# American Institute of Mathematical Sciences

doi: 10.3934/dcdss.2022122
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 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. Huang, L. 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. Huang, X. Long, L. 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. Huang, R. Su, J. 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. Huang, X. 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. Huang, L. 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. Huang, Z. Yang, T. 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. Huang, X. Zhao, J. 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. Li, S. 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. Li, H. 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. Rumer, O. Sheshukova, H. Dautel, OD. 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. Huang, L. 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. Huang, X. Long, L. 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. Huang, R. Su, J. 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. Huang, X. 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. Huang, L. 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. Huang, Z. Yang, T. 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. Huang, X. Zhao, J. 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. Li, S. 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. Li, H. 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. Rumer, O. Sheshukova, H. Dautel, OD. 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.
Equilibrium points $0<E_1<$ $E_2\le\gamma$ when $f(\gamma)\le\gamma$ ($\gamma = 2, \frac{\bar{r}}{a} = \frac{e^2}{2.5})$
Equilibrium points $0<E_1<$ $\gamma<E_2$ when $f(\gamma)>\gamma$ ($\gamma = 2, \frac{\bar{r}}{a} = 5$)
Numerical solutions $M(t)$ to system (4) under (61) and (62) with initial value $\varphi(\zeta)\equiv 0.2, 0.3, 0.5, 1.2, \sin(t)+1.8, 3$
Numerical solutions $M(t)$ to system (4) under (63) and (64) with initial value $\varphi(\zeta)\equiv 0.1, 0.2, 0.23, 1.5, 0.8\sin(t)+2.2, 2.9$
 [1] Serge Nicaise, Cristina Pignotti, Julie Valein. Exponential stability of the wave equation with boundary time-varying delay. Discrete and Continuous Dynamical Systems - S, 2011, 4 (3) : 693-722. doi: 10.3934/dcdss.2011.4.693 [2] Jim M. Cushing. The evolutionary dynamics of a population model with a strong Allee effect. Mathematical Biosciences & Engineering, 2015, 12 (4) : 643-660. doi: 10.3934/mbe.2015.12.643 [3] Dianmo Li, Zhen Zhang, Zufei Ma, Baoyu Xie, Rui Wang. Allee effect and a catastrophe model of population dynamics. Discrete and Continuous Dynamical Systems - B, 2004, 4 (3) : 629-634. doi: 10.3934/dcdsb.2004.4.629 [4] Xin-Guang Yang, Jing Zhang, Shu Wang. Stability and dynamics of a weak viscoelastic system with memory and nonlinear time-varying delay. Discrete and Continuous Dynamical Systems, 2020, 40 (3) : 1493-1515. doi: 10.3934/dcds.2020084 [5] Carlos Nonato, Manoel Jeremias dos Santos, Carlos Raposo. Dynamics of Timoshenko system with time-varying weight and time-varying delay. Discrete and Continuous Dynamical Systems - B, 2022, 27 (1) : 523-553. doi: 10.3934/dcdsb.2021053 [6] Baowei Feng, Carlos Alberto Raposo, Carlos Alberto Nonato, Abdelaziz Soufyane. Analysis of exponential stabilization for Rao-Nakra sandwich beam with time-varying weight and time-varying delay: Multiplier method versus observability. Mathematical Control and Related Fields, 2022  doi: 10.3934/mcrf.2022011 [7] Chuangxia Huang, Lihong Huang, Jianhong Wu. Global population dynamics of a single species structured with distinctive time-varying maturation and self-limitation delays. Discrete and Continuous Dynamical Systems - B, 2022, 27 (4) : 2427-2440. doi: 10.3934/dcdsb.2021138 [8] Manoj Kumar, Syed Abbas. Diffusive size-structured population model with time-varying diffusion rate. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2022128 [9] Xin-Guang Yang. An Erratum on "Stability and dynamics of a weak viscoelastic system with memory and nonlinear time-varying delay" (Discrete Continuous Dynamic Systems, 40(3), 2020, 1493-1515). Discrete and Continuous Dynamical Systems, 2022, 42 (3) : 1493-1494. doi: 10.3934/dcds.2021161 [10] Abdelfettah Hamzaoui, Nizar Hadj Taieb, Mohamed Ali Hammami. Practical partial stability of time-varying systems. Discrete and Continuous Dynamical Systems - B, 2022, 27 (7) : 3585-3603. doi: 10.3934/dcdsb.2021197 [11] Cecilia Cavaterra, M. Grasselli. Robust exponential attractors for population dynamics models with infinite time delay. Discrete and Continuous Dynamical Systems - B, 2006, 6 (5) : 1051-1076. doi: 10.3934/dcdsb.2006.6.1051 [12] Mokhtar Kirane, Belkacem Said-Houari, Mohamed Naim Anwar. Stability result for the Timoshenko system with a time-varying delay term in the internal feedbacks. Communications on Pure and Applied Analysis, 2011, 10 (2) : 667-686. doi: 10.3934/cpaa.2011.10.667 [13] Ling Zhang, Xiaoqi Sun. Stability analysis of time-varying delay neural network for convex quadratic programming with equality constraints and inequality constraints. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2022035 [14] Aowen Kong, Carlos Nonato, Wenjun Liu, Manoel Jeremias dos Santos, Carlos Raposo. Equivalence between exponential stabilization and observability inequality for magnetic effected piezoelectric beams with time-varying delay and time-dependent weights. Discrete and Continuous Dynamical Systems - B, 2022, 27 (6) : 2959-2978. doi: 10.3934/dcdsb.2021168 [15] Quan Hai, Shutang Liu. Mean-square delay-distribution-dependent exponential synchronization of chaotic neural networks with mixed random time-varying delays and restricted disturbances. Discrete and Continuous Dynamical Systems - B, 2021, 26 (6) : 3097-3118. doi: 10.3934/dcdsb.2020221 [16] Miljana JovanoviĆ, Marija KrstiĆ. Extinction in stochastic predator-prey population model with Allee effect on prey. Discrete and Continuous Dynamical Systems - B, 2017, 22 (7) : 2651-2667. doi: 10.3934/dcdsb.2017129 [17] Pengmiao Hao, Xuechen Wang, Junjie Wei. Global Hopf bifurcation of a population model with stage structure and strong Allee effect. Discrete and Continuous Dynamical Systems - S, 2017, 10 (5) : 973-993. doi: 10.3934/dcdss.2017051 [18] Yangzi Hu, Fuke Wu. The improved results on the stochastic Kolmogorov system with time-varying delay. Discrete and Continuous Dynamical Systems - B, 2015, 20 (5) : 1481-1497. doi: 10.3934/dcdsb.2015.20.1481 [19] Hyeong-Ohk Bae, Seung Yeon Cho, Jane Yoo, Seok-Bae Yun. Effect of time delay on flocking dynamics. Networks and Heterogeneous Media, 2022  doi: 10.3934/nhm.2022027 [20] Serge Nicaise, Julie Valein, Emilia Fridman. Stability of the heat and of the wave equations with boundary time-varying delays. Discrete and Continuous Dynamical Systems - S, 2009, 2 (3) : 559-581. doi: 10.3934/dcdss.2009.2.559

2021 Impact Factor: 1.865