doi: 10.3934/dcdsb.2021242
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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 and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021242
References:
[1]

W. C. Allee, Animal Aggregations. A Study in General Sociology, University of Chicago Press, 1931.

[2]

L. BerezanskyE. Braverman and L. Idels, Nicholson's blowflies differential equations revisited: Main results and open problems, Appl. Math. Model., 34 (2010), 1405-1417.  doi: 10.1016/j.apm.2009.08.027.

[3]

Buedo-Fernández. Buedo-Fernández and E. Liz, On the stability properties of a delay differential neoclassical model of economic growth, Electron. J. Qual. Theory Differ. Equ., (2018), 1-14.  doi: 10.14232/ejqtde.2018.1.43.

[4]

X. ChangJ. Shi and J. Zhang, Dynamics of a scalar population model with delayed Allee effect, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 28 (2018), 1850153.  doi: 10.1142/S0218127418501535.

[5]

F. Courchamp, L. Berec and J. Gascoigne, Allee Effects in Ecology and Conservation, Oxford University Press, 2008. doi: 10.1093/acprof:oso/9780198570301.001.0001.

[6]

S. Gourley and S. Ruan, Dynamics of the diffusive Nicholson's blowflies equation with distributed delay, Proc. Roy. Soc. Edinburgh Sect. A, 130 (2000), 1275-1291.  doi: 10.1017/S0308210500000688.

[7]

W. S. C. GurneyS. P. Blythe and R. M. Nisbet, Nicholson's blowflies revisited, Nature, 287 (1980), 17-21.  doi: 10.1038/287017a0.

[8] J. K. Hale and S. V. Lunel, Introduction to Functional-Differential Equations, Springer-Verlag, New York, 1993.  doi: 10.1007/978-1-4612-4342-7.
[9]

A. HastingsK. C. AbbottK. CuddingtonT. FrancisG. GellnerY.-C. LaiA. MorozovS. PetrovskiiK. Scranton and M. L. Zeeman, Transient phenomena in ecology, Science, 361 (2018), 6406.  doi: 10.1126/science.aat6412.

[10]

C. HuangX. ZhaoJ. Cao and F. Alsaadi, Global dynamics of neoclassical growth model with multiple pairs of variable delays, Nonlinearity, 33 (2020), 6819-6834.  doi: 10.1088/1361-6544/abab4e.

[11]

C. HuangJ. Wang and L. Huang, Asymptotically almost periodicity of delayed Nicholson-type system involving patch structure, Electron. J. Differ. Equ, 2020 (2020), 1-17. 

[12]

C. HuangZ. YangT. Yi and X. Zou, On the basins of attraction for a class of delay differential equations with non-monotone bistable nonlinearities, J. Differential Equations, 256 (2014), 2101-2114.  doi: 10.1016/j.jde.2013.12.015.

[13] Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, Inc., Boston, MA, 1993. 
[14]

A. Lasota, Ergodic problems in biology, Dynamical Systes Astérisque, Soc. Math. France, Paris, 2 (1977), 239-250. 

[15]

M. Li and H. Shu, Global dynamics of an in-host viral model with intracellular delay, Bull. Math. Biol., 72 (2010), 1492-1505.  doi: 10.1007/s11538-010-9503-x.

[16]

E. Liz and A. Ruiz-Herrera, Delayed population models with Allee effects and exploitation, Math. Biosci. Eng., 12 (2015), 83-97.  doi: 10.3934/mbe.2015.12.83.

[17]

X. Long, Novel stability criteria on a patch structure Nicholson's blowflies model with multiple pairs of time-varying delays, AIMS Math., 5 (2020), 7387-7401.  doi: 10.3934/math.2020473.

[18]

Z. Long and Y. Tan, Global attractivity for lasota-wazewska-type system with patch structure and multiple time-varying delays, Complexity, 2020 (2020). doi: 10.1155/2020/1947809.

[19]

M. C. Mackey, Unified hypothesis for the origin of aplastic anemia and periodic hematopoiesis, Blood, 51 (1978), 941-956. 

[20]

M. C. Mackey and L. Glass, Oscillation and chaos in physiological control systems, Science, 197 (1977), 287-289.  doi: 10.1126/science.267326.

[21]

A. Matsumoto and F. Szidarovszky, Asymptotic behavior of a delay differential neoclassical growth model, Sustainability, 5 (2013), 440-455.  doi: 10.3390/su5020440.

[22]

A. Matsumoto and F. Szidarovszky, Delay differential neoclassical growth model, J. Econ. Beha. Organ., 78 (2011), 272-289.  doi: 10.1016/j.jebo.2011.01.014.

