Dynamics of a Prey–Predator System with Foraging Facilitation in Predators

The dynamics of a prey–predator system with foraging facilitation among predators are investigated. The analysis involves the computation of many semi-algebraic systems of large degrees. We apply the pseudo-division reduction, real-root isolation technique and complete discrimination system of polynomial to obtain the parameter conditions for the exact number of equilibria and their qualitative properties as well as do a complete investigation of bifurcations including saddle-node, transcritical, pitchfork, Hopf and Bogdanov–Takens bifurcations. Moreover, numerical simulations are presented to support our theoretical results.

Bifurcation Analysis of an Enzyme Reaction System with General Power of Autocatalysis

In this paper, we study the local bifurcations of an enzyme reaction system with positive parameters [Formula: see text], [Formula: see text], [Formula: see text] and integer [Formula: see text]. This system is orbitally equivalent to a polynomial differential system of order [Formula: see text]. Although not all coordinates of its equilibria can be calculated because of the high order polynomial, parameter conditions for the existence of each equilibrium are given. Moreover, qualitative properties of each equilibrium are determined. It shows that various bifurcations, including saddle-node bifurcation, Bogdanov–Takens bifurcation and Hopf bifurcation, may occur in this system as the parameters are varied. With Lyapunov quantities, the maximum order of the weak focus in this system is proved to be at most two by resultant elimination and real-root isolation. Furthermore, parameter conditions of its exact order are obtained. Finally, numerical simulations demonstrate the theoretical results.