A DISCRETE-TIME DYNAMICAL SYSTEM OF WILD MOSQUITO POPULATION WITH ALLEE EFFECTS

The paper proves (i) global extinction when β ≤ µ(1 + γµ/α) and (ii) a bistable regime when β > µ(1 + γµ/α) with forward-invariant rectangles Ω1, Ω2: trajectories in Ω1 go to (0,0), while in Ω2 one has x_n → ∞ and y_n → α/µ. The candidate solution establishes the same two conclusions via a different, concise energy-identity approach using z_{n+1} − z_n = βy_n^2/(γ+y_n) − µy_n and invariance of Ω1, Ω2. Aside from a minor algebraic slip in a continuity argument and an overbroad closing claim about “complete” global dynamics beyond Ω1∪Ω2, the model’s reasoning matches the paper’s results.