[23]

A. Y. MorozovM. Banerjee and S. V. Petrovskii, Long-term transients and complex dynamics of a stage-structured population with time delay and the {A}llee effect, J. Theoret. Biol., 396 (2016), 116-124.  doi: 10.1016/j.jtbi.2016.02.016.

[24]

A. MorozovK. AbbottK. CuddingtonT. FrancisG. GellnerA. HastingsY.-C. LaiS. PetrovskiiK. Scranton and M. L. Zeeman, Long transients in ecology: Theory and applications, Physics of Life Reviews, 32 (2020), 1-40.  doi: 10.1016/j.plrev.2019.09.004.

[25]

A. Nicholson, An outline of the dynamics of animal populations, Aust. J. Zool., 2 (1954), 9-65.  doi: 10.1071/ZO9540009.

[26]

S. Pilyugin and P. Waltman, Multiple limit cycles in the chemostat with variable yield, Math. Biosci., 182 (2003), 151-166.  doi: 10.1016/S0025-5564(02)00214-6.

[27]

C. Qian and Y. Hu, Novel stability criteria on nonlinear density-dependent mortality Nicholson's blowflies systems in asymptotically almost periodic environments, J. Inequal. Appl., 2020 (2020), 1-18.  doi: 10.1186/s13660-019-2275-4.

[28]

G. Röst and J. Wu, Domain-decomposition method for the global dynamics of delay differential equations with unimodal feedback, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 463 (2007), 2655-2669.  doi: 10.1098/rspa.2007.1890.

[29]

S. Ruan, Delay differential equations in single species dynamics, Springer, Dordrecht, 205 (2006), 477-517.  doi: 10.1007/1-4020-3647-7_11.

[30]

H. ShuL. Wang and J. Wu, Global dynamics of Nicholson's blowflies equation revisited: Onset and termination of nonlinear oscillations, J. Differential Equations, 255 (2013), 2565-2586.  doi: 10.1016/j.jde.2013.06.020.

[31] H. Smith, An Introduction to Delay Differential Equations with Applications to the Life Sciences, Springer, New York, 2011.  doi: 10.1007/978-1-4419-7646-8.
[32]

J. W.-H. So and J. S. Yu, Global attractivity and uniform persistence in Nicholson's blowflies, Differ. Equat. Dyn. Sys., 2 (1994), 11-18. 

[33]

P. StephensW. Sutherland and R. Freckleton, What is the Allee effect?, Oikos, 87 (1999), 185-190.  doi: 10.2307/3547011.

[34]

L. SullivanB. LiT. MillerM. Neubert and A. Shaw, Density dependence in demography and dispersal generates fluctuating invasion speeds, Proc. Natl. Acad. Sci. U.S.A., 114 (2017), 5053-5058.  doi: 10.1073/pnas.1618744114.

[35]

Y. Tan, Dynamics analysis of Mackey-Glass model with two variable delays, Math. Biosci. Eng., 17 (2020), 4513-4526.  doi: 10.3934/mbe.2020249.

[36]

A. J. Terry, Impulsive adult culling of a tropical pest with a stage-structured life cycle, Nonlinear Anal. Real World Appl., 11 (2010), 645-664.  doi: 10.1016/j.nonrwa.2009.01.005.

[37]

M. Ważewska-Czyżewska and A. Lasota, Mathematical problems of the dynamics of a system of red blood cells, Mat. Stos., 6 (1976), 23-40. 

[38]

J. Wei and M. Y. Li, Hopf bifurcation analysis in a delayed Nicholson blowflies equation, Nonlinear Anal., 60 (2005), 1351-1367.  doi: 10.1016/j.na.2003.04.002.

[39]

J. Wu, Symmetric functional differential equations and neural networks with memory, Trans. Amer. Math. Soc., 350 (1998), 4799-4838.  doi: 10.1090/S0002-9947-98-02083-2.

[40]

Y. XuQ. Cao and X. Guo, Stability on a patch structure Nicholson's blowflies system involving distinctive delays, Appl. Math. Lett., 105 (2020), 106340.  doi: 10.1016/j.aml.2020.106340.

[41]

H. ZhangQ. Cao and H. Yang, Asymptotically almost periodic dynamics on delayed Nicholson-type system involving patch structure, J. Inequal. Appl., 2020 (2020), 1-27.  doi: 10.1186/s13660-020-02366-0.

