# American Institute of Mathematical Sciences

April  2021, 26(4): 1763-1781. doi: 10.3934/dcdsb.2021005

## On a stage-structured population model in discrete periodic habitat: III. unimodal growth and delay effect

 School of Mathematics, Harbin Institute of Technology, Harbin, Heilongjiang 150001, China

* Corresponding author. G. Shang's current address is Advanced Microscopy and Instrumentation Research Center, Harbin Institute of Technology, Harbin, China

Dedicated to Professor Sze-Bi Hsu on the occasion of his 70th birthday and retirement

Received  April 2020 Revised  November 2020 Published  April 2021 Early access  December 2020

For a stage-structured population model in periodic discrete habitat, with periodic initial values it reduces to a system of two differential equations with time delay. Assuming the birth rate is of unimodal type, we obtain the influence of time delay on the local and global dynamics. In particular, large delay leads to population vanishing. As delay decreases, we found three critical values of delay for the emergence of different dynamics, by appealing to a combination of monotone dynamical system theory, Hopf bifurcation theory and the fluctuation method. Numerical simulations are also performed to illustrate the results.

Citation: Thazin Aye, Guanyu Shang, Ying Su. On a stage-structured population model in discrete periodic habitat: III. unimodal growth and delay effect. Discrete and Continuous Dynamical Systems - B, 2021, 26 (4) : 1763-1781. doi: 10.3934/dcdsb.2021005
##### References:
 [1] T. Aye, J. Fang and Y. Pan, On a population model in discrete periodic habitat. I. Spreading speed and optimal dispersal strategy}, J. Diff. Eqns., 269 (2020), 9653-9679.  doi: 10.1016/j.jde.2020.06.050. [2] T. Aye, J. Fang and Y. Pan, On a population model in discrete periodic habitat. II. Allee effect and propagation failure, Preprint. [3] E. Beretta and Y. Kuang, Geometric stability switch criteria in delay differential systems with delay dependent parameters, SIAM J. Math. Anal., 33 (2002), 1144-1165.  doi: 10.1137/S0036141000376086. [4] S. Chen and J. Shi, Global dynamics of the diffusive Lotka-Volterra competition model with stage structure, Calc. Var. Partial Differential Equations, 59 (2020), 33, 19 pp. doi: 10.1007/s00526-019-1693-y. [5] J. Fang, S. A. Gourley and Y. Lou, Stage-structured models of intra- and inter-specific competition within age classes, J. Diff. Eqns., 260 (2016), 1918-1953.  doi: 10.1016/j.jde.2015.09.048. [6] S. A. Gourley and J. Wu, Delayed non-local diffusive systems in biological invasion and disease spread, Nonlinear Dynamics and Evolution Equations, Fields Inst. Commun., Amer. Math. Soc., Providence, RI, 48 (2006), 137â€"200. doi: 10.1090/fic/048/06. [7] W. S. C. Gurney, S. P. Blythe and R. M. Bisbet, Bicholson's blowflies revisited, Nature, 287 (1980), 17-21.  doi: 10.1038/287017a0. [8] J. Hale, Theory of Functional Differential Equations, Springer, New York-Hedelberg, 1977. doi: 10.1007/978-1-4612-9892-2. [9] B. D. Hassard, N. D. Kazarinoff and Y. H. Wan, Theory and Applications of Hopf Bifurcation, Cambridge University Press, Cambridge, 1981. [10] S.-B. Hsu and X.-Q. Zhao, Spreading speeds and traveling waves for nonmonotone integrodifference equations,, SIAM J. Math. Anal., 40 (2008), 776-789.  doi: 10.1137/070703016. [11] A. J. Nicholson, Compensatory reactions of population to stresses, and their evolutionary significance, Aust. J. Zool., 2 (1954), 1-8.  doi: 10.1071/zo9540001. [12] S. Ruan and J. Wei, On the zeros of transcendental functions with applications to stability of delay differential equations with two delays, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 10 (2003), 863-874. [13] H. Shu, L. Wang and J. Wu, Global dynamics of Nicholson's blowflies equation revisited: Onset and termination of nonlinear oscillations, J. Diff. Eqns., 255 (2013), 2565-2586.  doi: 10.1016/j.jde.2013.06.020. [14] H. Shu, L. Wang and J. Wu, Bounded global Hopf branches for stage-structured differential equations with unimodal feedback, Nonlinearity, 30 (2017), 943-964.  doi: 10.1088/1361-6544/aa5497. [15] H. L. Smith, Monotone Dynamical System. An Introduction to the Theory of Competitive and Cooperative Systems, Amer. Math. Soc., 1995. doi: 10.1090/surv/041. [16] H. L. Smith and H. R. Thieme, Strongly order preserving semiflows generated by functional-differential equations, J. Diff. Eqns., 93 (1991), 332-363.  doi: 10.1016/0022-0396(91)90016-3. [17] J. W.-H. So, J. Wu and X. Zou, Structured population on two patches: Modelling dispersal and delay, J. Math. Biol., 43 (2001), 37-51.  doi: 10.1007/s002850100081. [18] N. Sun and J. Fang, Propagation dynamics of Fisher-KPP equation with time delay and free boundaries, Calc. Var. Partial Differential Equations, 58 (2019), Paper No. 148, 38 pp. doi: 10.1007/s00526-019-1599-8. [19] H. R. Thieme and X.-Q. Zhao, Asymptotic speeds of spread and traveling waves for integral equations and delayed reaction diffusion models, J. Diff. Eqns., 195 (2003), 430-470.  doi: 10.1016/S0022-0396(03)00175-X. [20] 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.  doi: 10.1137/18M1236162. [21] J. Wei, Bifurcation analysis in a scalar delay differential equation, Nonlinearity, 20 (2007), 2483-2498.  doi: 10.1088/0951-7715/20/11/002. [22] Y. Yuan and X.-Q. Zhao, Global stability for non-monotone delay equations with application to a model of blood cell production, J. Diff. Eqns., 252 (2012), 2189-2209.  doi: 10.1016/j.jde.2011.08.026. [23] X.-Q. Zhao, Global attractivity in a class of nonmonotone reaction-diffusion equations with time delay, Can. Appl. Math. Q., 17 (2009), 271-281.

