Stability and Bifurcation: Discrete Differential Algebraic Planar Model with Square Root Response

Abstract

The stability and existence of bifurcation analysis for a two spices discrete time model by introducing square root functional response and step size is examined in this work. Forward Euler scheme method is applied to formulate the discrete model from the continuous model, particularly to explore the rich dynamical behavior of the proposed model. Because the model has square root response function, trivial and axial equilibrium positions are singular. In order to discuss the stability of the trivial and axial equilibrium positions, a transformation is applied. Moreover, we explore the stability of the interior equilibrium position in a discrete two spices model using jury conditions. The numerical experiments are performed for distinct parameter values and also time series and phase line diagrams are presented. We also apply bifurcation theory to find whether the model of spices undergoes periodic doubling bifurcation at its axial and interior equilibrium positions. Numerical examples are provided and they exhibit rich dynamics in both species, including, period – 2, 4, 8 & 16 orbits, periodic windows and Non periodic orbit(chaos).