Dynamical behavior of Pielou’s difference system with exponential term

The paper studies the 2D system y_{n+1} = a z_n/(p+z_n) e^{-y_n}, z_{n+1} = b y_n/(q+y_n) e^{-z_n} and claims, among other things, that the zero equilibrium is a global attractor if ab < min(p,q) (Theorem 3.3(i)) and even reiterates this in the Conclusions (item (ii)) . This is false. A simple two-step comparison gives y_{n+2} ≤ (ab/pq) y_n and z_{n+2} ≤ (ab/pq) z_n; hence (0,0) is globally asymptotically stable if and only if ab < pq. The candidate solution gives this sharp condition and exhibits a counterexample to the paper’s claim (e.g., p=q=0.2, a=b=0.25 has ab<min(p,q) yet ab>pq, producing a positive equilibrium and not global convergence to the origin). The paper also contains internal inconsistencies: it asserts persistence for all a,b∈(0,1) (Theorem 2.1 and Conclusions (i)) while simultaneously providing parameters under which the zero equilibrium is globally asymptotically stable (Example 4.2 and Conclusions (ii)), which contradicts persistence . By contrast, the candidate solution’s positive-equilibrium regime (ab>pq) aligns with the paper’s existence/uniqueness result (Theorem 3.1) and uses standard bounding rectangles and mixed-monotonicity arguments compatible with the paper’s local-stability hypotheses e^{y_*}>a, e^{z_*}>b (Theorem 3.2(ii)) and comparison inequalities (Theorem 3.3(ii)) to conclude global convergence under stated assumptions . Net: the paper’s global-zero condition is wrong and its persistence claim conflicts with its own examples; the model’s analysis correctly identifies ab<pq as the global-zero regime and provides a coherent path to global convergence in the ab>pq regime.