Dynamics of a two-dimensional system of rational difference equations of Leslie--Gower type

Abstract

We investigate global dynamics of the following systems of difference equationswhere the parameters a 1 , b 1 , A 1 , g 2 , A 2 , B 2 are positive numbers, and the initial conditions x 0 and y 0 are arbitrary nonnegative numbers. We show that this system has rich dynamics which depends on the region of parametric space. We show that the basins of attractions of different locally asymptotically stable equilibrium points or non-hyperbolic equilibrium points are separated by the global stable manifolds of either saddle points or non-hyperbolic equilibrium points. We give examples of a globally attractive non-hyperbolic equilibrium point and a semi-stable non-hyperbolic equilibrium point. We also give an example of two local attractors with precisely determined basins of attraction. Finally, in some regions of parameters, we give an explicit formula for the global stable manifold.

Basins of Attraction for Two-Species Competitive Model with Quadratic Terms and the Singular Allee Effect

“…The global dynamics of system (6) was completed in [7]. Several variations of system (6) where the competition of two species was modeled by linear fractional difference equations were considered in [8][9][10][11][12][13][14]. An interesting fact is that none of these models exhibited the Allee effect.…”

Global Dynamics and Bifurcations Analysis of a Two-Dimensional Discrete-Time Lotka-Volterra Model

“…Analyzing the behavior of solutions of a higher-order nonlinear difference equation is very interesting and attracted many researchers in recent times. Behavior of solutions means studying the equilibrium point, boundedness and persistence, existence and uniqueness of positive equilibrium point, local and global stability, periodicity nature of such difference equations or systems of difference equations (see [8][9][10][11][12][13][14][15][16] and references cited therein).…”

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].…”