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February  2022, 27(2): 1091-1119. doi: 10.3934/dcdsb.2021082

Discrete-time dynamics of structured populations via Feller kernels

Horst R. Thieme

School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ 85287-1804, USA

Received  August 2020 Revised  January 2021 Published  February 2022 Early access  March 2021

Feller kernels are a concise means to formalize individual structural transitions in a structured discrete-time population model. An iteroparous populations (in which generations overlap) is considered where different kernels model the structural transitions for neonates and for older individuals. Other Feller kernels are used to model competition between individuals. The spectral radius of a suitable Feller kernel is established as basic turnover number that acts as threshold between population extinction and population persistence. If the basic turnover number exceeds one, the population shows various degrees of persistence that depend on the irreducibility and other properties of the transition kernels.

Keywords: Integro-difference equations, integral projection models, uniform persistence, ordered normed vector spaces, spectral radius, stability, extinction, basic reproduction number, basic turnover number, compact attractor, eigenmeasure, census.
Mathematics Subject Classification: Primary: 92D25, 47B65, 47G10, 47H07, 47J25; Secondary: 28C15, 47N60.
Citation: Horst R. Thieme. Discrete-time dynamics of structured populations via Feller kernels. Discrete and Continuous Dynamical Systems - B, 2022, 27 (2) : 1091-1119. doi: 10.3934/dcdsb.2021082
References:
[1]

C. D. Aliprantis and K. C. Border, Infinite Dimensional Analysis. A Hitchhiker's Guide, 3rd edition, Springer, Berlin Heidelberg 1999, 2006

[2]

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.  doi: 10.1080/17513758.2010.491583.

[3]

J. M. Cushing and Y. Zhou, The net reproductive value and stability in matrix population models, Nat. Res. Mod., 8 (1994), 297-333.  doi: 10.1111/j.1939-7445.1994.tb00188.x.

[4]

O. Diekmann, M. Gyllenberg, J. A. J. Metz and H. Thieme, The 'cumulative' formulation of (physiologically) structured population models, in Evolution Equations, Control Theory, and Biomathematics (eds. Ph. Clément and G. Lumer), Marcel Dekker, 1994,145–154

[5]

O. Diekmann, J. A. P. Heesterbeek and 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 doi: 10.1007/BF00178324.

[6]

R. M. Dudley, Convergence of Baire measures, Stud. Math., 27 (1966), 251-268, Correction to "Convergence of Baire measures", Stud. Math., 51 (1974), 275. doi: 10.4064/sm-27-3-251-268.

[7]

R. M. Dudley, Real Analysis and Probability, sec. ed., Cambridge University Press, Cambridge 2002 doi: 10.1017/CBO9780511755347.

[8]

E. A. Eager, R. Rebarber and B. Tenhumberg, Modeling and analysis of a density-dependent stochastic integral projection model for a disturbance specialist plant and its seed bank, Bull. Math. Biol., 76 (2014), 1809-1834.  doi: 10.1007/s11538-014-9978-y.

[9]

S. P. Ellner, D. Z. Childs and M. Rees, Data-driven Modelling of Structured Populations: A Practical Guide to the Integral Projection Model, Springer, Cham, 2016. doi: 10.1007/978-3-319-28893-2.

[10]

S. P. Ellner and M. Rees, Stochastic stable population growth in integral projection models: Theory and application, J. Math. Biol., 54 (2007), 227-256.  doi: 10.1007/s00285-006-0044-8.

[11]

P. Gwiazda, A. Marciniak-Czochra and H. R. Thieme, Measures under the flat norm as ordered normed vector space, Positivity, 22 (2018), 139-140.  doi: 10.1007/s11117-017-0503-z.

[12]

S. C. Hille and 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.  doi: 10.1007/s00020-008-1652-z.

[13]

W. Jin and H. R. Thieme, An extinction/persistence threshold for sexually reproducing populations: The cone spectral radius, Discrete and Continuous Dynamical System - B, 21 (2016), 447-470.  doi: 10.3934/dcdsb.2016.21.447.

[14]

M. Kot and W. M. Schaffer, Discrete-time growth-dispersal models, Math. Biosci., 80 (1986) 109–136. doi: 10.1016/0025-5564(86)90069-6.

[15]

U. Krause, Positive Dynamical Systems in Siscrete Time. Theory, Models, and Applications, De Gruyter Studies in Mathematics, 62, De Gruyter, Berlin, 2015 doi: 10.1515/9783110365696.

[16]

M. A. Lewis, N. G. Marculis and Z. Shen, Integrodifference equations in the presence of climate change: Persistence criterion, travelling waves and inside dynamics, J. Math. Biol., 77 (2018), 1649-1687.  doi: 10.1007/s00285-018-1206-1.

[17]

B. Li, M. A. Lewis and H. F. Weinberger, Existence of traveling waves for integral recursions with nonmonotone growth functions, J. Math. Biol., 58 (2009), 323-338.  doi: 10.1007/s00285-008-0175-1.

