## Global asymptotic behavior of a two-dimensional difference equation modelling competition.(English)Zbl 1019.39006

Consider the system of difference equations $x_{n+1}=\frac{x_n}{a+cy_n},\quad y_{n+1}=\frac{y_n}{b+dx_n}\quad n=0,1,\dots \quad (x_0\geq 0,\;y_0\geq 0)\tag{1}$ where $$a$$ and $$b$$ are in $$(0,1),c$$ and $$d$$ are positive numbers. The global asymptotic behavior of solutions of (1) is investigated. The authors show that the stable manifold of (1) separates the positive quadrant into basins of attraction of two types of asymptotic behavior. An explicit equation for the stable manifold is obtained in the case where $$a=b$$.
Reviewer: Fozi Dannan (Doha)

### MSC:

 39A11 Stability of difference equations (MSC2000) 39B05 General theory of functional equations and inequalities
Full Text:

### References:

 [1] Clark, D.; Kulenović, M.R.S., On a coupled system of rational difference equations, Comput. math. appl., 43, 849-867, (2002) · Zbl 1001.39017 [2] Dancer, E.N.; Hess, P., Stability of fixed points for order-preserving discrete-time dynamical systems, J. reine. angew. math., 419, 125-139, (1991) · Zbl 0728.58018 [3] Franke, J.E.; Yakubu, A.-A., Mutual exclusion verses coexistence for discrete competitive systems, J. math. biol., 30, 161-168, (1991) · Zbl 0735.92023 [4] Franke, J.E.; Yakubu, A.-A., Geometry of exclusion principles in discrete systems, J. math. anal. appl., 168, 385-400, (1992) · Zbl 0778.93012 [5] Hale, J.; Kocak, H., Dynamics and bifurcations, (1991), Springer New York · Zbl 0745.58002 [6] Hassell, M.P.; Comins, H.N., Discrete time models for two-species competition, Theoret. population biol., 9, 202-221, (1976) · Zbl 0338.92020 [7] Hess, P., Periodic – parabolic boundary value problems and positivity, Pitman research notes in mathematics series, Vol. 247, (1991), Longman Scientific & Technical Essex, UK · Zbl 0731.35050 [8] Hess, P.; Lazer, A.C., On an abstract competition model and applications, Nonlinear anal. TMA, 16, 917-940, (1991) · Zbl 0743.35033 [9] Lakshmikantham, V.; Triggiante, D., Theory of difference equations, (1988), Academic Press Boston [10] Robinson, C., Stability, symbolic dynamics, and chaos, (1995), CRC Press Boca Raton, FL · Zbl 0853.58001 [11] Selgrade, J.F.; Ziehe, M., Convergence to equilibrium in a genetic model with differential viability between the sexes, J. math. biol., 25, 477-490, (1987) · Zbl 0634.92008 [12] Smith, H.L., Monotone dynamical systems: an introduction to the theory of competitive and cooperative systems, (1995), American Mathematical Society Providence, RI · Zbl 0821.34003
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.