Permanence for the Discrete Competition Model with Infinite Deviating Arguments

Sufficient conditions are obtained for the permanence of the following discrete model of competition: x1(k + 1) = x1(k)exp {r1(k)[(K1(k) + α1(k)∑+∞ s=0 J2(s)x2(k − s))/(1 +∑+∞ s=0 J2(s)x2(k − s)) − x1(k − δ1(k))]}; x2(k + 1) = x2(k)exp {r2(k)[(K2(k) + α2(k)∑+∞ s=0 J1(s)x1(k − s))/(1 + ∑+∞ s=0 J1(s)x1(k − s)) − x2(k − δ2(k))]}, where ri, Ki, αi, Ji, and δi, i = 1, 2, are nonnegative sequences bounded above and below by positive constants, and Ki > αi, i = 1, 2.

Li and Xu [1] studied the following two-species integrodifferential model of mutualism: By applying the coincidence degree theory, they showed that system (1) admits at least one positive -periodic solution.
Chen and You [2] argued that a general nonautonomous nonperiodic system is more appropriate, and for the general nonautonomous case, they showed that the system is permanent.For more background and biological adjustments of system (1), one could refer to [1][2][3][4][5][6] and the references cited therein.For more work on mutualism model, one could refer to  and the references cited therein.Li and Yang [31] and Li [32] argued that the discrete time models governed by difference equations are more appropriate than the continuous ones when the populations have nonoverlapping generations; corresponding to system (1), they proposed the following two-species discrete model of mutualism with infinite deviating arguments: where   (),  = 1, 2, is the density of mutualism species  at the th generation; {  ()}, {  ()}, {  ()}, {  ()}, and {  ()},  = 1, 2, are bounded nonnegative sequences such that They showed that, under the above assumption, system (2) is permanent.
It brings to our attention the fact that the main results of [31,32] deeply depend on the assumption   >   ,  = 1, 2. Now, an interesting issue is proposed: is it possible for us to investigate the persistent property of system (2) under the assumption   >   ,  = 1, 2?
From the point of view of biology, in the sequel, we shall consider (2) together with the initial conditions: Then system (2) has a unique positive solution ( 1 (),  2 ()) +∞ =0 satisfying the initial condition (4).From now on, we assume that the coefficients of system (2) satisfy the following.
(A) {  ()}, {  ()}, {  ()}, {  ()}, and {  ()},  = 1, 2, are bounded nonnegative sequences such that We mention here that such an assumption implies that the relationship between two species is competition; indeed, under the assumption (), the first equation in system (1) can be rewritten as follows: Similar to the above analysis, the second equation in system (2) can be rewritten as follows: From ( 6) and ( 7), one could easily see that both species have negative effect on the other species; that is, the relationship between two species is competition.Concerned with the persistent property of systems ( 2) and ( 4), we have the following result.

Proof of the Main Result
Now we state several lemmas which will be useful in proving our main result.