doi: 10.3934/dcdss.2022122
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Bistable dynamics on a tick population equation incorporating Allee effect and two different time-varying delays

1. 

School of Mathematics and Statistics, Changsha University of Science and Technology, Hunan Provincial Key Laboratory of Mathematical Modeling and Analysis in Engineering, Changsha 410114, Hunan, China

2. 

College of Data Science, Jiaxing University, Jiaxing, Zhejiang 314001, China

3. 

School of Mathematics, Southeast University, Nanjing 210096, China

4. 

Yonsei Frontier Lab, Yonsei University, Seoul 03722, South Korea

5. 

Department of Mathematics, Nazarbayev University, Nur-Sultan 010000, Kazakhstan

* Corresponding author: Chuangxia Huang and Jinde Cao

Received  December 2021 Revised  March 2022 Early access May 2022

Fund Project: This work is partially supported by the National Natural Science Foundation of China (No. 11971076), the Science Committee of the Ministry of Education and Science of the Republic of Kazakhstan Grant OR11466188 (Dynamical Analysis and Synchronization of Complex Neural Networks with Its Applications), Nazarbayev University under Collaborative Research Program (No. 11022021CRP1509), the Postgraduate Scientific Research Innovation Project of Hunan Province (No. CX20210818) and Jiaxing public welfare research program (No. 2022AD30113)

We study the bistable dynamic behaviors for a tick population model involving Allee effect and multiple different time-varying delays. Utilizing some basic inequality techniques and dynamics theory, the positive invariant sets and exponential stability conditions of the zero equilibrium and larger positive equilibrium for the addressed model are presented. In addition, some numerical examples are shown to verify the correctness and novelty of the theoretical results.

Citation: Chuangxia Huang, Xiaojin Guo, Jinde Cao, Ardak Kashkynbayev. Bistable dynamics on a tick population equation incorporating Allee effect and two different time-varying delays. Discrete and Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2022122
References:
[1]

L. Berezansky and E. Braverman, On exponential stability of a linear delay differential equation with an oscillating coefficient, Appl. Math. Lett., 22 (2009), 1833-1837.  doi: 10.1016/j.aml.2009.07.007.

[2]

S. 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), Paper No. 43, 14 pp. doi: 10.14232/ejqtde.2018.1.43.

[3]

Q. Cao, G. Wang and C. Qian, New results on global exponential stability for a periodic Nicholson's blowflies model involving time-varying delays, Adv. Difference Equ., (2020), Paper No. 43, 12 pp. doi: 10.1186/s13662-020-2495-4.

[4]

Q. Cao, G. Wang, H. Zhang and S. Gong, New results on global asymptotic stability for a nonlinear density-dependent mortality Nicholson's blowflies model with multiple pairs of time-varying delays, J. Inequal. Appl., (2020), Paper No. 7, 12 pp. doi: 10.1186/s13660-019-2277-2.

[5]

X. Chang and J. Shi, Bistable and oscillatory dynamics of Nicholson's blowflies equation with Allee effect, Discrete Contin. Dyn. Syst. Ser., 2021. doi: 10.3934/dcdsb.2021242.

[6]

X. Ding, Global asymptotic stability of a scalar delay Nicholson's blowflies equation in periodic environment, Electron. J. Qual. Theory Differ. Equ., 14 (2022), Paper No. 14, 10 pp. doi: 10.14232/ejqtde.2022.1.14.

[7]

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

[8]

C. HuangL. Huang and J. Wu, Global population dynamics of a single species structured with distinctive time-varying maturation and self-limitation delays, Discrete Contin. Dyn. Syst. Ser. B, 27 (2022), 2427-2440.  doi: 10.3934/dcdsb.2021138.

[9]

C. HuangX. LongL. Huang and S. Fu, Stability of almost periodic Nicholson's blowflies model involving patch structure and mortality terms, Canad. Math. Bull., 63 (2020), 405-422.  doi: 10.4153/S0008439519000511.