[18]

F. Lutscher, Integrodifference Equations in Spatial Ecology, Springer, Cham, 2019 doi: 10.1007/978-3-030-29294-2.

[19]

F. Lutscher and S. V. Petrovskii, The importance of census times in discrete-time growth-dispersal models, J. Biol. Dyn., 2 (2008), 55-63.  doi: 10.1080/17513750701769899.

[20] J. N. McDonald and N. A. Weiss, A Course in Real Analysis, Academic Press, an Diego, 1999. 
[21]

T. E. X. Miller, A. K. Shaw, B. D. Inouye and M. G. Neubert, Sex-biased dispersal and the speed of two-sex invasions, Amer. Nat., 177 (2011), 549-561.  doi: 10.1086/659628.

[22]

J. Musgrave and F. Lutscher, Integrodifference equations in patchy landscapes II: Population level consequences, J. Math. Biol., 69 (2014), 617-658.  doi: 10.1007/s00285-013-0715-1.

[23]

C. Poetzsche, Numerical dynamics of integrodifference equations: Global attractivity in a C0-setting, SIAM J. Numer. Anal., 57 (2019), 2121-2141.  doi: 10.1137/18M1214469.

[24]

V. M. Shurenkov, On the relationship between spectral radii and Perron Roots, Chalmers Univ Tech and Göteborg Univ (preprint)

[25]

H. L. Smith and H. R. Thieme, Dynamical Systems and Population Persistence, American Mathematical Society, Providence 2011 doi: 10.1090/gsm/118.

[26]

H. R. Thieme, On a class of Hammerstein integral equations, Manuscripta Math., 29 (1979), 49-84.  doi: 10.1007/BF01309313.

[27]

H. R. Thieme, Eigenfunctionals of homogeneous order-preserving maps with applications to sexually reproducing populations, J. Dynamics and Differential Equations, 28 (2016), 1115-1144.  doi: 10.1007/s10884-015-9463-9.

[28]

H. R. Thieme, From homogeneous eigenvalue problems to two-sex population dynamics, J. Math. Biol., 75 (2017), 783-804.  doi: 10.1007/s00285-017-1114-9.

[29]

H. R. Thieme, Discrete-time population dynamics on the state space of measures, Math. Biosci. Engin., 17 (2020), 1168-1217.  doi: 10.3934/mbe.2020061.

[30]

H. R. Thieme, Persistent discrete-time dynamics on measures, in Progress on Difference Equations and Discrete Dynamical Systems, (eds. Stephen Baigent, Saber Elaydi and Martin Bohner), 59-100, Springer Proceedings in Mathematics & Statistics 341, Springer Nature Switzerland AG, 2020

[31]

H. R. Thieme, Discrete-time population dynamics of spatially distributed semelparous two-sex populations, (preprint)

[32]

R. Wu and X.-Q. Zhao, Propagation dynamics for a spatially periodic integrodifference competition model, J. Differential Equations, 264 (2018), 6507-6534.  doi: 10.1016/j.jde.2018.01.039.

[33]

X.-Q. Zhao, Dynamical Systems in Population Biology, Springer, New York, 2003 doi: 10.1007/978-0-387-21761-1.

show all references

References:
[1]

C. D. Aliprantis and K. C. Border, Infinite Dimensional Analysis. A Hitchhiker's Guide, 3rd edition, Springer, Berlin Heidelberg 1999, 2006

[2]

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.  doi: 10.1080/17513758.2010.491583.

[3]

J. M. Cushing and Y. Zhou, The net reproductive value and stability in matrix population models, Nat. Res. Mod., 8 (1994), 297-333.  doi: 10.1111/j.1939-7445.1994.tb00188.x.

[4]

O. Diekmann, M. Gyllenberg, J. A. J. Metz and H. Thieme, The 'cumulative' formulation of (physiologically) structured population models, in Evolution Equations, Control Theory, and Biomathematics (eds. Ph. Clément and G. Lumer), Marcel Dekker, 1994,145–154

[5]

O. Diekmann, J. A. P. Heesterbeek and 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 doi: 10.1007/BF00178324.

[6]

R. M. Dudley, Convergence of Baire measures, Stud. Math., 27 (1966), 251-268, Correction to "Convergence of Baire measures", Stud. Math., 51 (1974), 275. doi: 10.4064/sm-27-3-251-268.

[7]

R. M. Dudley, Real Analysis and Probability, sec. ed., Cambridge University Press, Cambridge 2002 doi: 10.1017/CBO9780511755347.

[8]

E. A. Eager, R. Rebarber and B. Tenhumberg, Modeling and analysis of a density-dependent stochastic integral projection model for a disturbance specialist plant and its seed bank, Bull. Math. Biol., 76 (2014), 1809-1834.  doi: 10.1007/s11538-014-9978-y.

[9]

S. P. Ellner, D. Z. Childs and M. Rees, Data-driven Modelling of Structured Populations: A Practical Guide to the Integral Projection Model, Springer, Cham, 2016. doi: 10.1007/978-3-319-28893-2.

