# American Institute of Mathematical Sciences

doi: 10.3934/dcdsb.2021242
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

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

 1 Department of Mathematics, Harbin University of Science and Technology, Harbin, Heilongjiang, 150080, China 2 Department of Mathematics, William & Mary, Williamsburg, Virginia, 23187-8795, USA

*Corresponding author: Junping Shi

Received  October 2021 Revised  August 2021 Early access October 2021

Fund Project: 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.

Citation: Xiaoyuan Chang, Junping Shi. Bistable and oscillatory dynamics of Nicholson's blowflies equation with Allee effect. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021242
##### References:

show all references

##### References:
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 $\beta = 2$
The existence of positive equilibria of (3) when $k = 3$ and $\beta>\beta_* = 1.847$
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
. 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 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)
. 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 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 $u_2 = 3.7093$
. 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 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 $u_2 = 4.5364$
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 $u_2 = 4.5364$
. 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 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 $u_2 = 5.5432$
; 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 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 $u_2 = 5.5432$
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 $0\leq t\leq 1000$.
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 $0\leq t\leq 1000$
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
 [1] Tarik Mohammed Touaoula. Global stability for a class of functional differential equations (Application to Nicholson's blowflies and Mackey-Glass models). Discrete & Continuous Dynamical Systems, 2018, 38 (9) : 4391-4419. doi: 10.3934/dcds.2018191 [2] Chuangxia Huang, Hua Zhang, Lihong Huang. Almost periodicity analysis for a delayed Nicholson's blowflies model with nonlinear density-dependent mortality term. Communications on Pure & Applied Analysis, 2019, 18 (6) : 3337-3349. doi: 10.3934/cpaa.2019150 [3] Pengmiao Hao, Xuechen Wang, Junjie Wei. Global Hopf bifurcation of a population model with stage structure and strong Allee effect. Discrete & Continuous Dynamical Systems - S, 2017, 10 (5) : 973-993. doi: 10.3934/dcdss.2017051 [4] Na Min, Mingxin Wang. Hopf bifurcation and steady-state bifurcation for a Leslie-Gower prey-predator model with strong Allee effect in prey. Discrete & Continuous Dynamical Systems, 2019, 39 (2) : 1071-1099. doi: 10.3934/dcds.2019045 [5] Darya V. Verveyko, Andrey Yu. Verisokin. Application of He's method to the modified Rayleigh equation. Conference Publications, 2011, 2011 (Special) : 1423-1431. doi: 10.3934/proc.2011.2011.1423 [6] Fatihcan M. Atay. Delayed feedback control near Hopf bifurcation. Discrete & Continuous Dynamical Systems - S, 2008, 1 (2) : 197-205. doi: 10.3934/dcdss.2008.1.197 [7] Yuncherl Choi, Taeyoung Ha, Jongmin Han, Doo Seok Lee. Bifurcation and final patterns of a modified Swift-Hohenberg equation. Discrete & Continuous Dynamical Systems - B, 2017, 22 (7) : 2543-2567. doi: 10.3934/dcdsb.2017087 [8] Kerioui Nadjah, Abdelouahab Mohammed Salah. Stability and Hopf bifurcation of the coexistence equilibrium for a differential-algebraic biological economic system with predator harvesting. Electronic Research Archive, 2021, 29 (1) : 1641-1660. doi: 10.3934/era.2020084 [9] Cuilian You, Yangyang Hao. Stability in mean for fuzzy differential equation. Journal of Industrial & Management Optimization, 2019, 15 (3) : 1375-1385. doi: 10.3934/jimo.2018099 [10] R. Ouifki, M. L. Hbid, O. Arino. Attractiveness and Hopf bifurcation for retarded differential equations. Communications on Pure & Applied Analysis, 2003, 2 (2) : 147-158. doi: 10.3934/cpaa.2003.2.147 [11] Qingyan Shi, Junping Shi, Yongli Song. Hopf bifurcation and pattern formation in a delayed diffusive logistic model with spatial heterogeneity. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 467-486. doi: 10.3934/dcdsb.2018182 [12] Aiyong Chen, Xinhui Lu. Orbital stability of elliptic periodic peakons for the modified Camassa-Holm equation. Discrete & Continuous Dynamical Systems, 2020, 40 (3) : 1703-1735. doi: 10.3934/dcds.2020090 [13] Byungsoo Moon. Orbital stability of periodic peakons for the generalized modified Camassa-Holm equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (12) : 4409-4437. doi: 10.3934/dcdss.2021123 [14] Tao Wang. Global dynamics of a non-local delayed differential equation in the half plane. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2475-2492. doi: 10.3934/cpaa.2014.13.2475 [15] Yuanxian Hui, Genghong Lin, Jianshe Yu, Jia Li. A delayed differential equation model for mosquito population suppression with sterile mosquitoes. Discrete & Continuous Dynamical Systems - B, 2020, 25 (12) : 4659-4676. doi: 10.3934/dcdsb.2020118 [16] Yanbin Tang, Ming Wang. A remark on exponential stability of time-delayed Burgers equation. Discrete & Continuous Dynamical Systems - B, 2009, 12 (1) : 219-225. doi: 10.3934/dcdsb.2009.12.219 [17] Nakao Hayashi, Pavel I. Naumkin, Isahi Sánchez-Suárez. Asymptotics for the modified witham equation. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1407-1448. doi: 10.3934/cpaa.2018069 [18] Ana Paula S. Dias, Paul C. Matthews, Ana Rodrigues. Generating functions for Hopf bifurcation with $S_n$-symmetry. Discrete & Continuous Dynamical Systems, 2009, 25 (3) : 823-842. doi: 10.3934/dcds.2009.25.823 [19] Qizhen Xiao, Binxiang Dai. Heteroclinic bifurcation for a general predator-prey model with Allee effect and state feedback impulsive control strategy. Mathematical Biosciences & Engineering, 2015, 12 (5) : 1065-1081. doi: 10.3934/mbe.2015.12.1065 [20] Tomás Caraballo, Renato Colucci, Luca Guerrini. Bifurcation scenarios in an ordinary differential equation with constant and distributed delay: A case study. Discrete & Continuous Dynamical Systems - B, 2019, 24 (6) : 2639-2655. doi: 10.3934/dcdsb.2018268

2020 Impact Factor: 1.327