Dr. Xiaoqiang Zhao's Recent Papers (2017-Present)

Infinite Dimensional Dynamical Systems

• L. Zhang and X.-Q. Zhao, Spreading Properties and Forced Traveling Waves of Reaction-Diffusion Equations in a Time-heterogeneous Shifting Environment, J. Math. Pures Appl., in press, 2025.
• T. Yi and X.-Q. Zhao, Spatio-temporal Dynamics for Non-monotone Semiflows with Limiting Systems Having Spreading Speeds, J. Differential Equations, 436(2025), 113267. https://doi.org/10.1016/j.jde.2025.113267 (2023 arXiv version 2)
• X.-Q. Zhao, The Linear Stability and Basic Reproduction Numbers for Autonomous FDEs, Discrete and Continuous Dynamical Systems (Ser. S), 17(2024), 708-719. https://www.aimsciences.org/article/doi/10.3934/dcdss.2023082
• F.-B. Wang, L. Zhang and X.-Q. Zhao, Basic Reproduction Ratios for Time-Periodic Homogeneous Evolution Systems, SIAM J. Appl. Math., 83(2023), 1806-1831. https://doi.org/10.1137/22M1531865
• T. Yi and X.-Q. Zhao, Global Dynamics of Evolution Systems with Asymptotic Annihilation, Discrete and Continuous Dynamical Systems (Ser. A), 43(2023), 2693-2720. https://www.aimsciences.org/article/doi/10.3934/dcds.2023025
• M. Huang, S.-L. Wu and X.-Q. Zhao, The Principal Eigenvalue for Partially Degenerate and Periodic Reaction-Diffusion Systems with Time Delay (AppendixA. Basic reproduction ratios for abstract FDEs), J. Differential Equations, 371(2023), 396-449. https://doi.org/10.1016/j.jde.2023.06.024
• X.-Q. Zhao, Some Remarks on Lattice and Nonlocal Dispersal Evolution Systems, Discrete and Continuous Dynamical Systems (Ser. S), 16(2023), 787-796. https://www.aimsciences.org/article/doi/10.3934/dcdss.2022137
• L. Zhang and X.-Q. Zhao, Asymptotic Behavior of The Basic Reproduction Ratio for Periodic Reaction-Diffusion Systems, SIAM J. Math. Anal., 53(2021), 6873-6909. https://doi.org/10.1137/20M1366344
• Y. Zhang and X.-Q. Zhao, Uniqueness and Stability of Bistable Waves for Monotone Semiflows, Proc. Amer. Math. Soc., 149(2021), 4287-4302. https://doi.org/10.1090/proc/15506
• J. Fang, R. Peng and X.-Q. Zhao, Propagation Dynamics of a Reaction-Diffusion Equation in a Time-Periodic Shifting Environment, J. Math. Pures Appl., 147(2021), 1-28. https://doi.org/10.1016/j.matpur.2021.01.001
• T. Yi and X.-Q. Zhao, Propagation Dynamics for Monotone Evolution Systems without Spatial Translation Invariance, J. Functional Analysis, 279(2020), 108722. https://doi.org/10.1016/j.jfa.2020.108722
• L. Qiang, B.-G. Wang and X.-Q. Zhao, Basic Reproduction Ratios for Almost Periodic Compartmental Models with Time Delay, J. Differential Equations, 269(2020), 4440-4476. https://doi.org/10.1016/j.jde.2020.03.027
• L. Zhang, Z.-C. Wang and X.-Q. Zhao, Propagation Dynamics of a Time Periodic and Delayed Reaction-Riffusion Population Model without Quasi-monotonicity, Trans. Amer. Math. Soc., 372(2019), 1751-1782. https://doi.org/10.1090/tran/7709
• X. Liang, L. Zhang and X.-Q. Zhao, The Principal Eigenvalue for Periodic Nonlocal Dispersal Systems with Time Delay, J. Differential Equations, 266(2019), 2100-2124. https://doi.org/10.1016/j.jde.2018.08.022
• Y. Lou, X.-Q. Zhao and P. Zhou, Global Dynamics of a Lotka-Volterra Competition-Diffusion-Advection System in Heterogeneous Environments, J. Math. Pures Appl., 121(2019), 47-82. https://doi.org/10.1016/j.matpur.2018.06.010
• X. Liang, L. Zhang and X.-Q. Zhao, The Principal Eigenvalue for Degenerate Periodic Reaction-Diffusion Systems, SIAM J. Math. Anal., 49(2017), 3603-3636. https://doi.org/10.1137/16M1108832
• X.-Q. Zhao, Notes on Solution Maps of Abstract FDEs, The first version of this short paper is at http://arxiv.org/pdf/1701.00730v1.pdf, and it was published as Section 3.5, page 112-116, in X.-Q. Zhao's book "Dynamical Systems in Population Biology" (second edition), Springer, 2017.
