Multiple Attractors for a Competitive System of Rational Difference Equations in the Plane

Abstract

We investigate global dynamics of the following systems of difference equationsxn+1=β1xn/(B1xn+yn),yn+1=(α2+γ2yn)/(A2+xn),n=0,1,2,…, where the parametersβ1,B1,β2,α2,γ2,A2are positive numbers, and initial conditionsx0andy0are arbitrary nonnegative numbers such thatx0+y0>0. We show that this system has up to three equilibrium points with various dynamics which depends on the part of parametric space. We show that the basins of attractions of different locally asymptotically stable equilibrium points or nonhyperbolic equilibrium points are separated by the global stable manifolds of either saddle points or of nonhyperbolic equilibrium points. We give an example of globally attractive nonhyperbolic equilibrium point and semistable non-hyperbolic equilibrium point.

Bifurcation and Global Dynamics of a Leslie-Gower Type Competitive System of Rational Difference Equations with Quadratic Terms

“…Competitive and cooperative systems have been investigated by many authors; see [1][2][3][7][8][9][10][11][12][13][14][15][16]. Special attention to discrete competitive and cooperative systems in the plane was given in [1-3, 16, 17].…”

Global dynamics of (1,2)- type systems of difference equations

“…This made the study of qualitative behavior of difference equations an active area of research (see [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17] and references cited therein). For instance, Touafek and Elsayed [18,19] investigated the behavior of following systems of difference equations:…”

Stability and Neimark–Sacker bifurcation of a ratio‐dependence predator–prey model

“…Yang and Li [10] studied the permanence of species for a delayed discrete ratio-dependent predator-prey model with monotonic functional response. Study of discrete dynamical behavior of systems is usually focussed on boundedness and persistence, existence and uniqueness of equilibria, periodicity, and there local and global stability (see for example, [11][12][13][14][15][16][17][18][19][20][21][22][23][24][25]), but there are few articles that discuss the dynamical behavior of discrete-time predator-prey models for exploring the possibility of bifurcation and chaos phenomena [26][27][28][29][30].…”