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
[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 L^{1} 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 R_{0} 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 R_{0} 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. |