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

• *Corresponding author: Junping Shi

X. Chang is supported by NSFC-11901140, NSFHLJ-A2018009 and UNPYSCT-2018206. J. Shi is supported by US-NSF DMS-1853598

• The bistable dynamics of a modified Nicholson's blowflies delay differential equation with Allee effect is analyzed. The stability and basins of attraction of multiple equilibria are studied by using Lyapunov-LaSalle invariance principle. The existence of multiple periodic solutions are shown using local and global Hopf bifurcations near positive equilibria, and these solutions generate long transient oscillatory patterns and asymptotic stable oscillatory patterns.

Mathematics Subject Classification: Primary: 34K18, 34K20; Secondary: 37N25, 92D25.

 Citation: • • 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 $\tau = \tau^{}_0$ and HB curve $\tau = \tau^{}_0$ represent the Hopf bifurcation curves $\tau = \tau^{}_0$ at $u_1$ and $\tau = \tau^{}_0$ 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

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