[10]

S. P. Ellner and M. Rees, Stochastic stable population growth in integral projection models: Theory and application, J. Math. Biol., 54 (2007), 227-256.  doi: 10.1007/s00285-006-0044-8.

[11]

P. Gwiazda, A. Marciniak-Czochra and H. R. Thieme, Measures under the flat norm as ordered normed vector space, Positivity, 22 (2018), 139-140.  doi: 10.1007/s11117-017-0503-z.

[12]

S. C. Hille and 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.  doi: 10.1007/s00020-008-1652-z.

[13]

W. Jin and H. R. Thieme, An extinction/persistence threshold for sexually reproducing populations: The cone spectral radius, Discrete and Continuous Dynamical System - B, 21 (2016), 447-470.  doi: 10.3934/dcdsb.2016.21.447.

[14]

M. Kot and W. M. Schaffer, Discrete-time growth-dispersal models, Math. Biosci., 80 (1986) 109–136. doi: 10.1016/0025-5564(86)90069-6.

[15]

U. Krause, Positive Dynamical Systems in Siscrete Time. Theory, Models, and Applications, De Gruyter Studies in Mathematics, 62, De Gruyter, Berlin, 2015 doi: 10.1515/9783110365696.

[16]

M. A. Lewis, N. G. Marculis and Z. Shen, Integrodifference equations in the presence of climate change: Persistence criterion, travelling waves and inside dynamics, J. Math. Biol., 77 (2018), 1649-1687.  doi: 10.1007/s00285-018-1206-1.

[17]

B. Li, M. A. Lewis and H. F. Weinberger, Existence of traveling waves for integral recursions with nonmonotone growth functions, J. Math. Biol., 58 (2009), 323-338.  doi: 10.1007/s00285-008-0175-1.

[18]

F. Lutscher, Integrodifference Equations in Spatial Ecology, Springer, Cham, 2019 doi: 10.1007/978-3-030-29294-2.

[19]

F. Lutscher and S. V. Petrovskii, The importance of census times in discrete-time growth-dispersal models, J. Biol. Dyn., 2 (2008), 55-63.  doi: 10.1080/17513750701769899.

[20] J. N. McDonald and N. A. Weiss, A Course in Real Analysis, Academic Press, an Diego, 1999. 
[21]

T. E. X. Miller, A. K. Shaw, B. D. Inouye and M. G. Neubert, Sex-biased dispersal and the speed of two-sex invasions, Amer. Nat., 177 (2011), 549-561.  doi: 10.1086/659628.

[22]

J. Musgrave and F. Lutscher, Integrodifference equations in patchy landscapes II: Population level consequences, J. Math. Biol., 69 (2014), 617-658.  doi: 10.1007/s00285-013-0715-1.

[23]

C. Poetzsche, Numerical dynamics of integrodifference equations: Global attractivity in a C0-setting, SIAM J. Numer. Anal., 57 (2019), 2121-2141.  doi: 10.1137/18M1214469.

[24]

V. M. Shurenkov, On the relationship between spectral radii and Perron Roots, Chalmers Univ Tech and Göteborg Univ (preprint)

[25]

H. L. Smith and H. R. Thieme, Dynamical Systems and Population Persistence, American Mathematical Society, Providence 2011 doi: 10.1090/gsm/118.

[26]

H. R. Thieme, On a class of Hammerstein integral equations, Manuscripta Math., 29 (1979), 49-84.  doi: 10.1007/BF01309313.

[27]

H. R. Thieme, Eigenfunctionals of homogeneous order-preserving maps with applications to sexually reproducing populations, J. Dynamics and Differential Equations, 28 (2016), 1115-1144.  doi: 10.1007/s10884-015-9463-9.

[28]

H. R. Thieme, From homogeneous eigenvalue problems to two-sex population dynamics, J. Math. Biol., 75 (2017), 783-804.  doi: 10.1007/s00285-017-1114-9.

[29]

H. R. Thieme, Discrete-time population dynamics on the state space of measures, Math. Biosci. Engin., 17 (2020), 1168-1217.  doi: 10.3934/mbe.2020061.

[30]

H. R. Thieme, Persistent discrete-time dynamics on measures, in Progress on Difference Equations and Discrete Dynamical Systems, (eds. Stephen Baigent, Saber Elaydi and Martin Bohner), 59-100, Springer Proceedings in Mathematics & Statistics 341, Springer Nature Switzerland AG, 2020

[31]

H. R. Thieme, Discrete-time population dynamics of spatially distributed semelparous two-sex populations, (preprint)

[32]

R. Wu and X.-Q. Zhao, Propagation dynamics for a spatially periodic integrodifference competition model, J. Differential Equations, 264 (2018), 6507-6534.  doi: 10.1016/j.jde.2018.01.039.

[33]

X.-Q. Zhao, Dynamical Systems in Population Biology, Springer, New York, 2003 doi: 10.1007/978-0-387-21761-1.

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