show all references

##### References:
 [1] T. Aye, J. Fang and Y. Pan, On a population model in discrete periodic habitat. I. Spreading speed and optimal dispersal strategy}, J. Diff. Eqns., 269 (2020), 9653-9679.  doi: 10.1016/j.jde.2020.06.050. [2] T. Aye, J. Fang and Y. Pan, On a population model in discrete periodic habitat. II. Allee effect and propagation failure, Preprint. [3] E. Beretta and Y. Kuang, Geometric stability switch criteria in delay differential systems with delay dependent parameters, SIAM J. Math. Anal., 33 (2002), 1144-1165.  doi: 10.1137/S0036141000376086. [4] S. Chen and J. Shi, Global dynamics of the diffusive Lotka-Volterra competition model with stage structure, Calc. Var. Partial Differential Equations, 59 (2020), 33, 19 pp. doi: 10.1007/s00526-019-1693-y. [5] J. Fang, S. A. Gourley and Y. Lou, Stage-structured models of intra- and inter-specific competition within age classes, J. Diff. Eqns., 260 (2016), 1918-1953.  doi: 10.1016/j.jde.2015.09.048. [6] S. A. Gourley and J. Wu, Delayed non-local diffusive systems in biological invasion and disease spread, Nonlinear Dynamics and Evolution Equations, Fields Inst. Commun., Amer. Math. Soc., Providence, RI, 48 (2006), 137â€"200. doi: 10.1090/fic/048/06. [7] W. S. C. Gurney, S. P. Blythe and R. M. Bisbet, Bicholson's blowflies revisited, Nature, 287 (1980), 17-21.  doi: 10.1038/287017a0. [8] J. Hale, Theory of Functional Differential Equations, Springer, New York-Hedelberg, 1977. doi: 10.1007/978-1-4612-9892-2. [9] B. D. Hassard, N. D. Kazarinoff and Y. H. Wan, Theory and Applications of Hopf Bifurcation, Cambridge University Press, Cambridge, 1981. [10] S.-B. Hsu and X.-Q. Zhao, Spreading speeds and traveling waves for nonmonotone integrodifference equations,, SIAM J. Math. Anal., 40 (2008), 776-789.  doi: 10.1137/070703016. [11] A. J. Nicholson, Compensatory reactions of population to stresses, and their evolutionary significance, Aust. J. Zool., 2 (1954), 1-8.  doi: 10.1071/zo9540001. [12] S. Ruan and J. Wei, On the zeros of transcendental functions with applications to stability of delay differential equations with two delays, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 10 (2003), 863-874. [13] H. Shu, L. Wang and J. Wu, Global dynamics of Nicholson's blowflies equation revisited: Onset and termination of nonlinear oscillations, J. Diff. Eqns., 255 (2013), 2565-2586.  doi: 10.1016/j.jde.2013.06.020. [14] H. Shu, L. Wang and J. Wu, Bounded global Hopf branches for stage-structured differential equations with unimodal feedback, Nonlinearity, 30 (2017), 943-964.  doi: 10.1088/1361-6544/aa5497. [15] H. L. Smith, Monotone Dynamical System. An Introduction to the Theory of Competitive and Cooperative Systems, Amer. Math. Soc., 1995. doi: 10.1090/surv/041. [16] H. L. Smith and H. R. Thieme, Strongly order preserving semiflows generated by functional-differential equations, J. Diff. Eqns., 93 (1991), 332-363.  