doi: 10.3934/dcdsb.2019162

Boundary perturbations and steady states of structured populations

1. 

Department of Mathematics, Universitat Autònoma de Barcelona, Bellaterra, 08193, Spain

2. 

Division of Computing Science and Mathematics, University of Stirling, Stirling, FK94LA, UK

* Corresponding author: József Z. Farkas

Received  July 2018 Revised  February 2019 Published  July 2019

In this work we establish conditions which guarantee the existence of (strictly) positive steady states of a nonlinear structured population model. In our framework, the steady state formulation amounts to recasting the nonlinear problem as a family of eigenvalue problems, combined with a fixed point problem. Amongst other things, our formulation requires us to control the growth behaviour of the spectral bound of a family of linear operators along positive rays. For the specific class of model we consider here this presents a considerable challenge. We are going to show that the spectral bound of the family of operators, arising from the steady state formulation, can be controlled by perturbations in the domain of the generators (only). These new boundary perturbation results are particularly important for models exhibiting fertility controlled dynamics. As an important by-product of the application of the boundary perturbation results we employ here, we recover (using a recent theorem by H. R. Thieme) the familiar net reproduction number (or function) for models with single state at birth, which include for example the classic McKendrick (linear) and Gurtin-McCamy (non-linear) age-structured models.

Citation: Àngel Calsina, József Z. Farkas. Boundary perturbations and steady states of structured populations. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2019162
References:
[1]

W. Arendt and C. J. K. Batty, Principal eigenvalues and perturbation, Oper. Theory Adv. Appl., 75 (1995), 39-55. Google Scholar

[2]

W. Arendt and C. J. K. Batty, Domination and ergodicity for positive semigroups, Proc. Amer. Math. Soc., 114 (1992), 743-747. doi: 10.1090/S0002-9939-1992-1072082-3. Google Scholar

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C. BarrilÀ. Calsina and J. Ripoll, On the reproduction number of a gut microbiota model, Bull. Math. Biol., 79 (2017), 2727-2746. doi: 10.1007/s11538-017-0352-8. Google Scholar

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À. CalsinaO. Diekmann and J. Z. Farkas, Structured populations with distributed recruitment: From PDE to delay formulation, Math. Methods Appl. Sci., 39 (2016), 5175-5191. doi: 10.1002/mma.3898. Google Scholar

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À. Calsina and J. Z. Farkas, Steady states in a structured epidemic model with Wentzell boundary condition, J. Evol. Equ., 12 (2012), 495-512. doi: 10.1007/s00028-012-0142-6. Google Scholar

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À. Calsina and J. Z. Farkas, Positive steady states of evolution equations with finite dimensional nonlinearities, SIAM J. Math. Anal., 46 (2014), 1406-1426. doi: 10.1137/130931199. Google Scholar

[8]

À. Calsina and J. Z. Farkas, On a strain-structured epidemic model, Nonlinear Anal. Real World Appl., 31 (2016), 325-342. doi: 10.1016/j.nonrwa.2016.01.014. Google Scholar

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À. Calsina and J. M. Palmada, Steady states of a selection-mutation model for an age structured population, J. Math. Anal. Appl., 400 (2013), 386-395. doi: 10.1016/j.jmaa.2012.11.042. Google Scholar

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Ph. Clément, H. J. A. M. Heijmans, S. Angenent, C. J. van Duijn and B. de Pagter, One-parameter Semigroups, North-Holland Publishing Co., Amsterdam, 1987. Google Scholar

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M. G. Crandall and P. H. Rabinowitz, Bifurcation, perturbation of simple eigenvalues and linearized stability, Arch. Rational Mech. Anal., 52 (1973), 161-180. doi: 10.1007/BF00282325. Google Scholar

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M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. Functional Analysis, 8 (1971), 321-340. doi: 10.1016/0022-1236(71)90015-2. Google Scholar

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J. M. Cushing, The dynamics of hierarchical age-structured populations, J. Math. Biol., 32 (1994), 705-729. doi: 10.1007/BF00163023. Google Scholar

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J. M. Cushing, An Introduction to Structured Population Dynamics, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, 1998. doi: 10.1137/1.9781611970005. Google Scholar

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J. M. Cushing and O. Diekmann, The many guises of $R_0$ (a didactic note), J. Theoret. Biol., 404 (2016), 295-302. doi: 10.1016/j.jtbi.2016.06.017. Google Scholar

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G. Da Prato and P. Grisvard, Maximal regularity for evolution equations by interpolation and extrapolation, J. Funct. Anal., 58 (1984), 107-124. doi: 10.1016/0022-1236(84)90034-X. Google Scholar

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W. Desch and W. Schappacher, On relatively bounded perturbations of linear $C_0$-semigroups, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 11 (1984), 327-341. Google Scholar

