Bistable and oscillatory dynamics of Nicholson's blowflies equation with Allee effect

Figure 1.  The graph of functions $ f(u) = \beta u^ke^{-u} $ represented by the cyan curve, and $ \tilde{f}(u) = u $ represented by the magenta curve. Here, $ k = 3 $ and $ \beta = 2 $

Figure 2.  The existence of positive equilibria of (3) when $ k = 3 $ and $ \beta>\beta_* = 1.847 $

Figure 3.  Dynamics of system (3) with $ k = 4 $ in the $ \beta-\tau $ plane. The critical values are $ \beta_* = 0.7439 $, $ \hat \beta = 0.8531 $ and $ \beta^* = 1.1873 $. We choose eleven points in $ \beta-\tau $ plane to perform the numerical simulations in Section 5: $ P_1 = (0.7, 1) $, $ P_2 = (0.7, 10) $, $ P_3 = (0.8, 1) $, $ P_4 = (0.8, 10) $, $ P_5 = (1, 1) $, $ P_6 = (1, 7) $, $ P_7 = (1, 8) $, $ P_8 = (1.5, 1) $, $ P_{9} = (1.5, 1.94) $, $ P_{10} = (1.5, 3) $, $ P_{11} = (1.5, 10). $ Here L.A.S. stands for locally asymptotically stable, and G.A.S. stands for globally asymptotically stable. HB curve $ \tau = \tau^{[1]}_0 $ and HB curve $ \tau = \tau^{[2]}_0 $ represent the Hopf bifurcation curves $ \tau = \tau^{[1]}_0 $ at $ u_1 $ and $ \tau = \tau^{[2]}_0 $ at $ u_2 $, respectively

Figure 4.  The global stability of $ u_0 $ of system (3) with $ k = 4 $, $ \beta = 0.7<\beta_* $. Left panel: $ \tau = 1 $ and $ (\beta, \tau) = P_1 $; right panel: $ \tau = 10 $ and $ (\beta, \tau) = P_2 $ as in Figure 3. Initial condition: $ \phi(t) = 1<u_1 $ (red), $ \phi(t) = 3\in(u_1, u_3) $ (blue), and $ \phi(t) = 9.3>u_3 $ (green)

Figure 5.  The dynamics of system (3) with $ k = 4 $ and $ \beta = 0.8 $. Upper row: $ \tau = 1 $ and $ (\beta, \tau) = P_3 $; lower row: $ \tau = 10 $ and $ (\beta, \tau) = P_4 $ as in Figure 3. Initial condition: (left column) $ \phi(t) = 1<u_1 $ (red), $ \phi(t) = 3\in(u_1, u_3) $ (blue), and $ \phi(t) = 6.2>u_3 $ (green); (right column) $ \phi(t) = 6.3 $ (blue). Here positive equilibria are $ u_1 = 2.3817 $ and $ u_2 = 3.7093 $

Figure 6.  The dynamics of system (3) with $ k = 4 $ and $ \beta = 1 $. First row: $ \tau = 1 $ and $ (\beta, \tau) = P_5 $; Second row: $ \tau = 7 $ and $ (\beta, \tau) = P_6 $; Third row: $ \tau = 8 $ and $ (\beta, \tau) = P_7 $ as in Figure 3. Initial condition: (left column) $ \phi(t) = 1<u_1 $ (red), $ \phi(t) = 3\in(u_1, u_3) $ (blue), and $ \phi(t) = 9.2>u_3 $ (green); (right column) $ \phi(t) = 9.3 $ (blue). Here positive equilibria are $ u_1 = 1.8572 $ and $ u_2 = 4.5364 $

Figure 7.  Transient oscillation dynamics of system (3) with $ k = 4 $, $ \beta = 1 $ and $ \tau = 50 $. Upper left: initial condition $ \phi(t) = 1<u_1 $ (red), $ \phi(t) = 3\in(u_1, u_3) $ (blue), and $ \phi(t) = 9.2>u_3 $ (green). Upper right: initial condition $ \phi(t) = 9.3 $ (blue). Lower row: snapshots of the solution with $ \phi(t) = 9.2 $ over different time intervals. Here positive equilibria are $ u_1 = 1.8572 $ and $ u_2 = 4.5364 $

Figure 8.  Oscillatory dynamics of (3) with $ k = 4 $ and $ \beta = 1.5 $. Left panel: $ \tau = 1 $ and $ (\beta, \tau) = P_8 $; right panel: $ \tau = 3 $ and $ (\beta, \tau) = P_{10} $ as in Figure 3. Initial condition: $ \phi(t) = 1<u_1 $ (red), and $ \phi(t) = 3\in(u_1, u_3) $ (blue). Here positive equilibria are $ u_1 = 1.3871 $ and $ u_2 = 5.5432 $

Figure 9.  Two-frequency oscillations with asymmetric peaks and the transient oscillation dynamics of (3) with $ k = 4 $ and $ \beta = 1.5 $. First row: $ \tau = 10 $ and $ (\beta, \tau) = P_{11} $ in Figure 3; Second row: $ \tau = 50 $; Third row: snapshots of the solution with the initial condition $ \phi(t) = 9.3 $ and $ \tau = 50 $ over different time intervals. Initial condition: $ \phi(t) = 1<u_1 $ (red), $ \phi(t) = 3\in(u_1, u_3) $ (blue), and $ \phi(t) = 9.3 $ (green). Here positive equilibria are $ u_1 = 1.3871 $ and $ u_2 = 5.5432 $

Figure 10.  The $ u(t)-u(t-\tau) $ phase planes of system (3) with $ k = 4 $. Upper left: $ \beta = 0.8 $, $ \tau = 2 $; Upper right: $ \beta = 0.8 $, $ \tau = 50 $; Middle left: $ \beta = 1 $, $ \tau = 2 $; Middle right: $ \beta = 1 $, $ \tau = 50 $; Bottom left: $ \beta = 1.5 $, $ \tau = 2 $. Bottom right: $ \beta = 1.5 $, $ \tau = 50 $. Solution orbits are shown for $ 0\leq t\leq 1000 $.

Figure 11.  The $ u(t)-u(t-\tau) $ phase plane of system (3) with $ k = 4 $, $ \beta = 1.5 $, $ \tau = 50 $ and $ \phi = 9 $. Solution orbits are shown for $ 0\leq t\leq 1000 $

Figure 12.  The domains of attraction of stable states on $ \phi-\tau $ plane with $ k = 4 $, where $ \phi $ is a constant initial condition. Upper left: $ \beta = 0.8\in(\beta_*, \hat\beta) $; Upper right: $ \beta = 1\in(\hat\beta, \beta^*) $; and lower: $ \beta = 1.5>\beta^* $. Region I: converging to $ u_0 = 0 $; Region II: converging to $ u_2>0 $; and Region III: converging to $ u_0 = 0 $ for upper row, and oscillating for lower row