Asymptotic behavior of a system of linear fractional difference equations
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We investigate the global asymptotic behavior of solutions of the system of difference equations xn+1 = (a + xn )/(b + yn ), yn+1 = (d + yn )/(e + xn ), n = 0,1,..., where the parameters a, b, d, and e are positive numbers and the initial conditions x0 and y0 are arbitrary nonnegative numbers. We obtain some asymptotic results for the positive equilibrium of this system. 1. Introduction and preliminaries Consider the following system of difference equations xn+1 =
a + xn , b + yn
yn+1 =
d + yn , e + xn
n = 0,1,...,
(1.1)
where the parameters a, b, d, and e are positive numbers and the initial conditions x0 and y0 are arbitrary nonnegative numbers. In a modelling setting, system (1.1) of nonlinear difference equations represents the rule by which two discrete, competitive populations reproduce from one generation to the next. The phase variables xn and yn denote population sizes during the nth generation and the sequence or orbit {(xn , yn ) : n = 0,1,2,... } depicts how the populations evolve over time. Competition between the two populations is reflected by the fact that the transition function for each population is a decreasing function of the other population size. Hassell and Comins [6] studied 2-species competition with rational transition functions of a similar type. They discussed equilibrium stability and illustrated oscillatory and even chaotic behavior. Franke and Yakubu [4, 5] also investigated interspecific competition with rational transition functions. They established results about population exclusion where one population always goes extinct, but their assumptions included selfrepression and precluded the existence of any equilibria in the interior of the positive quadrant. A simple competition model that allows unbounded growth of a population size has been discussed in [1, 2], where it was assumed that a = d = 0, that is, xn+1 =
xn , b + yn
yn+1 =
yn , e + xn
Copyright © 2005 Hindawi Publishing Corporation Journal of Inequalities and Applications 2005:2 (2005) 127–143 DOI: 10.1155/JIA.2005.127
n = 0,1,....
(1.2)
128
Asymptotic behavior of a system of difference equations
Our goal in this paper is to investigate the effect of the parameters a and d on the global behavior of solutions of system (1.2). We will show that the parameters a and d can have stabilizing effect for the global behavior of solutions of system (1.2), in the sense that the unique positive equilibrium of (1.1) can become the global attractor of all positive solutions of this system for certain values of a and d. The techniques that will be used in this paper and the results that will be obtained are essentially different from the corresponding techniques and results in [1, 2]. A more general system of the form xn+1 =
a + xn , b + cxn + yn
yn+1 =
d + yn , e + xn + f y n
n = 0,1,...,
(1.3)
has been investigated in [9]. Here we will obtain more precise results for the special case of system (1.1) by using the monotonicity properties of the map in (1.1). We will give some basic results on the equilibrium points and thei
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