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O. DiekmannM. GyllenbergH. HuangM. KirkilionisJ. 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. Google Scholar

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O. DiekmannM. Gyllenberg and J. A. J. Metz, Steady-state analysis of structured population models, Theoretical Population Biology, 63 (2003), 309-338. doi: 10.1016/S0040-5809(02)00058-8. Google Scholar

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K. -J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Springer-Verlag, New York, 2000. Google Scholar

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J. Z. Farkas, Net reproduction functions for nonlinear structured population models, Math. Model. Nat. Phenom., 13 (2018), Art. 32, 12 pp. doi: 10.1051/mmnp/2018036. Google Scholar

[22]

J. Z. FarkasD. M. Green and P. Hinow, Semigroup analysis of structured parasite populations, Math. Model. Nat. Phenom., 5 (2010), 94-114. doi: 10.1051/mmnp/20105307. Google Scholar

[23]

J. Z. Farkas and T. Hagen, Stability and regularity results for a size-structured population model, J. Math. Anal. Appl., 328 (2007), 119-136. doi: 10.1016/j.jmaa.2006.05.032. Google Scholar

[24]

J. Z. Farkas and P. Hinow, Steady states in hierarchical structured populations with distributed states at birth, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), 2671-2689. doi: 10.3934/dcdsb.2012.17.2671. Google Scholar

[25]

G. Greiner, Perturbing the boundary conditions of a generator, Houston J. Math., 13 (1987), 213-229. Google Scholar

[26]

M. E. Gurtin and R. C. MacCamy, Non-linear age-dependent population dynamics, Arch. Rational Mech. Anal., 54 (1974), 281-300. doi: 10.1007/BF00250793. Google Scholar

[27]

M. Iannelli and F. Milner, The Basic Approach to Age-Structured Population Dynamics, Lecture Notes on Mathematical Modelling in the Life Sciences. Springer-Verlag, Dordrecht, 2017. doi: 10.1007/978-94-024-1146-1. Google Scholar

[28]

T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, Berlin-Heidelberg, 1995. Google Scholar

[29]

P. Magal and S. G. Ruan, Theory and Applications of Abstract Semilinear Cauchy Problems, Applied Mathematical Sciences. Vol. 201. Springer, Switzerland, 2018. doi: 10.1007/978-3-030-01506-0. Google Scholar

[30]

P. Magal and S. G. Ruan, Center manifolds for semilinear equations with non-dense domain and applications to Hopf bifurcation in age structured models, Mem. Amer. Math. Soc., 202 (2009), 71 pp. doi: 10.1090/S0065-9266-09-00568-7. Google Scholar

[31]

J. A. J. Metz and O. Diekmann, The Dynamics of Physiologically Structured Populations, Springer-Verlag, Berlin, 1986. doi: 10.1007/978-3-662-13159-6. Google Scholar

[32]

R. OlendorfF. H. RoddD. PunzalanA. E. HoudeC. HurtD. N. Reznick and K. A. Hughes, Frequency-dependent survival in natural guppy populations, Nature, 441 (2006), 633-636. doi: 10.1038/nature04646. Google Scholar

[33]

H. H. Schäfer, Banach Lattices and Positive Operators, Springer-Verlag, Berlin, 1974. Google Scholar

[34]

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

[35]

H. R. Thieme, Remarks on resolvent positive operators and their perturbation, Discrete Contin. Dynam. Systems, 4 (1998), 73-90. doi: 10.3934/dcds.1998.4.73. Google Scholar

[36]

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. Google Scholar

[37]

G. F. Webb, Theory of Nonlinear Age-Dependent Population Dynamics, Marcel Dekker, lnc., New York, 1985. Google Scholar

show all references

References:
[1]

W. Arendt and C. J. K. Batty, Principal eigenvalues and perturbation, Oper. Theory Adv. Appl., 75 (1995), 39-55. Google Scholar

[2]

W. Arendt and C. J. K. Batty, Domination and ergodicity for positive semigroups, Proc. Amer. Math. Soc., 114 (1992), 743-747. doi: 10.1090/S0002-9939-1992-1072082-3. Google Scholar

[3]

W. Arendt, A. Grabosch, G. Greiner, U. Groh, H. P. Lotz, U. Moustakas, R. Nagel, F. Neubrander and U. Schlotterbeck, One-Parameter Semigroups of Positive Operators, Springer-Verlag, Berlin, 1986. doi: 10.1007/BFb0074922. Google Scholar

[4]

C. BarrilÀ. Calsina and J. Ripoll, On the reproduction number of a gut microbiota model, Bull. Math. Biol., 79 (2017), 2727-2746. doi: 10.1007/s11538-017-0352-8. Google Scholar