[10]

C. HuangR. SuJ. Cao and S. Xiao, Asymptotically stable high-order neutral cellular neural networks with proportional delays and $D$ operators, Math. Comput. Simulation, 171 (2020), 127-135.  doi: 10.1016/j.matcom.2019.06.001.

[11]

C. Huang and Y. Tan, Global behavior of a reaction-diffusion model with time delay and Dirichlet condition, J. Differential Equations, 271 (2021), 186-215.  doi: 10.1016/j.jde.2020.08.008.

[12]

C. HuangX. Yang and J. Cao, Stability analysis of Nicholson's blowflies equation with two different delays, Math. Comput. Simulation, 171 (2020), 201-206.  doi: 10.1016/j.matcom.2019.09.023.

[13]

C. HuangL. Yang and J. Cao, Asymptotic behavior for a class of population dynamics, AIMS Math., 5 (2020), 3378-3390.  doi: 10.3934/math.2020218.

[14]

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.

[15]

C. Huang, J. Zhang and J. Cao, Delay-dependent attractivity on a tick population dynamics model incorporating two distinctive time-varying delays, Proc. R. Soc. A-Math. Phys. Eng. Sci., 477 (2021), 20210302, 12 pp.

[16]

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

[17]

X. Li and P. Li, Input-to-state stability of nonlinear systems: Event-triggered impulsive control, IEEE Trans. Automat. Control, 67 (2022), 1460-1465.  doi: 10.1109/TAC.2021.3063227.

[18]

X. LiS. Song and J. Wu, Exponential stability of nonlinear systems with delayed impulses and applications, IEEE Trans. Automat. Control, 64 (2019), 4024-4034.  doi: 10.1109/TAC.2019.2905271.

[19]

X. LiH. Zhu and S. Song, Input-to-state stability of nonlinear systems using observer-based event-triggered impulsive control, IEEE Trans. Syst, Man, Cybernet: Syst, 51 (2011), 6892-6900. 

[20]

B. Liu, Asymptotic behavior of solutions to a class of non-autonomous delay differential equations, J. Math. Anal. Appl., 446 (2017), 580-590.  doi: 10.1016/j.jmaa.2016.09.001.

[21]

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.

[22]

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.

[23]

X. Long and S. Gong, New results on stability of Nicholsons blowflies equation with multiple pairs of time-varying delays, Appl. Math. Lett., 100 (2020), 106027, 6 pp. doi: 10.1016/j.aml.2019.106027.

[24]

Z. Long and Y. Tan, Global attractivity for Lasota-Wazewska-type system with patch structure and multiple time-varying delays, Complexity, 2020 (2020), 1947809. 

[25]

W. E. Ricker, Computation and Interpretation of Biological Statistics of Fish Populabtions, Bulletin of the Fisheries Research Board of Canada, 1975.

[26]

L. RumerO. SheshukovaH. DautelOD. Mantke and M. Niedrig, Differentiation of medically important Euro-Asian tick species ixodes ricinus, ixodes persulcatus, ixodes hexagonus, and sermacentor reticulatus by polymerase chain reaction, Vector Borne Zoonotic Dis, 11 (2011), 899-905. 

[27]

H. L. Smith, Monotone Dynamical Systems, Math. Surveys Monogr., Amer. Math. Soc. Providence, RI, 1995.

[28]

W. Wang, The exponential convergence for a delay differential neoclassical growth model with variable delay, Nonlinear Dynam., 86 (2016), 1875-1883.  doi: 10.1007/s11071-016-3001-0.

[29]

Y. Xu, Q. Cao and X. Guo, Stability on a patch structure Nicholson's blowflies system involving distinctive delays, Appl. Math. Lett., 105 (2020), 106340, 7 pp. doi: 10.1016/j.aml.2020.106340.

