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.

Methods for Transformation of Rectangular Spatial Coordinates to Geodetic Coordinates

The article gives a brief analysis of methods and algorithms for the transformation of spatial rectangular coordinates to curvilinear coordinates - geodetic latitude, geodetic longitude, geodetic height. Two algorithms for solving the equation for determining longitude are considered. Three formulas used to calculate the height are analyzed, with an estimate of their errors due to the approximate latitude. The shortcomings of mathematical solutions to these problems are revealed. A study of different approaches and methods for solving the transcendental equation for determining the latitude, based on the theory of separation of the root of the equation, is performed. Using this technique, iterative processes were performed to calculate the reduced latitude , using trigonometric identities, by introducing an auxiliary angle and transforming it to an algebraic quartic equation, which Borkowski solves by the Ferrari's method. The determination of the root isolation interval allowed using the chord method (proportional parts) to determine the latitude. In all cases, estimates of the convergence of the iterative processes that facilitate the comparative analysis of the proposed solutions are obtained. By further decreasing the separation interval of the root, the accuracy of the non-iterative determination of the latitude is improved by the Newton method.