[42]

Z. Zheng and J. Zhou, The structure of the solution of delay differential equations with one unstable positive equilibrium, Nonlinear Dyn. Syst. Theory, 14 (2014), 187-207. 

show all references

References:
[1]

W. C. Allee, Animal Aggregations. A Study in General Sociology, University of Chicago Press, 1931.

[2]

L. BerezanskyE. Braverman and L. Idels, Nicholson's blowflies differential equations revisited: Main results and open problems, Appl. Math. Model., 34 (2010), 1405-1417.  doi: 10.1016/j.apm.2009.08.027.

[3]

Buedo-Fernández. Buedo-Fernández and E. Liz, On the stability properties of a delay differential neoclassical model of economic growth, Electron. J. Qual. Theory Differ. Equ., (2018), 1-14.  doi: 10.14232/ejqtde.2018.1.43.

[4]

X. ChangJ. Shi and J. Zhang, Dynamics of a scalar population model with delayed Allee effect, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 28 (2018), 1850153.  doi: 10.1142/S0218127418501535.

[5]

F. Courchamp, L. Berec and J. Gascoigne, Allee Effects in Ecology and Conservation, Oxford University Press, 2008. doi: 10.1093/acprof:oso/9780198570301.001.0001.

[6]

S. Gourley and S. Ruan, Dynamics of the diffusive Nicholson's blowflies equation with distributed delay, Proc. Roy. Soc. Edinburgh Sect. A, 130 (2000), 1275-1291.  doi: 10.1017/S0308210500000688.

[7]

W. S. C. GurneyS. P. Blythe and R. M. Nisbet, Nicholson's blowflies revisited, Nature, 287 (1980), 17-21.  doi: 10.1038/287017a0.

[8] J. K. Hale and S. V. Lunel, Introduction to Functional-Differential Equations, Springer-Verlag, New York, 1993.  doi: 10.1007/978-1-4612-4342-7.
[9]

A. HastingsK. C. AbbottK. CuddingtonT. FrancisG. GellnerY.-C. LaiA. MorozovS. PetrovskiiK. Scranton and M. L. Zeeman, Transient phenomena in ecology, Science, 361 (2018), 6406.  doi: 10.1126/science.aat6412.

[10]

C. HuangX. ZhaoJ. Cao and F. Alsaadi, Global dynamics of neoclassical growth model with multiple pairs of variable delays, Nonlinearity, 33 (2020), 6819-6834.  doi: 10.1088/1361-6544/abab4e.

[11]

C. HuangJ. Wang and L. Huang, Asymptotically almost periodicity of delayed Nicholson-type system involving patch structure, Electron. J. Differ. Equ, 2020 (2020), 1-17. 

[12]

C. HuangZ. YangT. Yi and X. Zou, On the basins of attraction for a class of delay differential equations with non-monotone bistable nonlinearities, J. Differential Equations, 256 (2014), 2101-2114.  doi: 10.1016/j.jde.2013.12.015.

[13] Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, Inc., Boston, MA, 1993. 
[14]

A. Lasota, Ergodic problems in biology, Dynamical Systes Astérisque, Soc. Math. France, Paris, 2 (1977), 239-250. 

[15]

M. Li and H. Shu, Global dynamics of an in-host viral model with intracellular delay, Bull. Math. Biol., 72 (2010), 1492-1505.  doi: 10.1007/s11538-010-9503-x.

[16]

E. Liz and A. Ruiz-Herrera, Delayed population models with Allee effects and exploitation, Math. Biosci. Eng., 12 (2015), 83-97.  doi: 10.3934/mbe.2015.12.83.

[17]

X. Long, Novel stability criteria on a patch structure Nicholson's blowflies model with multiple pairs of time-varying delays, AIMS Math., 5 (2020), 7387-7401.  doi: 10.3934/math.2020473.

[18]

Z. Long and Y. Tan, Global attractivity for lasota-wazewska-type system with patch structure and multiple time-varying delays, Complexity, 2020 (2020). doi: 10.1155/2020/1947809.

[19]

M. C. Mackey, Unified hypothesis for the origin of aplastic anemia and periodic hematopoiesis, Blood, 51 (1978), 941-956. 

[20]

M. C. Mackey and L. Glass, Oscillation and chaos in physiological control systems, Science, 197 (1977), 287-289.  doi: 10.1126/science.267326.

[21]

A. Matsumoto and F. Szidarovszky, Asymptotic behavior of a delay differential neoclassical growth model, Sustainability, 5 (2013), 440-455.  doi: 10.3390/su5020440.

[22]

A. Matsumoto and F. Szidarovszky, Delay differential neoclassical growth model, J. Econ. Beha. Organ., 78 (2011), 272-289.  doi: 10.1016/j.jebo.2011.01.014.