[30]

X. Zhang, F. Scarabel, X.-S. Wang and J. Wu, Global continuation of periodic oscillations to a diapause rhythm, J. Dyn. Differ. Equ., 2020. doi: 10.1007/s10884-020-09856-1.

[31]

X. Zhang and J. Wu, Critical diapause portion for oscillations: Parametric trigonometric functions and their applications for Hopf bifurcation analyses, Math. Methods Appl. Sci., 42 (2019), 1363-1376.  doi: 10.1002/mma.5424.

show all references

References:
[1]

L. Berezansky and E. Braverman, On exponential stability of a linear delay differential equation with an oscillating coefficient, Appl. Math. Lett., 22 (2009), 1833-1837.  doi: 10.1016/j.aml.2009.07.007.

[2]

S. 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), Paper No. 43, 14 pp. doi: 10.14232/ejqtde.2018.1.43.

[3]

Q. Cao, G. Wang and C. Qian, New results on global exponential stability for a periodic Nicholson's blowflies model involving time-varying delays, Adv. Difference Equ., (2020), Paper No. 43, 12 pp. doi: 10.1186/s13662-020-2495-4.

[4]

Q. Cao, G. Wang, H. Zhang and S. Gong, New results on global asymptotic stability for a nonlinear density-dependent mortality Nicholson's blowflies model with multiple pairs of time-varying delays, J. Inequal. Appl., (2020), Paper No. 7, 12 pp. doi: 10.1186/s13660-019-2277-2.

[5]

X. Chang and J. Shi, Bistable and oscillatory dynamics of Nicholson's blowflies equation with Allee effect, Discrete Contin. Dyn. Syst. Ser., 2021. doi: 10.3934/dcdsb.2021242.

[6]

X. Ding, Global asymptotic stability of a scalar delay Nicholson's blowflies equation in periodic environment, Electron. J. Qual. Theory Differ. Equ., 14 (2022), Paper No. 14, 10 pp. doi: 10.14232/ejqtde.2022.1.14.

[7]

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

[8]

C. HuangL. Huang and J. Wu, Global population dynamics of a single species structured with distinctive time-varying maturation and self-limitation delays, Discrete Contin. Dyn. Syst. Ser. B, 27 (2022), 2427-2440.  doi: 10.3934/dcdsb.2021138.

[9]

C. HuangX. LongL. Huang and S. Fu, Stability of almost periodic Nicholson's blowflies model involving patch structure and mortality terms, Canad. Math. Bull., 63 (2020), 405-422.  doi: 10.4153/S0008439519000511.

[10]

C. HuangR. SuJ. Cao and S. Xiao, Asymptotically stable high-order neutral cellular neural networks with proportional delays and $D$ operators, Math. Comput. Simulation, 171 (2020), 127-135.  doi: 10.1016/j.matcom.2019.06.001.

[11]

C. Huang and Y. Tan, Global behavior of a reaction-diffusion model with time delay and Dirichlet condition, J. Differential Equations, 271 (2021), 186-215.  doi: 10.1016/j.jde.2020.08.008.

[12]

C. HuangX. Yang and J. Cao, Stability analysis of Nicholson's blowflies equation with two different delays, Math. Comput. Simulation, 171 (2020), 201-206.  doi: 10.1016/j.matcom.2019.09.023.

[13]

C. HuangL. Yang and J. Cao, Asymptotic behavior for a class of population dynamics, AIMS Math., 5 (2020), 3378-3390.  doi: 10.3934/math.2020218.

[14]

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.

[15]

C. Huang, J. Zhang and J. Cao, Delay-dependent attractivity on a tick population dynamics model incorporating two distinctive time-varying delays, Proc. R. Soc. A-Math. Phys. Eng. Sci., 477 (2021), 20210302, 12 pp.

