Global asymptotic behavior of a two-dimensional difference equation modelling competition

We investigate the global asymptotic behavior of solutions of the system of difference equations x n+1= x n a+cy n , y n+1= y n b+dx n , n=0,1,…, where the parameters a and b are in (0,1), c and d are arbitrary positive numbers and the initial conditions x 0 and y 0 are arbitrary nonnegative numbers...

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Bibliographic Details
Published inNonlinear analysis Vol. 52; no. 7; pp. 1765 - 1776
Main Authors Clark, Dean, Kulenović, M.R.S., F. Selgrade, James
Format Journal Article
LanguageEnglish
Published Oxford Elsevier Ltd 01.03.2003
Elsevier
Subjects
Asymptotic behavior
Exact sciences and technology
Finite differences and functional equations
Mathematical analysis
Mathematics
Monotonicity
Sciences and techniques of general use
Stability
Stable manifold
Stable manifold
Stability
Monotonicity
Asymptotic behavior
Competition
Two dimensional equation
Manifold
Difference equation
Modeling
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Summary:We investigate the global asymptotic behavior of solutions of the system of difference equations x n+1= x n a+cy n , y n+1= y n b+dx n , n=0,1,…, where the parameters a and b are in (0,1), c and d are arbitrary positive numbers and the initial conditions x 0 and y 0 are arbitrary nonnegative numbers. We show that the stable manifold of this system separates the positive quadrant into basins of attraction of two types of asymptotic behavior. In the case where a= b we find an explicit equation for the stable manifold.
ISSN:0362-546X
1873-5215
DOI:10.1016/S0362-546X(02)00294-8