intensively. In this study, we aim at constructing a general modeling framework of a state-dependent impulsive system and generalizing analytical methods for its dynamical behavior. To do this, we propose a state-dependent impulsive model based on the generalized Kolmogorov model. The existence and global stability of the semitrivial periodic solution (STPS) is discussed initially. By defining a one-parameter family of discrete maps for the proposed system, we completely investigated the transcritical, pitchfork, and backward bifurcations of this generalized state-dependent impulsive model near the STPS with respect to all of the key parameters. Consequently, we showed the existence and stability of a positive order-1 periodic solution near the STPS. Further, we discussed the existence and stability of a positive order-2 periodic solution via flip bifurcation. Finally, we showed that the main results can be easily applied to specialized models in the fields of integrated pest management and the control of infectious disease.