Search Advanced

Mathematical Biosciences and Engineering

2020, Volume 17, Issue 2: 1168-1217. doi: 10.3934/mbe.2020061
Previous Article Next Article
Research article Special Issues

Discrete-time population dynamics on the state space of measures

  • School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ 85287-1804, USA
  • Received: 16 July 2019 Accepted: 29 October 2019 Published: 15 November 2019

  • If the individual state space of a structured population is given by a metric space S, measures μ on the σ-algebra of Borel subsets T of S offer a modeling tool with a natural interpretation: μ(T) is the number of individuals with structural characteristics in the set T. A discrete-time population model is given by a population turnover map F on the cone of finite nonnegative Borel measures that maps the structural population distribution of a given year to the one of the next year. Under suitable assumptions, F has a first order approximation at the zero measure (the extinction fixed point), which is a positive linear operator on the ordered vector space of real measures and can be interpreted as a basic population turnover operator. For a semelparous population, it can be identified with the next generation operator. A spectral radius can be defined by the usual Gelfand formula.We investigate in how far it serves as a threshold parameter between population extinction and population persistence. The variation norm on the space of measures is too strong to give the basic turnover operator enough compactness that its spectral radius is an eigenvalue associated with a positive eigenmeasure. A suitable alternative is the flat norm (also known as (dual) bounded Lipschitz norm), which, as a trade-off, makes the basic turnover operator only continuous on the cone of nonnegative measures but not on the whole space of real measures.

    Citation: Horst R. Thieme. Discrete-time population dynamics on the state space of measures[J]. Mathematical Biosciences and Engineering, 2020, 17(2): 1168-1217. doi: 10.3934/mbe.2020061

    Related Papers:

  • If the individual state space of a structured population is given by a metric space S, measures μ on the σ-algebra of Borel subsets T of S offer a modeling tool with a natural interpretation: μ(T) is the number of individuals with structural characteristics in the set T. A discrete-time population model is given by a population turnover map F on the cone of finite nonnegative Borel measures that maps the structural population distribution of a given year to the one of the next year. Under suitable assumptions, F has a first order approximation at the zero measure (the extinction fixed point), which is a positive linear operator on the ordered vector space of real measures and can be interpreted as a basic population turnover operator. For a semelparous population, it can be identified with the next generation operator. A spectral radius can be defined by the usual Gelfand formula.We investigate in how far it serves as a threshold parameter between population extinction and population persistence. The variation norm on the space of measures is too strong to give the basic turnover operator enough compactness that its spectral radius is an eigenvalue associated with a positive eigenmeasure. A suitable alternative is the flat norm (also known as (dual) bounded Lipschitz norm), which, as a trade-off, makes the basic turnover operator only continuous on the cone of nonnegative measures but not on the whole space of real measures.


