DYNAMICAL SYSTEM OF A MOSQUITO POPULATION WITH DISTINCT BIRTH-DEATH RATES

The paper studies the discrete-time system W0: x' = x + βy − αx/(1+x), y' = y + αx/(1+x) − µy under 0 < α ≤ 1, β > 0, 0 < µ ≤ 1, proves R^2_+ invariance, and shows the global limit: (0,0) if β < µ; (+∞, α/µ) if β > µ (Theorem 1), using the identity s_n − s_{n−1} = (β − µ) y_{n−1} and auxiliary lemmas, including boundedness of y_n and eventual monotonicity (see (2.2)–(2.3), Theorem 1, and (2.6)–(2.8) in the PDF) . The candidate solution proves the same dichotomy with a more direct summation/variation-of-constants argument, also correctly treating the trivial initial condition (0,0). Minor paper caveat: Theorem 1, as stated, omits the explicit exception of the fixed initial state z(0) = (0,0), which does not flow to (+∞, α/µ) when β > µ. Aside from that clarity issue, both arguments are sound; they reach the same result by different routes.