[23]

A. Y. MorozovM. Banerjee and S. V. Petrovskii, Long-term transients and complex dynamics of a stage-structured population with time delay and the {A}llee effect, J. Theoret. Biol., 396 (2016), 116-124.  doi: 10.1016/j.jtbi.2016.02.016.

[24]

A. MorozovK. AbbottK. CuddingtonT. FrancisG. GellnerA. HastingsY.-C. LaiS. PetrovskiiK. Scranton and M. L. Zeeman, Long transients in ecology: Theory and applications, Physics of Life Reviews, 32 (2020), 1-40.  doi: 10.1016/j.plrev.2019.09.004.

[25]

A. Nicholson, An outline of the dynamics of animal populations, Aust. J. Zool., 2 (1954), 9-65.  doi: 10.1071/ZO9540009.

[26]

S. Pilyugin and P. Waltman, Multiple limit cycles in the chemostat with variable yield, Math. Biosci., 182 (2003), 151-166.  doi: 10.1016/S0025-5564(02)00214-6.

[27]

C. Qian and Y. Hu, Novel stability criteria on nonlinear density-dependent mortality Nicholson's blowflies systems in asymptotically almost periodic environments, J. Inequal. Appl., 2020 (2020), 1-18.  doi: 10.1186/s13660-019-2275-4.

[28]

G. Röst and J. Wu, Domain-decomposition method for the global dynamics of delay differential equations with unimodal feedback, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 463 (2007), 2655-2669.  doi: 10.1098/rspa.2007.1890.

[29]

S. Ruan, Delay differential equations in single species dynamics, Springer, Dordrecht, 205 (2006), 477-517.  doi: 10.1007/1-4020-3647-7_11.

[30]

H. ShuL. Wang and J. Wu, Global dynamics of Nicholson's blowflies equation revisited: Onset and termination of nonlinear oscillations, J. Differential Equations, 255 (2013), 2565-2586.  doi: 10.1016/j.jde.2013.06.020.

[31] H. Smith, An Introduction to Delay Differential Equations with Applications to the Life Sciences, Springer, New York, 2011.  doi: 10.1007/978-1-4419-7646-8.
[32]

J. W.-H. So and J. S. Yu, Global attractivity and uniform persistence in Nicholson's blowflies, Differ. Equat. Dyn. Sys., 2 (1994), 11-18. 

[33]

P. StephensW. Sutherland and R. Freckleton, What is the Allee effect?, Oikos, 87 (1999), 185-190.  doi: 10.2307/3547011.

[34]

L. SullivanB. LiT. MillerM. Neubert and A. Shaw, Density dependence in demography and dispersal generates fluctuating invasion speeds, Proc. Natl. Acad. Sci. U.S.A., 114 (2017), 5053-5058.  doi: 10.1073/pnas.1618744114.

[35]

Y. Tan, Dynamics analysis of Mackey-Glass model with two variable delays, Math. Biosci. Eng., 17 (2020), 4513-4526.  doi: 10.3934/mbe.2020249.

[36]

A. J. Terry, Impulsive adult culling of a tropical pest with a stage-structured life cycle, Nonlinear Anal. Real World Appl., 11 (2010), 645-664.  doi: 10.1016/j.nonrwa.2009.01.005.

[37]

M. Ważewska-Czyżewska and A. Lasota, Mathematical problems of the dynamics of a system of red blood cells, Mat. Stos., 6 (1976), 23-40. 

[38]

J. Wei and M. Y. Li, Hopf bifurcation analysis in a delayed Nicholson blowflies equation, Nonlinear Anal., 60 (2005), 1351-1367.  doi: 10.1016/j.na.2003.04.002.

[39]

J. Wu, Symmetric functional differential equations and neural networks with memory, Trans. Amer. Math. Soc., 350 (1998), 4799-4838.  doi: 10.1090/S0002-9947-98-02083-2.

[40]

Y. XuQ. Cao and X. Guo, Stability on a patch structure Nicholson's blowflies system involving distinctive delays, Appl. Math. Lett., 105 (2020), 106340.  doi: 10.1016/j.aml.2020.106340.

[41]

H. ZhangQ. Cao and H. Yang, Asymptotically almost periodic dynamics on delayed Nicholson-type system involving patch structure, J. Inequal. Appl., 2020 (2020), 1-27.  doi: 10.1186/s13660-020-02366-0.

[42]

Z. Zheng and J. Zhou, The structure of the solution of delay differential equations with one unstable positive equilibrium, Nonlinear Dyn. Syst. Theory, 14 (2014), 187-207. 

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
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