    加载中


    [1] O. Diekmann, M. Gyllenberg, J. A. J. Metz, H. R. Thieme, The 'cumulative' formulation of (physiologically) structured population models. Evolution Equations, Control Theory, and Biomathematics (Ph. Clément, G. Lumer; eds.), 145-154, Marcel Dekker, 1994.
    [2] C. D. Aliprantis, K. C. Border, Infinite Dimensional Analysis. A Hitchhiker's Guide, 3rd edition, Springer, Berlin Heidelberg 1999, 2006.
    [3] H. Bauer, Probability Theory and Elements of Measure Theory, sec. ed., Academic Press, London, 1981.
    [4] R. M. Dudley, Real Analysis and Probability, sec. ed., Cambridge University Press, Cambridge, 2002.
    [5] N. Dunford, J. T. Schwartz, Linear Operators. Part I. General Theory, John Wiley, Classics Library Edition, New York, 1988.
    [6] J. N. McDonald, N. A. Weiss, A Course in Real Analysis, Academic Press, San Diego, 1999.
    [7] H. R. Thieme, Mathematics in Population Biology, Princeton University Press, Princeton, 2003.
    [8] A. de Roos, L. Persson, Population and Community Ecology of Ontogenic Development, Princeton University Press, Princeton, 2013.
    [9] A. Lasota, M. C. Mackey, Chaos, Fractals and Noise, Springer, New York, 1994.
    [10] A. S. Ackleh, R. M. Colombo, S. C. Hille, A. Muntean, (guest editors), Modeling with measures, Math. Biosci. Eng. 12 (2015), special issue.
    [11] O. Diekmann, M. Gyllenberg, H. Huang, M. Kirkilionis, J. A. J. Metz, H. R. Thieme, On the formulation and analysis of general deterministic structured population models. II. Nonlinear theory, J. Math. Biol., 43 (2001), 157-189.
    [12] O. Diekmann, M. Gyllenberg, J. A. J. Metz, H. R. Thieme, On the formulation and analysis of general deterministic structured population models. I. Linear theory, J. Math. Biol., 36 (1998), 349-338.
    [13] H. J. A. M. Heijmans, Markov semigroups and structured population dynamics, Aspects of Positivity in Functional Analysis (R. Nagel, U. Schlotterbeck, M.P.H. Wolff, eds.), 199-208, Elsevier, Amsterdam, 1986.
    [14] A. S. Ackleh, V. K. Chellamuthu, K. Ito, Finite difference approximations for measure-valued solutions of a hierarchically size-structured population model, Math. Biosci. Eng., 12 (2015), 233-258.
    [15] H. R. Thieme, Well-posedness of physiologically structured population models for Daphnia magna (How biological concepts can benefit by abstract mathematical analysis), J. Math. Biol., 26 (1988), 299-317.
    [16] H. L. Smith, H. R. Thieme, Persistence and global stability for a class of discrete time structured population models, Disc. Cont. Dyn. Syst., 33 (2013), 4627-4646.
    [17] A. S. Ackleh, J. Cleveland, H. R. Thieme, Selection-mutation differential equations: Long-time behavior of measure-valued solutions, J. Diff. Eqn., 261 (2016), 1472-1505.
    [18] A. S. Ackleh, B. G. Fitzpatrick, H. R. Thieme, Rate distributions and survival of the fittest: a formulation on the space of measures, Discrete Contin. Dyn-B, 5 (2005), 917-928.
    [19] J.-E. Busse, P. Gwiazda, A. Marciniak-Czochra, Mass concentration in a nonlocal model of clonal selection, J. Math. Biol., 73 (2016), 1001-1033.
    [20] H. R. Thieme, J. Yang, An endemic model with variable re-infection rate and applications to influenza, John A. Jacquez memorial volume, Math. Biosci., 180 (2002), 207-235.
    [21] J. H. M. Evers, S. C. Hille, A. Muntean, Mild solutions to a measure-valued mass evolution problem with flux boundary conditions, J. Differ. Equ., 259 (2015), 1068-1097.
    [22] S. C. Hille, D. T. H. Worm, Continuity properties of Markov semigroups and their restrictions to invariant L1 spaces, Semigroup Forum, 79 (2009), 575-600.
    [23] J. Cleveland, A. S. Ackleh, Evolutionary game theory on measure spaces: well-posedness. Nonlinear Anal. Real World Appl., 14 (2013), 785-797.
    [24] J. Cleveland, Basic stage structure measure valued evolutionary game model, Math. Biosci. Eng., 12 (2015), 291-310.