• Z.-C. Wang, L. Zhang and X.-Q. Zhao, Time Periodic Traveling Waves for a Periodic and Diffusive SIR Epidemic Model, J. Dyn. Diff. Equat., 30(2018), 379-403. https://doi.org/10.1007/s10884-016-9546-2
• J. Fang, X. Yu and X.-Q. Zhao, Traveling Waves and Spreading Speeds for Time-Space Periodic Monotone Systems, J. Funct. Anal., 272(2017), 4222-4262. http://dx.doi.org/10.1016/j.jfa.2017.02.028
• W. Ding, F. Hamel and X.-Q. Zhao, Bistable Pulsating Fronts for Reaction-Diffusion Equations in a Periodic Habitat, Indiana University Mathematics Journal, 66(2017), 1189-1265.

Spatial Dynamics of Evolution Systems

• X. Lin, H. Ye and X.-Q. Zhao, Global Dynamics of Nonlocal Diffusion Systems on Time-Varying Domains, in reiew, 2025. https://arxiv.org/abs/2502.05860
• T. Hou, Y. Wang and X.-Q. Zhao, Spatio-Temporal Dynamics for Cooperative Reaction-Diffusion Systems with Asymptotic Annihilation, J. Differential Equations, 432(2025), 113234. https://doi.org/10.1016/j.jde.2025.113234
• W.-T. Li, M.-Z. Xin and X.-Q. Zhao, Spatio-Temporal Dynamics of a Lattice Prey-Predator System with Nonlocal Diffusion in a Periodic Habitat, Proc. Roy. Soc. Edinburgh (Sect. A), 2025, in press.
• T. Hou, Y. Wang and X.-Q. Zhao, Global Dynamics for a Class of Nonlocal Evolution Systems in a Periodic Shifting Environment, J. Nonlinear Science 35, 48(2025). https://doi.org/10.1007/s00332-025-10146-7
• L. Jiang and X.-Q. Zhao, Threshold Dynamics for a Class of Time-delayed Reaction Diffusion Equations in a Periodic Shifting Habitat, J. Dyn. Diff. Equat., 2025. https://doi.org/10.1007/s10884-025-10415-9
• X. Chen, R. Cui and X.-Q. Zhao, Asymptotic Profiles of Positive Periodic Solutions for a Class of Reaction-Diffusion Equations, Proc. Amer. Math. Soc., 2025, in press.
• W.-T. Li, M.-Z. Xin and X.-Q. Zhao, Spatio-Temporal Dynamics of Nonlocal Dispersal Systems in Time-Space Periodic Habitats, J. Differential Equations, 416(2025), 2000-2042. https://doi.org/10.1016/j.jde.2024.11.001
• Y. Li and X.-Q. Zhao, The Propagation Dynamics for Three Species Competitive-Cooperative Reaction-Diffusion Systems, Calculus of Variations and PDEs 64, 28(2025). https://doi.org/10.1007/s00526-024-02887-2
• Z. Li and X.-Q. Zhao, Traveling Waves for a Time-delayed Nonlocal Reaction-Diffusion Model of Within-host Viral Infections, J. Differential Equations, (413)2024, 240-287. https://doi.org/10.1016/j.jde.2024.08.056
• K.-Y. Lam, X.-Q. Zhao and M. Zhu, Global Dynamics of Reaction-Diffusion Systems With a Time-varying Domain, SIAM Journal on Applied Mathematics, 84(2024), 1742-1765. https://doi.org/10.1137/23M1582990
• H. Nie, J. Xing and X.-Q. Zhao, Global Dynamics of an Aedes Aegypti Population Model in Closed Advective Environments, J. Differential Equations, 407(2024), 236-268. https://doi.org/10.1016/j.jde.2024.06.018
• L. Jiang, T. Yi and X.-Q. Zhao, Propagation Dynamics for a Class of Integro-difference Equations in a Shifting Environment, J. Differential Equations, 380(2024), 491-515. https://doi.org/10.1016/j.jde.2023.10.053
• T. Hou, Y. Wang and X.-Q. Zhao, Propagation Dynamics of Cooperative Reaction-Diffusion Systems in a Periodic Shifting Environment, J. Differential Equations, 379(2024), 468-496. https://doi.org/10.1016/j.jde.2023.10.029
• G. Lin, X. Wang and X.-Q. Zhao, Propagation Phenomena of a Vector-Host Disease Model, J. Differential Equations, 378(2024), 757-791. https://doi.org/10.1016/j.jde.2023.10.016
• S. Fan and X.-Q. Zhao, Propagation Dynamics of Two-Species Competition Models in a Periodic Discrete Habitat, J. Differential Equations, 377(2023), 544-577. https://doi.org/10.1016/j.jde.2023.09.004
