Global dynamics of a competitive system of rational difference equations

2014 ◽  
Vol 38 (18) ◽  
pp. 4786-4796 ◽  
Author(s):  
A. Q. Khan ◽  
M. N. Qureshi
Keyword(s):  
Difference Equations ◽  
Global Dynamics ◽  
Competitive System ◽  
Rational Difference Equations

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 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 Global Dynamics of a 3 × 6 System of Difference Equations

2019 ◽  
Vol 2019 ◽  
pp. 1-14 ◽  
Author(s):  
S. M. Qureshi ◽  
A. Q. Khan
Keyword(s):  
Difference Equations ◽  
Global Dynamics ◽  
Asymptotically Stable ◽  
System Of Difference Equations ◽  
Globally Asymptotically Stable ◽  
Rational Difference Equations ◽  
Dynamical Properties ◽  
Theoretical Results

In the proposed work, global dynamics of a 3×6 system of rational difference equations has been studied in the interior of R+3. It is proved that system has at least one and at most seven boundary equilibria and a unique +ve equilibrium under certain parametric conditions. By utilizing method of Linearization, local dynamical properties about equilibria have been investigated. It is shown that every +ve solution of the system is bounded, and equilibrium P0 becomes a globally asymptotically stable if α1<α2,α4<α5, α7<α8. It is also shown that every +ve solution of the system converges to P0. Finally theoretical results are verified numerically.

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 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.

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