    [25] S. C. Hille, D. T. H. Worm, Embedding of semigroups of Lipschitz maps into positive linear semigroups on ordered Banach spaces generated by measures, Integral Equations Operator Theory, 63 (2009), 351-371.
    [26] P. Gwiazda, A. Marciniak-Czochra, H. R. Thieme, Measures under the flat norm as ordered normed vector space, Positivity, 22 (2018), 105-138; Correction: Positivity, 22 (2018), 139-140.
    [27] M. A. Krasnosel'skij, Positive Solutions of Operator Equations, Noordhoff, Groningen, 1964.
    [28] M. A. Krasnosel'skij, J. A. Lifshits, A. V. Sobolev, Positive Linear Systems: The Method of Positive Operators, Heldermann Verlag, Berlin, 1989.
    [29] U. Krause, Positive Dynamical Systems in Discrete Time. Theory, Models and Applications, De Gruyter, Berlin, 2015.
    [30] H. H. Schaefer, Positive Transformationen in lokalkonvexen halbgeordneten Vektorräumen, Math. Ann., 129 (1955), 323-329.
    [31] H. H. Schaefer, Halbgeordnete lokalkonvexe Vektorräume. II, Math. Ann., 138 (1959), 259-286.
    [32] H. H. Schaefer, Some spectral properties of positive linear operators, Pacific J. Math., 9 (1960), 847-860.
    [33] H. H. Schaefer, Topological Vector Spaces, Macmillan, New York, 1966.
    [34] H. H. Schaefer, Banach Lattices and Positive Operators, Springer, Berlin Heidelberg, 1974.
    [35] F. F. Bonsall, Linear operators in complete positive cones, Proc. London Math. Soc., 8 (1958), 53-75.
    [36] W. Jin, H. R. Thieme, An extinction/persistence threshold for sexually reproducing populations: the cone spectral radius, Discrete Cont. Dyn- B, 21 (2016), 447-470.
    [37] B. Lemmens, R. D. Nussbaum, Nonlinear Perron-Frobenius Theory, Cambridge University Press, Cambridge, 2012.
    [38] J. Mallet-Paret, R. D. Nussbaum, Eigenvalues for a class of homogeneous cone maps arising from max-plus operators, Discr. Cont. Dyn-A, 8 (2002), 519-562.
    [39] J. Mallet-Paret, R. D. Nussbaum, Generalizing the Krein-Rutman theorem, measures of noncompactness and the fixed point index, J. Fix. Point Theory A., 7 (2010), 103-143.
    [40] R. D. Nussbaum, Eigenvectors of nonlinear positive operators and the linear Krein-Rutman theorem, Fixed Point Theory (E. Fadell and G. Fournier, eds.), 309-331, Springer, Berlin New York, 1981.
    [41] R. D. Nussbaum, Eigenvectors of order-preserving linear operators, J. London Math. Soc., 2 (1998), 480-496.
    [42] H. R. Thieme, Comparison of spectral radii and Collatz-Wielandt numbers for homogeneous maps, and other applications of the monotone companion norm on ordered normed vector spaces, arXiv:1406.6657.
    [43] H. R. Thieme, Spectral radii and Collatz-Wielandt numbers for homogeneous order-preserving maps and the monotone companion norm, Ordered Structures and Applications (M. de Jeu, B. de Pagter, O. van Gaans, M. Veraar, eds.), 415-467, Birkhäuser/Springer, Cham 2016.
    [44] H. R. Thieme, Eigenfunctionals of homogeneous order-preserving maps with applications to sexually reproducing populations, J. Dyn. Differ. Equ., 28 (2016), 1115-1144.
    [45] H. R. Thieme, Eigenvectors of homogeneous order-bounded order-preserving maps, Discrete Cont. Dyn-B, 22 (2017), 1073-1097.
    [46] H. R. Thieme, From homogeneous eigenvalue problems to two-sex population dynamics, J. Math. Biol., 75 (2017), 783-804.
    [47] W. Jin, H. L. Smith, H. R. Thieme, Persistence versus extinction for a class of discrete-time structured population models, J. Math. Biol., 72 (2016), 821-850.
    [48] R. M. Dudley, Convergence of Baire measures, Stud. Math., 27 (1966), 251-268, Correction to "Convergence of Baire measures", Stud. Math., 51 (1974), 275.
    [49] J. A. Carrillo, R. M. Colombo, P. Gwiazda, A. Ulikowska, Structured populations, cell growth and measure valued balance laws, J. Diff. Eq., 252 (2012), 3245-3277.
    [50] P. Gwiazda, T. Lorenz, A. Marciniak-Czochra, A nonlinear structured population model: Lipschitz continuity of measure-valued solutions with respect to model ingredients, J. Diff. Eqns, 248 (2010), 2703-2735.