• M. Huang, S.-L. Wu and X.-Q. Zhao, Propagation Dynamics for Time-Space Periodic and Partially Degenerate Reaction-Diffusion Systems with Time Delay, J. Dyn. Diff. Equat., 37 (2025), 475-507. https://doi.org/10.1007/s10884-023-10299-7
• Z. Li and X.-Q. Zhao, Global Dynamics of a Nonlocal Periodic Reaction-Diffusion Model of Chikungunya Disease, J. Dyn. Diff. Equat., 36 (2024), 3073-3107. https://doi.org/10.1007/s10884-023-10267-1
• L. Zhang and X.-Q. Zhao, Linear and Nonlinear Minimal Speed Selection of Traveling Waves for a Competitive System with Nonlocal Dispersal, Applied Mathematics and Computation, 435(2022), 127360. https://doi.org/10.1016/j.amc.2022.127360
• Z. Bai, Y. Lou and X.-Q. Zhao, Spatial Dynamics of Species with Annually Synchronized Emergence of Adults, J. Nonlinear Sci. 32, 78 (2022). https://doi.org/10.1007/s00332-022-09836-3
• M. Wang, Q. Zhang and X.-Q. Zhao,Traveling Waves and Spreading Properties for a Reaction-Diffusion Competition Model with Seasonal Succession, Nonlinearity, 35(2022), 134-169. https://doi.org/10.1088/1361-6544/ac37f6
• M. Huang, S.-L. Wu and X.-Q. Zhao, Propagation Dynamics for Time-Periodic and Partially Degenerate Reaction-Diffusion Systems, SIAM J. Math. Anal., 54(2022), 1860-1897.
• M. Wang, Q. Zhang and X.-Q. Zhao, Dynamics for a Diffusive Competition Model with Seasonal Succession and Different Free Boundaries, J. Differential Equations, 285(2021), 536-582. https://doi.org/10.1016/j.jde.2021.03.006
• F. Li and X.-Q. Zhao, Global Dynamics of a Nonlocal Periodic Reaction-Diffusion Model of Bluetongue Disease, J. Differential Equations, 272(2021), 127-163. https://doi.org/10.1016/j.jde.2020.09.019
• G.-B. Zhang and X.-Q. Zhao, Propagation Phenomena for a Two-species Lotka-Volterra Strong Competition System with Nonlocal Dispersal, Calculus of Variations and PDEs, (2020) 59:10 https://doi.org/10.1007/s00526-019-1662-5
• G.-B. Zhang and X.-Q. Zhao, Propagation Dynamics of a Nonlocal Dispersal Fisher-KPP Equation in a Time-periodic Shifting Habitat, J. Differential Equations, 268(2020), 2852-2885. https://doi.org/10.1016/j.jde.2019.09.044
• W.-T. Li, J.-B. Wang and X.-Q. Zhao, Propagation Dynamics in a Time Periodic Nonlocal Dispersal Model with Stage Structure, J. Dyn. Diff. Equat., 32(2020), 1027-1064. https://doi.org/10.1007/s10884-019-09760-3
• L. Zhang, Z.-C. Wang and X.-Q. Zhao, Time Periodic Traveling Wave Solutions for a Kermack-McKendrick Epidemic Model with Diffusion and Seasonality, J. Evol. Equ., 2019. https://doi.org/10.1007/s00028-019-00544-2
• J.-B. Wang and X.-Q. Zhao, Uniqueness and Global Stability of Forced Waves in a Shifting Environment, Proc. Amer. Math. Soc., 147(2019), 1467-1481. https://doi.org/10.1090/proc/14235
• R. Wu and X.-Q. Zhao, Propagation Dynamics for a Spatially Periodic Integrodifference Competition Model, J. Differential Equations, 264(2018), 6507-6534. https://doi.org/10.1016/j.jde.2018.01.039
• Y. Tian and X.-Q. Zhao, Bistable Traveling Waves for a Competitive-Cooperative System with Nonlocal Delays, J. Differential Equations, 264(2018),5263-5299. https://doi.org/10.1016/j.jde.2018.01.010
• P. Zhou and X.-Q. Zhao, Evolution of Passive Movement in Advective Environments: General boundary condition, J. Differential Equations, 264(2018), 4176-4198. https://doi.org/10.1016/j.jde.2017.12.005
• Z. Yu and X. Zhao, Propagation Phenomena for CNNs with Asymmetric Templates and Distributed Delays, Discrete and Continuous Dynamical Systems (Ser. A), 38(2018), 905-939.
