Asymptotic Behavior of a System of Linear Fractional Difference Equations

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 competitionmodel 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 , n= 0,1, . . . . (1.2)