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. AyeJ. 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. FangS. 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. GurneyS. 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. HassardN. 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. ShuL. 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. ShuL. 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. SoJ. 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. WangR. 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. AyeJ. 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. FangS. 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. GurneyS. 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. HassardN. 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. ShuL. 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. ShuL. 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. SoJ. 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. WangR. 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. 

Figure 1.  Parameter regions for different dynamics in $ (\tau,p) $ plane
Figure 2.  Plot of $ S_0(\tau) $ and $ S_1(\tau) $
Figure 3.  (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
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