• J. Fang and X.-Q. Zhao, Notes on Nonlocal Dispersal Equations in a Periodic Habitat, 2017. https://arxiv.org/pdf/1711.00978.pdf
• W.-T. Li, J.-B. Wang and X.-Q. Zhao, Spatial Dynamics of a Nonlocal Dispersal Population Model in a Shifting Environment, J. Nonlinear Sci., 28(2018), 1189-1219. https://doi.org/10.1007/s00332-018-9445-2
• X. Yu and X.-Q. Zhao, Propagation Phenomena for A Reaction-Advection-Diffusion Competition Model in A Periodic Habitat, J. Dynamics and Differential Equations, 29(2017), 41-66. https://doi.org/10.1007/s10884-015-9426-1

Mathematical Ecology and Epidemiology

• X. San, Z.-C. Wang and X.-Q. Zhao, Traveling Waves for an Epidemic Patchy Model with Bilinear Incidence, J. Math. Biol., to appear, 2025.
• X. Wang and X.-Q. Zhao, Target Reproduction Numbers for Time-Delayed Population Systems, Mathematical Biosciences,381(2025), 109384. https://doi.org/10.1016/j.mbs.2025.109384
• D. Pang, Y. Xiao and X.-Q. Zhao, A Periodic Reaction-Diffusion Model of Hospital Infection with Crowding Effects, J. Math. Anal. Appl., 539(2024), 128487. https://doi.org/10.1016/j.jmaa.2024.128487
• Z. Li and X.-Q. Zhao, A Time-Space Periodic Population Growth model with Impulsive Birth, Z. Angew. Math. Phys. 75, 83 (2024). https://doi.org/10.1007/s00033-024-02222-x
• Z. Li and X.-Q. Zhao, Global Dynamics of a Time-delayed Nonlocal Reaction-Diffusion1 Model of Within-host Viral Infections, J. Math. Biol. 88, 38 (2024). https://doi.org/10.1007/s00285-024-02052-5
• F.-B. Wang, R. Wu and X.-Q. Zhao, A Nonlocal Reaction-Diffusion Model of West Nile Virus with Vertical Transmission, J. Nonlinear Sci. 34, 13(2024). https://doi.org/10.1007/s00332-023-09985-z
• T. Zhang and X.-Q. Zhao, Threshold Dynamics of an Almost Periodic Vector-borne Disease Model, J. Math. Biol. 87, 72 (2023). https://doi.org/10.1007/s00285-023-02002-7
• T. Zhang and X.-Q. Zhao, A Multi-host Schistosomiasis Model with Seasonality and Time-dependent Delays, Discrete and Continuous Dynamical Systems (Ser. B), 28 (2023), 2927-2959. https://doi.org/10.3934/dcdsb.2022198
• Y. Luo, Z. Teng and X.-Q. Zhao, Transmission Dynamics of a General Temporal-Spatial Vector-Host Epidemic Model with an Application to the Dengue Fever in Guangdong, China, Discrete and Continuous Dynamical Systems (Ser. B), 28 (2023), 134-169. https://doi.org/10.3934/dcdsb.2022069
• Z. Bai and X.-Q. Zhao, Threshold Dynamics of a Nonlocal and Time-Delayed West Nile Virus Model with Seasonality, Communications in Nonlinear Science and Numerical Simulation, 115 (2022), 106758. https://doi.org/10.1016/j.cnsns.2022.106758
• X. Wang, R. Wu and X.-Q. Zhao, A Reaction-Advection-Diffusion Model of Cholera Epidemics with Seasonality and Human Behavior Change, J. Math. Biol. 84, 34 (2022). https://doi.org/10.1007/s00285-022-01733-3
• D. Pang, Y. Xiao and X.-Q. Zhao, A Cross-Infection Model with Diffusion and Incubation Period, Discrete and Continuous Dynamical Systems (Ser. B), 27(2022), 6269-6294. https://doi.org/10.3934/dcdsb.2021316
• L. Qiang, B.-G. Wang and X.-Q. Zhao, A Stage-Structured Population Model with Time-Dependent Delay in an Almost Periodic Environment, J. Dyn. Diff. Equat., 34(2022), 341-364. https://doi.org/10.1007/s10884-020-09827-6
• R. Wu and X.-Q. Zhao, The Evolution Dynamics of An Impulsive Hybrid Population Model with Spatial Heterogeneity, Commun Nonlinear Sci Numer Simulat, 107(2022), 106181. https://doi.org/10.1016/j.cnsns.2021.106181
• L. Zhang and X.-Q. Zhao, Asymptotic Behavior of The Principal Eigenvalue and Basic Reproduction Ratio for Periodic Patch Models, Sci. China Math., 65(2022), 1363-1382. https://doi.org/10.1007/s11425-021-1894-2
• D. Pang, Y. Xiao and X.-Q. Zhao, A Cross-Infection Model with Diffusive Environmental Bacteria, J. Math. Anal. Appl., 505(2022), 125637. https://doi.org/10.1016/j.jmaa.2021.125637
• F. Li and X.-Q. Zhao, Global Dynamics of a Reaction-Diffusion Model of Zika Virus Transmission with Seasonality, Bulletin of Mathematical Biology 83, 43(2021). https://doi.org/10.1007/s11538-021-00879-3
• Y. Jin and X.-Q. Zhao, The Spatial Dynamics of a Zebra Mussel Model in River Environments, Discrete and Continuous Dynamical Systems (Ser. B), 26(2021), 1991-2010.
• X. Liu, Y. Wang and X.-Q. Zhao, Dynamics of a Periodic Chikungunya Model with Temperature and Rainfall effects, Commun Nonlinear Sci Numer Simulat, 90(2020), 105409. https://www.sciencedirect.com/science/article/pii/S1007570420302409
• X. Wang and X.-Q. Zhao, Target Reproduction Numbers for Reaction-Diffusion Population Models, Journal of Mathematical Biology, 81(2020), 625-647. https://doi.org/10.1007/s00285-020-01523-9
• Z. Bai, Y. Lou and X.-Q. Zhao, A Delayed Succession Model With Diffusion for the Impact of Diapause on Population Growth, SIAM Journal on Applied Mathematics, 80(2020), 1493-1519. https://doi.org/10.1137/19M1236448
• T. Zhang and X.-Q. Zhao, Mathematical Modeling for Schistosomiasis with Seasonal Influence: A Case Study in Hubei, China, SIAM Journal on Applied Dynamical Systems, 19(2020), 1438-1471. https://doi.org/10.1137/19M1280259
• X. Liu, Y. Wang and X.-Q. Zhao, Dynamics of a Climate-based Periodic Chikungunya Model with Incubation Period, Applied Mathematical Modelling, 80(2020), 151-168. https://doi.org/10.1016/j.apm.2019.11.038
• Z. Bai and X.-Q. Zhao, Basic Reproduction Ratios for Periodic and Time-delayed Compartmental Models with Impulses, Journal of Mathematical Biology, 80(2020), 1095-1117. https://doi.org/10.1007/s00285-019-01452-2
• F. Li and X.-Q. Zhao, Dynamics of a Periodic Bluetongue Model with a Temperature-Dependent Incubation Period, SIAM J. Appl. Math., 79(2019), 2479-2505. https://doi.org/10.1137/18M1218364
• F. Li, J. Liu and X.-Q. Zhao, A West Nile Virus Model with Vertical Transmission and Periodic Time Delays, J. Nonlinear Sci., 30(2020), 449-486. https://doi.org/10.1007/s00332-019-09579-8
• F.-B. Wang, R. Wu and X.-Q. Zhao, A West Nile Virus Transmission Model with Periodic Incubation Periods, SIAM J. Appl. Dyn. Syst., 18(2019), 1498-1535.
