GLOBAL DYNAMICS OF A MOSQUITO POPULATION SUPPRESSION MODEL UNDER A PERIODIC RELEASE STRATEGY

Let $ H(u, T) = h(u, T)-u $. Panels (A)-(D) are four schematic diagrams for excluding cases (a)-(d), respectively. Among them, panels (A) and (B) are inconsistent with (3.4). Panel (C) is inconsistent with the fact that $ Q(u, T) = 0 $ has exactly one root for $ u\in(0, A) $, and panel (D) is inconsistent with the facts that $ B_k(T)>0 $ and $ Q_k(u,T) $ is a quadratic polynomial.

The schematic diagram to depict the monotone increasing property of $ h(u, T)-u $ with respect to $ T $. We mention here that the blue curve manifests that the set $ \{u|h(u, T^{**}) = u\} $ contains only one element.

Let the parameters be defined in (3.13), and set $ c = 20,000\in(g^*, c^*) $. (A) shows the relation between $ H(u, T) $ and $ u $ when exploring the approximate value of $ T^{**} $, where $ T $ is evaluated at $ 14.2000, 14.3044 $ and $ 14.3400 $ separately. (B) shows the details of $ H(u,T) = 0 $ when $ u\in [0,540] $.

Let $ a, \mu, \xi $ and $ \bar{T} $ be set as in (3.13), and $ c = 20,000\in(g^*, c^*) $. By taking $ T = 14.2000\in(\bar{T}, T^{**}), T = 14.3044\approx T^{**} $ and $ T = 14.3400\in(T^{**}, T^*) $ in panels $ (A), (B) $ and $ (C) $, respectively, we obtain the above graphs of $ w(t) $ against $ t $ for examining the long-term behaviors of solutions of model (1.2)-(1.3). (A) indicates that the mosquito-free equilibrium is globally stable, (B) confirms that the system has a local stable equilibrium $ w(t) = 0 $ and a positive periodic solution, (C) displays two positive periodic solutions and a local stable equilibrium $ w(t) = 0 $.