doi: 10.1016/0022-0396(91)90016-3. [17] J. W.-H. So, J. Wu and X. Zou, Structured population on two patches: Modelling dispersal and delay, J. Math. Biol., 43 (2001), 37-51.  doi: 10.1007/s002850100081. [18] N. Sun and J. Fang, Propagation dynamics of Fisher-KPP equation with time delay and free boundaries, Calc. Var. Partial Differential Equations, 58 (2019), Paper No. 148, 38 pp. doi: 10.1007/s00526-019-1599-8. [19] H. R. Thieme and X.-Q. Zhao, Asymptotic speeds of spread and traveling waves for integral equations and delayed reaction diffusion models, J. Diff. Eqns., 195 (2003), 430-470.  doi: 10.1016/S0022-0396(03)00175-X. [20] 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.  doi: 10.1137/18M1236162. [21] J. Wei, Bifurcation analysis in a scalar delay differential equation, Nonlinearity, 20 (2007), 2483-2498.  doi: 10.1088/0951-7715/20/11/002. [22] Y. Yuan and X.-Q. Zhao, Global stability for non-monotone delay equations with application to a model of blood cell production, J. Diff. Eqns., 252 (2012), 2189-2209.  doi: 10.1016/j.jde.2011.08.026. [23] X.-Q. Zhao, Global attractivity in a class of nonmonotone reaction-diffusion equations with time delay, Can. Appl. Math. Q., 17 (2009), 271-281.
Parameter regions for different dynamics in $(\tau,p)$ plane
Plot of $S_0(\tau)$ and $S_1(\tau)$
(a) $\tau = 15\le \tau_1$ and the solution converges to the positive equilibrium; (b) $\tau = 20>\tau_1$ but close to $\tau_1$, and the solution converges to a periodic solution; (c) $\tau = 80<\tau_2$ but close to $\tau_2$, and the solution still converges to periodic solution which has a larger period than in (b); (d) $\tau = 90\in (\tau_2,\tau^*)$, the solution converges to the positive equilibrium
 [1] Fang Han, Bin Zhen, Ying Du, Yanhong Zheng, Marian Wiercigroch. Global Hopf bifurcation analysis of a six-dimensional FitzHugh-Nagumo neural network with delay by a synchronized scheme. Discrete and Continuous Dynamical Systems - B, 2011, 16 (2) : 457-474. doi: 10.3934/dcdsb.2011.16.457 [2] Songbai Guo, Wanbiao Ma. Global dynamics of a microorganism flocculation model with time delay. Communications on Pure and Applied Analysis, 2017, 16 (5) : 1883-1891. doi: 10.3934/cpaa.2017091 [3] Bernold Fiedler. Global Hopf bifurcation in networks with fast feedback cycles. Discrete and Continuous Dynamical Systems - S, 2021, 14 (1) : 177-203. doi: 10.3934/dcdss.2020344 [4] Runxia Wang, Haihong Liu, Fang Yan, Xiaohui Wang. Hopf-pitchfork bifurcation analysis in a coupled FHN neurons model with delay. Discrete and Continuous Dynamical Systems - S, 2017, 10 (3) : 523-542. doi: 10.3934/dcdss.2017026 [5] Zhihua Liu, Hui Tang, Pierre Magal. Hopf bifurcation for a spatially and age structured population dynamics model. Discrete and Continuous Dynamical Systems - B, 2015, 20 (6) : 1735-1757. doi: 10.3934/dcdsb.2015.20.1735 [6] Stephen