    [51] P. Gwiazda, A. Marciniak-Czochra, Structured population equations in metric spaces, J. Hyperbolic Differ. Equ., 7 (2010), 733-773.
    [52] R. Rudnicki, M. Tyran-Kaminska, Piecewise Deterministic processes in Biological Models, Springer, Cham, 2017.
    [53] O. Diekmann, J. A. P. Heesterbeek, J. A. J. Metz, On the definition and the computation of the basic reproduction number R0 in models for infectious diseases in heterogeneous populations, J. Math. Biol., 28 (1990), 365-382.
    [54] J. M. Cushing, Y. Zhou, The net reproductive value and stability in matrix population models, Nat. Res. Mod., 8 (1994), 297-333.
    [55] J. M. Cushing, On the relationship between r and R0 and its role in the bifurcation of stable equilibria of Darwinian matrix models, J. Biol. Dyn., 5 (2011), 277-297.
    [56] I. M. Gel'fand, Normierte Ringe, Mat. Sbornik N.S., 9 (1941), 3-24.
    [57] K. Yosida, Functional Analysis, sec. ed., Springer, Berlin Heidelberg 1965-1968.
    [58] K. D. Deimling, Nonlinear Functional Analysis, Springer, Berlin Heidelberg, 1985.
    [59] K.-H. Förster, B. Nagy, On the Collatz-Wielandt numbers and the local spectral radius of a nonnegative operator, Linear Algebra Appl., 120 (1980), 193-205.
    [60] G. Gripenberg, On the definition of the cone spectral radius, Proc. Amer. Math. Soc., 143 (2015), 1617-1625.
    [61] H. Inaba, Age-structured Population Dynamics in Demography and Epidemiology, Springer, 2017.
    [62] W. Jin, H. L. Smith, H. R. Thieme, Persistence and critical domain size for diffusing populations with two sexes and short reproductive season, J. Dyn. Differ. Equ., 28 (2016), 689-705.
    [63] W. Jin, H. R. Thieme, Persistence and extinction of diffusing populations with two sexes and short reproductive season, Discrete Cont. Dyn-B, 19 (2014), 3209-3218.
    [64] M. Akian, S. Gaubert, R. D. Nussbaum, A Collatz-Wielandt characterization of the spectral radius of order-preserving homogeneous maps on cones, arXiv:1112.5968v1 [math.FA].
    [65] H. J. A. M. Heijmans, B. de Pagter, Asymptotic behavior, One-Parameter Semigroups (Ph. Clément, H.J.A.M. Heimans, S. Angenent, C.J. van Duijn, B. de Pagter, eds.), 213-233, Centre for Mathematics and Computer Science, Amsterdam, 1987.
    [66] P. Gwiazda, P. Orliński, A. Ulikowska, Finite range method of approximation for balance laws in measure spaces, Kinet. Relat. Models, 10 (2017), 669-688.
    [67] A. C. Thompson, On the eigenvectors of some not-necessarily-linear transformations, Proc. London Math. Soc., 15 (1965), 577-598.
    [68] H. R. Thieme, Spectral bound and reproduction number for infinite dimensional population structure and time-heterogeneity, SIAM J. Appl. Math., 70 (2009), 188-211.
    [69] P. Jagers, The deterministic evolution of general branching populations. State of the Art in Probability and Statistics (M. de Gunst, C. Klaassen, A. van der Vaart, eds.), 384-398, Institute of Mathematical Statistics, Beachwood, OH, 2001.
    [70] L. Ambrosio, N. Gigli, G. Savaré, Gradient flows in metric spaces and in the space of probability measures, Birkhäuser, Basel, 2005.
    [71] B. Lemmens, R. D. Nussbaum, Continuity of the cone spectral radius, Proc. Amer. Math. Soc. 141 (2013), 2741-2754.
    [72] M. G. E. Krein, M. A. Rutman, Linear operators leaving invariant a cone in a Banach space (Russian), Uspehi Mat. Nauk (N.S.), 3 (1948), 3-95, English Translation, AMS Translation 1950 (1950), No. 26.
    [73] H. L. Smith, H. R. Thieme, Dynamical Systems and Population Persistence, AMS, Providence, 2011.
  • Reader Comments
  • © 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索
Mathematical Biosciences and Engineering
3.9

Metrics

Article views(4022) PDF downloads(460) Cited by(11)

Preview PDF
Download XML
Export Citation
Article outline

Other Articles By Authors

Related pages

Tools

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog