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Persistence and global stability for a class of discrete time structured population models

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  • We obtain sharp conditions distinguishing extinction from persistence and provide sufficient conditions for global stability of a positive fixed point for a class of discrete time dynamical systems on the positive cone of an ordered Banach space generated by a map which is, roughly speaking, a nonlinear, rank one perturbation of a linear contraction. Such maps were considered by Rebarber, Tenhumberg, and Towney (Theor. Pop. Biol. 81, 2012) as abstractions of a restricted class of density dependent integral population projection models modeling plant population dynamics. Significant improvements of their results are provided.
    Mathematics Subject Classification: Primary: 37N25, 92D25; Secondary: 37B25.

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  • [1]

    H. Caswell, "Matrix Population Models, Construction, Analysis, and Interpretation," 2nd ed., Sinauer Assoc. Inc., Sunderland MA, 2001.

    [2]

    J. M. Cushing, "An Introduction to Structured Population Dynamics," CBMS-NSF Regional Conf. Series in Applied Math. 71, SIAM, Philadelphia, PA, 1998.doi: 10.1137/1.9781611970005.

    [3]

    J. M. Cushing and Y. Zhou, The net reproductive value and stability in matrix population models, Nat. Res. Mod., 8 (1994), 297-333.

    [4]

    N. Davydova, O. Diekmann and S. van Gils, On circulant populations. I. The algebra of semelparity, Lin. Alg. Appl., 398 (2005), 185-243.doi: 10.1016/j.laa.2004.12.020.

    [5]

    W. Desch and W. Schappacher, Linearized stability for nonlinear semigroups, Differential Equations in Banach Spaces (A. Favini, E. Obrecht, eds.), 61-73, Lecture Notes in Mathematics 1223, Springer, Berlin Heidelberg, (1986).doi: 10.1007/BFb0099183.

    [6]

    O. Diekmann, N. Davydova and S. van Gils, On a boom and bust year class cycle, J. Difference Equ. Appl., 11 (2005), 327-335.doi: 10.1080/10236190412331335409.

    [7]

    O. Diekmann and Ph. Getto, Boundedness, global existence and continuous dependence for nonlinear dynamical systems describing physiologically structured populations, J. Diff. Equations, 215 (2005), 268-319.doi: 10.1016/j.jde.2004.10.025.

    [8]

    O. Diekmann, Ph. Getto and M. GyllenbergStability and bifurcation analysis of Volterra functional equations in the light of suns and stars, SIAM J. Math. Anal., 39 (2007/08), 1023-1069. doi: 10.1137/060659211.

    [9]

    O. Diekmann, M. Gyllenberg, H. Huang, M. Kirkilionis, J. A. J. Metz and H. R. Thieme, On the formulation and analysis of general deterministic structured population models. II. Nonlinear theory, J. Math. Biol., 43 (2001), 157-189.doi: 10.1007/s002850170002.

    [10]

    O. Diekmann, M. Gyllenberg and J. A. J. Metz, Steady-state analysis of structured population models, Theor. Pop. Biol., 63 (2003), 309-338.doi: 10.1016/S0040-5809(02)00058-8.

    [11]

    O. Diekmann, M. Gyllenberg, J. A. J. Metz and H. R. Thieme, On the formulation and analysis of general deterministic structured population models. I. Linear theory, J. Math. Biol., 36 (1998), 349-388.doi: 10.1007/s002850050104.

    [12]

    O. Diekmann, M. Gyllenberg, J. A. J. Metz, S. Nakaoka and A. M. de Roos, Daphnia revisited: Local stability and bifurcation theory for physiologically structured population models explained by way of an example, J. Math. Biol., 61 (2010), 277-318.doi: 10.1007/s00285-009-0299-y.

    [13]

    O. Diekmann, M. Gyllenberg and H. R. Thieme, Lack of uniqueness in transport equations with a nonlocal nonlinearity, Math. Models Methods Appl. Sci., 10 (2000), 581-591.doi: 10.1142/S0218202500000318.

    [14]

    O. Diekmann and S. A. van Gils, On the cyclic replicator equation and the dynamics of semelparous populations, SIAM J. Appl. Dyn. Sys., 8 (2009), 1160-1189.doi: 10.1137/080722734.

    [15]

    O. Diekmann, S. A. van Gils and S. M. Verduyn Lunel, "Delay Equations: Functional-, Complex-, and Nonlinear Analysis," Springer, New York, 1995.doi: 10.1007/978-1-4612-4206-2.

    [16]

    O. Diekmann, Y. Wang and P. Yan, Carrying simplices in discrete competitive systems and age-structured semelparous populations, Discrete Contin. Dyn. Syst., 20 (2008), 37-52.

    [17]

    J. K. Hale, "Asymptotic Behavior of Dissipative Systems," American Mathematical Society, Providence 1988.

    [18]

    H. J. A. M. Heijmans, Some results from spectral theory, in "One-Parameter Semigroups" (eds. Ph. Clément, et al.), North-Holland, Amsterdam, (1987), 282-291.

    [19]

    M. A. Krasnosel'skij, "Positive Solutions of Operator Equations," Noordhoff, Groningen, 1964.

    [20]

    M. A. Krasnosel'skij, Je. A. Lifshits and A. V. Sobolev, "Positive Linear Systems: The Method of Positive Operators," Heldermann Verlag, Berlin, 1989.

    [21]

    C.-K. Li and H. Schneider, Applications of Perron-Frobenious theory to population dynamics, J. Math. Biol., 44 (2002), 450-462.doi: 10.1007/s002850100132.

    [22]

    R. Rebarber, B. Tenhumberg and S. Townley, Global asymptotic stability of density dependent integral population projection models, Theoretical Population Biology, 81 (2012), 81-87.doi: 10.1016/j.tpb.2011.11.002.

    [23]

    H. H. Schaefer, "Topological Vector Spaces," Springer-Verlag, New York, 1971.

    [24]

    H. L. Smith, The discrete dynamics of monotonically decomposable maps, J. Math. Biology, 53 (2006), 747-758.doi: 10.1007/s00285-006-0004-3.

    [25]

    H. L. Smith and H. R. Thieme, "Dynamical Systems and Population Persistence," Graduate Studies in Mathematics, 118, American Mathematical Society, Providence R.I. 2011.

    [26]

    H. R. Thieme, Density-dependent regulation of spatially distributed populations and their asymptotic speed of spread, J. Math. Biol., 8 (1979), 173-187.doi: 10.1007/BF00279720.

    [27]

    H. R. Thieme, Well-posedness of physiologically structured population models for Daphnia magna (How biological concepts can benefit by abstract mathematical analysis), J. Math. Biology, 26 (1988), 299-317.doi: 10.1007/BF00277393.

    [28]

    H. R. Thieme, "Mathematics in Population Biology," Princeton University Press, Princeton 2003.

    [29]

    H. R. Thieme, Spectral bound and reproduction number for infinite dimensional population structure and time-heterogeneity, SIAM J. Appl. Math., 70 (2009), 188-211.doi: 10.1137/080732870.

    [30]

    K. Yosida, "Functional Analysis," Springer, Heidelberg, New York, 1968.

    [31]

    E. Zeidler., "Nonlinear Functional Analysis and Its Applications," I Springer-Verlag, New York, 1993.

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