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

2011 ◽  
Vol 2011 ◽  
pp. 1-35 ◽  
Author(s):  
S. Kalabušić ◽  
M. R. S. Kulenović ◽  
E. Pilav
Keyword(s):  
Equilibrium Point ◽  
Difference Equations ◽  
Initial Conditions ◽  
Equilibrium Points ◽  
Saddle Points ◽  
Global Dynamics ◽  
Competitive System ◽  
Multiple Attractors ◽  
Rational Difference Equations ◽  
Globally Attractive

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.

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

2017 ◽  
Vol 2017 ◽  
pp. 1-19
Author(s):  
V. Hadžiabdić ◽  
M. R. S. Kulenović ◽  
E. Pilav
Keyword(s):  
Difference Equations ◽  
Initial Conditions ◽  
Competition Model ◽  
Global Dynamics ◽  
Competitive System ◽  
Parametric Space ◽  
Rational Difference Equations

We investigate global dynamics of the following systems of difference equations xn+1=xn/A1+B1xn+C1yn, yn+1=yn2/A2+B2xn+C2yn2, n=0,1,…, where the parameters A1, A2, B1, B2, C1, and C2 are positive numbers and the initial conditions x0 and y0 are arbitrary nonnegative numbers. This system is a version of the Leslie-Gower competition model for two species. We show that this system has rich dynamics which depends on the part of parametric space.

scholarly journals Explicit Solutions and Bifurcations for a System of Rational Difference Equations

Mathematics ◽  
2019 ◽  
Vol 7 (1) ◽  
pp. 69
Author(s):  
Bashir Al-Hdaibat ◽  
Saleem Al-Ashhab ◽  
Ramadan Sabra
Keyword(s):  
Hypergeometric Function ◽  
Difference Equations ◽  
Explicit Solution ◽  
Initial Conditions ◽  
Global Dynamics ◽  
Explicit Solutions ◽  
Real Numbers ◽  
Positive Real ◽  
Rational Difference Equations

In this paper, we consider the explicit solution of the following system of nonlinear rational difference equations: x n + 1 = x n - 1 / x n - 1 + r , y n + 1 = x n - 1 y n / x n - 1 y n + r , with initial conditions x - 1 , x 0 and y 0 , which are arbitrary positive real numbers. By doing this, we encounter the hypergeometric function. We also investigate global dynamics of this system. The global dynamics of this system consists of two kind of bifurcations.

scholarly journals Global Behavior of Four Competitive Rational Systems of Difference Equations in the Plane

2009 ◽  
Vol 2009 ◽  
pp. 1-34 ◽  
Author(s):  
M. Garić-Demirović ◽  
M. R. S. Kulenović ◽  
M. Nurkanović
Keyword(s):  
Difference Equations ◽  
Stable Equilibrium ◽  
Equilibrium Points ◽  
Saddle Points ◽  
Global Behavior ◽  
Global Dynamics ◽  
Asymptotically Stable ◽  
Rational Systems ◽  
Global Stable Manifolds ◽  
Basins Of Attractions

We investigate the global dynamics of solutions of four distinct competitive rational systems of difference equations in the plane. We show that the basins of attractions of different locally asymptotically stable equilibrium points are separated by the global stable manifolds of either saddle points or nonhyperbolic equilibrium points. Our results give complete answer to Open Problem 2 posed recently by Camouzis et al. (2009).

scholarly journals Behavior of a Competitive System of Second-Order Difference Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Q. Din ◽  
T. F. Ibrahim ◽  
K. A. Khan
Keyword(s):  
Difference Equations ◽  
Initial Conditions ◽  
Positive Equilibrium ◽  
Positive Real ◽  
Competitive System ◽  
Order Difference ◽  
Rational Difference Equations ◽  
Positive Equilibrium Point ◽  
Boundedness And Persistence ◽  
Theoretical Results

We study the boundedness and persistence, existence, and uniqueness of positive equilibrium, local and global behavior of positive equilibrium point, and rate of convergence of positive solutions of the following system of rational difference equations:xn+1=(α1+β1xn-1)/(a1+b1yn),yn+1=(α2+β2yn-1)/(a2+b2xn), where the parametersαi,βi,ai, andbifori∈{1,2}and initial conditionsx0,x-1,y0, andy-1are positive real numbers. Some numerical examples are given to verify our theoretical results.

scholarly journals Global Dynamics of Rational Difference Equations xn+1=xn+xn-1/q+ynyn-1 and yn+1=yn+yn-1/p+xnxn-1

2017 ◽  
Vol 2017 ◽  
pp. 1-8
Author(s):  
Keying Liu ◽  
Peng Li ◽  
Weizhou Zhong
Keyword(s):  
Difference Equations ◽  
Boundary Point ◽  
Local Stability ◽  
Initial Conditions ◽  
Global Dynamics ◽  
Nonlinear Difference Equations ◽  
Rational Difference Equations ◽  
Isolated Points

Global dynamics of a system of nonlinear difference equations was investigated, which had five kinds of equilibria including isolated points and a continuum of nonhyperbolic equilibria along the coordinate axes. The local stability of these equilibria was analyzed which led to nine regions in the parameters space. The solution of the system converged to the equilibria or the boundary point (+∞,0) or (0,+∞) in each region depending on nonnegative initial conditions. These results completely described the behavior of the system.

scholarly journals Global Dynamics of Three Anticompetitive Systems of Difference Equations in the Plane

2013 ◽  
Vol 2013 ◽  
pp. 1-11 ◽  
Author(s):  
M. DiPippo ◽  
M. R. S. Kulenović
Keyword(s):  
Difference Equations ◽  
Initial Conditions ◽  
Basins Of Attraction ◽  
General Linear ◽  
Global Dynamics ◽  
Rational Difference Equations ◽  
Special Cases ◽  
Fractional System

We investigate the global dynamics of several anticompetitive systems of rational difference equations which are special cases of general linear fractional system of the forms ., where all parameters and the initial conditions are arbitrary nonnegative numbers, such that both denominators are positive. We find the basins of attraction of all attractors of these systems.

scholarly journals Analytical Study of the Difference Equation xn+1 = xnxn-6 / xn-5 (±1 – xnxn-6)

2019 ◽  
pp. 1-12
Author(s):  
Abdualrazaq Sanbo ◽  
Elsayed M. Elsayed ◽  
Faris Alzahrani
Keyword(s):  
Difference Equation ◽  
Difference Equations ◽  
Analytical Study ◽  
Initial Conditions ◽  
Specific Form ◽  
Positive Real ◽  
Rational Difference Equations ◽  
Special Cases ◽  
The Difference

This paper is devoted to find the form of the solutions of a rational difference equations with arbitrary positive real initial conditions. Specific form of the solutions of two special cases of this equation are given.

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