Periodic Solution Bifurcation and Spiking Dynamics of Impacting Predator–Prey Dynamical Model

A planar predator–prey impacting system model with a nonmonotonic functional response function is proposed and analyzed. The existence and stability of a boundary order-1 periodic solution were investigated and the threshold conditions for a transcritical bifurcation and stable switching were obtained, and also the definition and properties of the Poincaré map are discussed. The main results indicate that multiple discontinuous points of the Poincaré map could induce the coexistence of multiple order-1 periodic solutions. Numerical analyses reveal the complex dynamics of the model including periodic adding and halving bifurcations, which could result in multiple active phases, among them rapid spiking and quiescence phases which can switch from one to another and consequently create complex bursting patterns. The main results reveal that it is beneficial to restore the stability and balance of a ecosystem for species with group defence by moderately reducing population densities and the group defence capacity.

Impaired Diffusion Coupling-Source of Arrhythmia in Cell Systems

A detailed analytical and numerical analysis of a simple reaction-diffusion model of the source of non-homogeneities and arrhythmias in an originally homogeneous reaction system is presented. Solution, bifurcation and evolution diagrams are used to describe the behaviour of the model. It is shown that under certain conditions steady and/or oscillatory nonhomogeneous states are the only stable solutions of the model. These phenomena are essentially not dependent on a particular reaction kinetics. A possible relevance to some biological situations is discussed.