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Global dynamics of a di usive single species model with periodic delay

School of Mathematics and Statistics, Xidian University, Xi’an, Shaanxi 710071, P.R. China

The growth of the species population is greatly influenced by seasonally varying environments. By regarding the maturation age of the species as a periodic developmental process, we propose a time periodic and diffusive model in bounded domain. To analyze this model with periodic delay, we first define the basic reproduction ratio $\mathcal {R}_0$ of the spatially homogeneous model and then show that the species population will be extinct when $\mathcal {R}_0\leq 1$ while remains persistent and tends to periodic oscillation if $\mathcal {R}_0>1$. Finally, combining the comparison principle with the fact that solutions of the spatially homogeneous model are also solutions of our model subject to Neumann boundary condition, we establish the global dynamics of a threshold type for PDE model in terms of $\mathcal {R}_0$.
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1. L. M. Abia, O. Angulo and J. C. López-Marcos, Age-structured population models and their numerical solution, Ecol. Model., 188 (2005), 112–136.

2. K. Liu, Y. Lou and J. Wu, Analysis of an age structured model for tick populations subject to seasonal effects, J. Diff. Equat., 263 (2017), 2078–2112.

3. F. R. Sharpe and A. J. Lotka, A problem in age distributions, Philos. Mag., 21 (1911), 435–438.

4. A. G. M'Kendrick, Applications of the mathematics to medical problems, Proc. Edinburgh Math. Soc., 44 (1926), 98–130.

5. S. A. Gourley and J. Wu, Delayed non-local diffusive systems in biological invasion and disease spread, Fields Inst. Commun., 48 (2006), 137–200.

6. Y. Jin and X. Q. Zhao, Spatial dynamics of a nonlocal periodic reaction-diffusion model with stage structure, SIAM J. Math. Anal., 40 (2009), 2496–2516.

7. M. Kloosterman, S. A. Campbell and F. J. Poulin, An NPZ model with state-dependent delay due to size-structure in juvenile zooplankton, SIAM J. Appl. Math., 76 (2016), 551–577.

8. Y. Lou and X. Q. Zhao, A theoretical approach to understanding population dynamics with seasonal developmental durations, J. Nonlinear Sci., 27 (2017), 573–603.

9. J. A. Metz and O. Diekmann, The Dynamics of Physiologically Structured Populations, Lecture Notes in Biomathematics, Springer-Verlag, Heidelberg, Germany, 1986.

10. Y. Pan, J. Fang and J. Wei, Seasonal influence on age-structured invasive species with yearly generation, SIAM J. Appl. Math., 78 (2018), 1842–1862.

11. J. W. H. So, J. Wu and X. Zou, A reaction-diffusion model for a single species with age structure. I Travelling wavefronts on unbounded domains, Proc. R. Soc. Lond. A, 457 (2001), 1841–1853.

12. X.Wu, F. M. G. Magpantay and J.Wu, et al., Stage-structured population systems with temporally periodic delay, Math. Methods Appl. Sci., 38 (2015), 3464–3481.

13. F. Li and X. Q. Zhao, A periodic SEIRS epidemic model with a time-dependent latent period, J. Math. Biol., https://doi.org/10.1007/s00285-018-1319-6.

14. X. Wang and X. Q. Zhao, A malaria transmission model with temperature-dependent incubation period, Bull. Math. Biol., 79 (2017), 1155–1182.

15. X. Wang and X. Zou, Threshold dynamics of a temperature-dependent stage-structured mosquito population model with nested delays, Bull. Math. Biol., 80 (2018), 1962–1987.

16. R. Wu and X. Q. Zhao, A reaction-diffusion model of vector-borne disease with periodic delays, J. Nonlinear. Sci., https://doi.org/10.1007/s00332-018-9475-9.

17. Y. Jin and Mark A. Lewis, Seasonal influences on population spread and persistence in streams: spreading speeds, J. Math. Biol., 65 (2012), 403–439.

18. J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Springer, New York, 1993.

19. H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, Mathematical Surveys and Monographs, vol. 41. American Mathematical Society, 1995.

20. X. Q. Zhao, Basic reproduction ratios for periodic compartmental models with time delay, J. Dyn. Diff. Equat., 29 (2017), 67–82.

21. N. Bacaër and S. Guernaoui, The epidemic threshold of vector-borne diseases with seasonality, J. Math. Biol., 53 (2006), 421–436.

22. X. Q. Zhao, Dynamical Systems in Population Biology, 2nd edition, Springer, New York, 2017.

23. W. Walter, On strongly monotone flows, Annales Polonici Mathematici, 66 (1997), 269–274.

24. D. Daners and P. Koch Medina, Abstract Evolution Equations, Periodic Problems and Applications, Pitman Res. Notes Math. Ser., vol. 279, Longman, Harlow, UK, 1992.

25. R. H. Martin and H. L. Smith, Abstract functional differential equations and reaction-diffusion systems, Trans. Amer. Math. Soc., 321 (1990), 1–44.

26. K. Liu, Analysis of age-structured population growth for single species, Ph.D thesis, The Hong Kong Polytechnic University, 2018.

27. 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., DOI 10.1007/s10884-017-9601-7.

28. J. Fang, S. A. Gourley and Y. Lou, Stage-structured models of intra- and inter-specific competition within age classes, J. Diff. Equat., 260 (2016), 1918–1953.

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