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On an age structured population model with density-dependent dispersals between two patches

  • Received: 21 October 2018 Accepted: 13 May 2019 Published: 31 May 2019
  • Motivated by an age-structured population model over two patches that assumes constant dispersal rates, we derive a modified model that allows density-dependent dispersal, which contains both nonlinear dispersal terms and delayed non-local birth terms resulted from the mobility of the immature individuals between the patches. A biologically meaningful assumption that the dispersal rate during the immature period depends only on the mature population enables us investigate the model theoretically. Well-posedness is confirmed, criteria for existence of a positive equilibrium are obtained, threshold for extinction/persistence is established. Also addressed are a positive invariant set and global convergence of solutions under certain conditions. Although the levels of the density- dependent dispersals play no role in determining extinction/persistence, our numerical results show that they can affect, when the population is persistent, the long term dynamics including the temporal- spatial patterns and the final population sizes.

    Citation: Chang-Yuan Cheng, Shyan-Shiou Chen, Xingfu Zou. On an age structured population model with density-dependent dispersals between two patches[J]. Mathematical Biosciences and Engineering, 2019, 16(5): 4976-4998. doi: 10.3934/mbe.2019251

    Related Papers:

  • Motivated by an age-structured population model over two patches that assumes constant dispersal rates, we derive a modified model that allows density-dependent dispersal, which contains both nonlinear dispersal terms and delayed non-local birth terms resulted from the mobility of the immature individuals between the patches. A biologically meaningful assumption that the dispersal rate during the immature period depends only on the mature population enables us investigate the model theoretically. Well-posedness is confirmed, criteria for existence of a positive equilibrium are obtained, threshold for extinction/persistence is established. Also addressed are a positive invariant set and global convergence of solutions under certain conditions. Although the levels of the density- dependent dispersals play no role in determining extinction/persistence, our numerical results show that they can affect, when the population is persistent, the long term dynamics including the temporal- spatial patterns and the final population sizes.


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    [1] A. E. Abdllaoui, P. Auger, B. W. Kooi, et al., Effects of density-dependent migrations on stability of a two-patch predator-prey model, Math. Biosci., 210 (2007), 335–354.
    [2] S. A. Levin, Dispersion and population interactions, Amer. Natur., 108 (197), 207–228.
    [3] J. A. J. Metz and O. Diekmann, The Dynamics of Physiologically Structured Populations, Springer-Verlag, New York, 1986.
    [4] J. W. H. So, J. Wu and X. Zou, Structured population on two patches: modeling dispersal and delay, J. Math. Biol., 43 (2001), 37–51.
    [5] P. Weng, C. Xiao and X. Zou, Rich dynamics in a non-local population model over three patches, Nonlinear Dynam., 59 (2010), 161–172.
    [6] D. Xu, Global dynamics and Hopf bifurcation of a structured population model, Nonl. Anal. Real World Appl., 6 (2005), 461–476.
    [7] C. Yu, J. Wei and X. Zou, Bifurcation analysis in an age-structured model of a single species living in two identical patches, Appl. Math. Model., 34 (2010), 1068–1077.
    [8] A. J. Terry, Dynamics of a structured population on two patches, J. Math. Anal. Appl., 378 (2011), 1–15.
    [9] R. Cressman and V. Křivan, Two-patch population models with adaptive dispersal: the effects of varying dispersal speed, J. Math. Biol., 67 (2013), 329–358.
    [10] W. Wang and Y. Tacheuchi, Adaption of prey and predators between patches, J. Theo. Biol., 258 (2011), 603–613.
    [11] X. Zhang and W. Wang, Importance of dispersal adaption of two competitive populations between patches, Ecol. Model., 222 (2011), 11–20.
    [12] C.Huang, Z.Yang, T.Yi, et al., On the basins of attraction for a class of delay differential equations with non-monotone bistable nonlinearities, J. Diff. Eq., 256 (2014), 2101–2114.
    [13] G. Rost and J. Wu, Domain-decomposition method for the global dynamics of delay differential equations with unimodal feedback, Proc. R. Soc. A, 463 (2007), 2655–2665.
    [14] Y. Yuan and X. Q. Zhao, Global stability for non-monotone delay equations (with application to a model of blood cell production), J. Diff. Eq., 252 (2012), 2189–2209.
    [15] A. J. Terry, Impulsive culling of a structured population on two patches, J. Math. Biol., 61 (2010), 843–875.
    [16] H. C. J. Godfray, L. Partridge and P. H. Harvey, Clutch size, Annu. Rev. Ecol., 22 (1991), 409–429.
    [17] B. K. Sandercock, Incubation capacity and clutch size determination in two calidrine sandpipers: a test of the four-egg threshold, Oecologia, 110 (1997), 50–59.
    [18] B. A. Shanbhag, Reproductive strategies in the lizard, Calotes versicolor, Curr. Sci., 84 (2003), 646–652.
    [19] J. K. Hale and S. M. Verduyn Lunel, Introduction to functional differential equations, Springer, New York, 1993.
    [20] H. L. Smith, Monotone Dynamical Systems: An Introduction to the theory of Competitive and Cooperative Systems, Mathematical Surveys and Monographs, vol. 41, AMS, Providence, RI, 1995.
    [21] H. R. Thieme, Persistence under relaxed point-dissipativity (with application to an endemic model), SIAM J. Math. Anal., 24 (1993), 407–435.
    [22] M. A. McPeek and R. D. Holt, The evolution of dispersal in spatially and temporally varying environments, Amer. Natur., 140 (1992), 1010–1027.
    [23] J. Dockery, V. Hutson, K. Mischaikow, et al., The evolution of slow dispersal rates: a reaction-diffusion model, J. Math. Biol., 37 (1998), 61–83.
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