• R. Wu and X.-Q. Zhao, Spatial Invasion of a Birth Pulse Population with Nonlocal Dispersal, SIAM J. Appl. Math., 79(2019), 1075-1097.
• S.-B. Hsu, F.-B. Wang and X.-Q. Zhao, Mathematical Modeling and Analysis of Harmful Algal Blooms in Flowing Habitats, Mathematical Biosciences and Engineering, 16(2019), 6728-6752.
• A. Hurford, X. Wang and X.-Q. Zhao, Regional Climate Affects Salmon Lice Dynamics, Stage Structure and Management, Proc. R. Soc. B 286, 2019, 20190428. http://dx.doi.org/10.1098/rspb.2019.0428
• F. Li and X.-Q. Zhao, A Periodic SEIRS Epidemic Model with a Time-dependent Latent Period, Journal of Mathematical Biology, 78(2019), 1553-1579. https://doi.org/10.1007/s00285-018-1319-6
• X. Wang, X.-Q. Zhao and J. Wang, A Cholera Epidemic Model in a Spatiotemporally Heterogeneous Environment, J. Math. Anal. Appl., 468 (2018), 893-912.
• R. Wu and X.-Q. Zhao, A Reaction-Diffusion Model of Vector-Borne Disease with Periodic Delays, Journal of Nonlinear Science, 29(2019), 29-64. https://doi.org/10.1007/s00332-018-9475-9
• Z. Bai, R. Peng and X.-Q. Zhao, A Reaction-Diffusion Malaria Model with Seasonality and Incubation Period, Journal of Mathematical Biology, 77(2018), 201-228. https://doi.org/10.1007/s00285-017-1193-7
• X. Wang and X.-Q. Zhao, A Climate-based Malaria Model with The Use of Bed Nets, Journal of Mathematical Biology, 77(2018), 1-25. https://link.springer.com/content/pdf/10.1007%2Fs00285-017-1183-9.pdf
• S. Guo, W. Ma and X.-Q. Zhao, Global Dynamics of a Time-Delayed Microorganism Flocculation Model with Saturated Functional Responses, J. Dyn. Diff. Equat., 30(2018), 1247-1271. https://doi.org/10.1007/s10884-017-9605-3
• X. Liang, L. Zhang and X.-Q. Zhao, Basic Reproduction Ratios for Periodic Abstract Functional Differential Equations (with Application to a Spatial Model for Lyme Disease), J. Dyn. Diff. Equat., 31(2019), 1247-1278. https://doi.org/10.1007/s10884-017-9601-7
• X. Wang and X.-Q. Zhao, A Malaria Transmission Model with Temperature-Dependent Incubation Period, Bulletin of Mathematical Biology, 79(2017), 1155-1182. http://link.springer.com/article/10.1007/s11538-017-0276-3
• S.-B. Hsu, F.-B. Wang and X.-Q. Zhao, A Reaction-Diffusion Model of Harmful Algae and Zooplankton in An Ecosystem, J. Math. Anal. Appl., 451(2017), 659-677. http://dx.doi.org/10.1016/j.jmaa.2017.02.034
• X. Wang and X.-Q. Zhao, Dynamics of A Time-Delayed Lyme Disease Model with Seasonality, SIAM J. Appl. Dyn. Syst., 16(2017), 853-881. http://epubs.siam.org/doi/pdf/10.1137/16M1087916
• X. Wang and X.-Q. Zhao, A Periodic Vector-Bias Malaria Model with Incubation Period, SIAM J. Appl. Math., 77(2017), 181-201.
• Y. Lou and X.-Q. Zhao, A Theoretical Approach to Understanding Population Dynamics with Seasonal Developmental Durations, J. Nonlinear Sci., 27(2017), 573-603.
• X.-Q. Zhao, Basic Reproduction Ratios for Periodic Compartmental Models with Time Delay, J. Dynamics and Differential Equations, 29(2017), 67-82.