Semistability of two systems of difference equations using centre manifold theory

Abstract

In this paper, we study the stability of the zero equilibria of the following systems of difference equations: xn+1=axn+byne−xn,yn+1=cyn+dxne−yn and xn+1=ayn+bxne−yn,yn+1=cxn+dyne−xn where a, b, c and d are positive constants and the initial conditions x0 and y0 are positive numbers. We study the stability of those systems in the special case when one of the eigenvalues has absolute value equal to 1 and the other eigenvalue has absolute value less than 1, using centre manifold theory. Copyright © 2016 John Wiley & Sons, Ltd.

On the stability of some systems of exponential difference equations

“…From (2.6), we conclude that map h must satisfy the centre manifold equation (see [1, p. 34], [3, p. 243], [15], [16, p. 642] and [19]):…”

“…The complexity of the grassland ecosystem makes its study interesting but complicated. In addition, in [18], the authors studied the boundedness and the persistence of the positive solutions, the existence, Motivated by this discrete time model and recent studies of symmetric and close to symmetric systems of difference equations (see, e.g, [9,11,19,25,26]), in this paper, we will study the stability of the zero equilibria of the following systems: The results of this paper could be used to create more elaborate biological models to facilitate understanding the underlying ecological mechanisms. The results obtained for the systems (1.1) and (1.2) provide conditions for stability of the zero equilibria of those systems.…”

“…For some related cyclic systems of difference equations see [9,28,31] and [32], as well as some three-dimensional systems (see, e.g., [19,27,33]). Finally we note that, since difference equations have several applications in applied sciences, there exists a rich bibliography concerning theory and applications of difference equations (see ).…”

Local Stability in 3D Discrete Dynamical Systems: Application to a Ricker Competition Model

“…For general framework in the field of difference equations or discrete dynamical systems we refer to the books by Carr [6], Elaydi [9], and Kuznetzov [8] and for specific results and terminology the papers by Guzowska et al [13], Luis et al [16], and Psarros et al [17].…”

“…Several applications to planar systems of difference equations are provided in this paper. Luis et al [16] studied a nonlinear planar model with four parameters denominated planar Ricker competition model and Psarros et al [17] studied a three-dimensional model with six parameters.…”