[16]

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

[17]

X. Li and P. Li, Input-to-state stability of nonlinear systems: Event-triggered impulsive control, IEEE Trans. Automat. Control, 67 (2022), 1460-1465.  doi: 10.1109/TAC.2021.3063227.

[18]

X. LiS. Song and J. Wu, Exponential stability of nonlinear systems with delayed impulses and applications, IEEE Trans. Automat. Control, 64 (2019), 4024-4034.  doi: 10.1109/TAC.2019.2905271.

[19]

X. LiH. Zhu and S. Song, Input-to-state stability of nonlinear systems using observer-based event-triggered impulsive control, IEEE Trans. Syst, Man, Cybernet: Syst, 51 (2011), 6892-6900. 

[20]

B. Liu, Asymptotic behavior of solutions to a class of non-autonomous delay differential equations, J. Math. Anal. Appl., 446 (2017), 580-590.  doi: 10.1016/j.jmaa.2016.09.001.

[21]

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.

[22]

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.

[23]

X. Long and S. Gong, New results on stability of Nicholsons blowflies equation with multiple pairs of time-varying delays, Appl. Math. Lett., 100 (2020), 106027, 6 pp. doi: 10.1016/j.aml.2019.106027.

[24]

Z. Long and Y. Tan, Global attractivity for Lasota-Wazewska-type system with patch structure and multiple time-varying delays, Complexity, 2020 (2020), 1947809. 

[25]

W. E. Ricker, Computation and Interpretation of Biological Statistics of Fish Populabtions, Bulletin of the Fisheries Research Board of Canada, 1975.

[26]

L. RumerO. SheshukovaH. DautelOD. Mantke and M. Niedrig, Differentiation of medically important Euro-Asian tick species ixodes ricinus, ixodes persulcatus, ixodes hexagonus, and sermacentor reticulatus by polymerase chain reaction, Vector Borne Zoonotic Dis, 11 (2011), 899-905. 

[27]

H. L. Smith, Monotone Dynamical Systems, Math. Surveys Monogr., Amer. Math. Soc. Providence, RI, 1995.

[28]

W. Wang, The exponential convergence for a delay differential neoclassical growth model with variable delay, Nonlinear Dynam., 86 (2016), 1875-1883.  doi: 10.1007/s11071-016-3001-0.

[29]

Y. Xu, Q. Cao and X. Guo, Stability on a patch structure Nicholson's blowflies system involving distinctive delays, Appl. Math. Lett., 105 (2020), 106340, 7 pp. doi: 10.1016/j.aml.2020.106340.

[30]

X. Zhang, F. Scarabel, X.-S. Wang and J. Wu, Global continuation of periodic oscillations to a diapause rhythm, J. Dyn. Differ. Equ., 2020. doi: 10.1007/s10884-020-09856-1.

[31]

X. Zhang and J. Wu, Critical diapause portion for oscillations: Parametric trigonometric functions and their applications for Hopf bifurcation analyses, Math. Methods Appl. Sci., 42 (2019), 1363-1376.  doi: 10.1002/mma.5424.

Figure 1.  Equilibrium points $ 0<E_1< $ $ E_2\le\gamma $ when $ f(\gamma)\le\gamma $ ($ \gamma = 2, \frac{\bar{r}}{a} = \frac{e^2}{2.5}) $
Figure 2.  Equilibrium points $ 0<E_1< $ $ \gamma<E_2 $ when $ f(\gamma)>\gamma $ ($ \gamma = 2, \frac{\bar{r}}{a} = 5 $)
Figure 3.  Numerical solutions $M(t)$ to system (4) under (61) and (62) with initial value $\varphi(\zeta)\equiv 0.2, 0.3, 0.5, 1.2, \sin(t)+1.8, 3$
Figure 4.  Numerical solutions $ M(t) $ to system (4) under (63) and (64) with initial value $ \varphi(\zeta)\equiv 0.1, 0.2, 0.23, 1.5, 0.8\sin(t)+2.2, 2.9 $
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