Pankavich, Nathan Neri, Deborah Shutt. Bistable dynamics and Hopf bifurcation in a refined model of early stage HIV infection. Discrete and Continuous Dynamical Systems - B, 2020, 25 (8) : 2867-2893. doi: 10.3934/dcdsb.2020044 [7] Fabien Crauste. Global Asymptotic Stability and Hopf Bifurcation for a Blood Cell Production Model. Mathematical Biosciences & Engineering, 2006, 3 (2) : 325-346. doi: 10.3934/mbe.2006.3.325 [8] 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 [9] 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 [10] Ryan T. Botts, Ale Jan Homburg, Todd R. Young. The Hopf bifurcation with bounded noise. Discrete and Continuous Dynamical Systems, 2012, 32 (8) : 2997-3007. doi: 10.3934/dcds.2012.32.2997 [11] Matteo Franca, Russell Johnson, Victor Muñoz-Villarragut. On the nonautonomous Hopf bifurcation problem. Discrete and Continuous Dynamical Systems - S, 2016, 9 (4) : 1119-1148. doi: 10.3934/dcdss.2016045 [12] John Guckenheimer, Hinke M. Osinga. The singular limit of a Hopf bifurcation. Discrete and Continuous Dynamical Systems, 2012, 32 (8) : 2805-2823. doi: 10.3934/dcds.2012.32.2805 [13] Xiaoqin P. Wu, Liancheng Wang. Hopf bifurcation of a class of two coupled relaxation oscillators of the van der Pol type with delay. Discrete and Continuous Dynamical Systems - B, 2010, 13 (2) : 503-516. doi: 10.3934/dcdsb.2010.13.503 [14] Zuolin Shen, Junjie Wei. Hopf bifurcation analysis in a diffusive predator-prey system with delay and surplus killing effect. Mathematical Biosciences & Engineering, 2018, 15 (3) : 693-715. doi: 10.3934/mbe.2018031 [15] Qi An, Weihua Jiang. Spatiotemporal attractors generated by the Turing-Hopf bifurcation in a time-delayed reaction-diffusion system. Discrete and Continuous Dynamical Systems - B, 2019, 24 (2) : 487-510. doi: 10.3934/dcdsb.2018183 [16] Udhayakumar Kandasamy, Rakkiyappan Rajan. Hopf bifurcation of a fractional-order octonion-valued neural networks with time delays. Discrete and Continuous Dynamical Systems - S, 2020, 13 (9) : 2537-2559. doi: 10.3934/dcdss.2020137 [17] Hong Yang, Junjie Wei. Dynamics of spatially heterogeneous viral model with time delay. Communications on Pure and Applied Analysis, 2020, 19 (1) : 85-102. doi: 10.3934/cpaa.2020005 [18] Hooton Edward, Balanov Zalman, Krawcewicz Wieslaw, Rachinskii Dmitrii. Sliding Hopf bifurcation in interval systems. Discrete and Continuous Dynamical Systems, 2017, 37 (7) : 3545-3566. doi: 10.3934/dcds.2017152 [19] Qigang Yuan, Jingli Ren. Periodic forcing on degenerate Hopf bifurcation. Discrete and Continuous Dynamical Systems - B, 2021, 26 (5) : 2857-2877. doi: 10.3934/dcdsb.2020208 [20] Xiuli Sun, Rong Yuan, Yunfei Lv. Global Hopf bifurcations of neutral functional differential equations with state-dependent delay. Discrete and Continuous Dynamical Systems - B, 2018, 23 (2) : 667-700. doi: 10.3934/dcdsb.2018038

2021 Impact Factor: 1.497