On a Difference Equation with Exponentially Decreasing Nonlinearity

Figure 2

Some iterations for (2.1) with 𝛼=0.5,𝑝=10,𝑞=2 (a) and 𝛼=0.5,𝑝=50,𝑞=2 (b). In the former case all positive solutions converge (oscillatory) to the positive equilibrium 𝑥≈1.35, and in the latter case the solution converges to a stable two-cycle ≈ 2.86, 1.59.

(a)

(b)