[5]

À. CalsinaO. Diekmann and J. Z. Farkas, Structured populations with distributed recruitment: From PDE to delay formulation, Math. Methods Appl. Sci., 39 (2016), 5175-5191. doi: 10.1002/mma.3898. Google Scholar

[6]

À. Calsina and J. Z. Farkas, Steady states in a structured epidemic model with Wentzell boundary condition, J. Evol. Equ., 12 (2012), 495-512. doi: 10.1007/s00028-012-0142-6. Google Scholar

[7]

À. Calsina and J. Z. Farkas, Positive steady states of evolution equations with finite dimensional nonlinearities, SIAM J. Math. Anal., 46 (2014), 1406-1426. doi: 10.1137/130931199. Google Scholar

[8]

À. Calsina and J. Z. Farkas, On a strain-structured epidemic model, Nonlinear Anal. Real World Appl., 31 (2016), 325-342. doi: 10.1016/j.nonrwa.2016.01.014. Google Scholar

[9]

À. Calsina and J. M. Palmada, Steady states of a selection-mutation model for an age structured population, J. Math. Anal. Appl., 400 (2013), 386-395. doi: 10.1016/j.jmaa.2012.11.042. Google Scholar

[10]

Ph. Clément, H. J. A. M. Heijmans, S. Angenent, C. J. van Duijn and B. de Pagter, One-parameter Semigroups, North-Holland Publishing Co., Amsterdam, 1987. Google Scholar

[11]

M. G. Crandall and P. H. Rabinowitz, Bifurcation, perturbation of simple eigenvalues and linearized stability, Arch. Rational Mech. Anal., 52 (1973), 161-180. doi: 10.1007/BF00282325. Google Scholar

[12]

M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. Functional Analysis, 8 (1971), 321-340. doi: 10.1016/0022-1236(71)90015-2. Google Scholar

[13]

J. M. Cushing, The dynamics of hierarchical age-structured populations, J. Math. Biol., 32 (1994), 705-729. doi: 10.1007/BF00163023. Google Scholar

[14]

J. M. Cushing, An Introduction to Structured Population Dynamics, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, 1998. doi: 10.1137/1.9781611970005. Google Scholar

[15]

J. M. Cushing and O. Diekmann, The many guises of $R_0$ (a didactic note), J. Theoret. Biol., 404 (2016), 295-302. doi: 10.1016/j.jtbi.2016.06.017. Google Scholar

[16]

G. Da Prato and P. Grisvard, Maximal regularity for evolution equations by interpolation and extrapolation, J. Funct. Anal., 58 (1984), 107-124. doi: 10.1016/0022-1236(84)90034-X. Google Scholar

[17]

W. Desch and W. Schappacher, On relatively bounded perturbations of linear $C_0$-semigroups, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 11 (1984), 327-341. Google Scholar

[18]

O. DiekmannM. GyllenbergH. HuangM. KirkilionisJ. 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. Google Scholar

[19]

O. DiekmannM. Gyllenberg and J. A. J. Metz, Steady-state analysis of structured population models, Theoretical Population Biology, 63 (2003), 309-338. doi: 10.1016/S0040-5809(02)00058-8. Google Scholar

[20]

K. -J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Springer-Verlag, New York, 2000. Google Scholar

[21]

J. Z. Farkas, Net reproduction functions for nonlinear structured population models, Math. Model. Nat. Phenom., 13 (2018), Art. 32, 12 pp. doi: 10.1051/mmnp/2018036. Google Scholar

[22]

J. Z. FarkasD. M. Green and P. Hinow, Semigroup analysis of structured parasite populations, Math. Model. Nat. Phenom., 5 (2010), 94-114. doi: 10.1051/mmnp/20105307. Google Scholar

[23]

J. Z. Farkas and T. Hagen, Stability and regularity results for a size-structured population model, J. Math. Anal. Appl., 328 (2007), 119-136. doi: 10.1016/j.jmaa.2006.05.032. Google Scholar

[24]

J. Z. Farkas and P. Hinow, Steady states in hierarchical structured populations with distributed states at birth, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), 2671-2689. doi: 10.3934/dcdsb.2012.17.2671. Google Scholar

[25]

G. Greiner, Perturbing the boundary conditions of a generator, Houston J. Math., 13 (1987), 213-229. Google Scholar

[26]

M. E. Gurtin and R. C. MacCamy, Non-linear age-dependent population dynamics, Arch. Rational Mech. Anal., 54 (1974), 281-300. doi: 10.1007/BF00250793. Google Scholar

[27]

M. Iannelli and F. Milner, The Basic Approach to Age-Structured Population Dynamics, Lecture Notes on Mathematical Modelling in the Life Sciences. Springer-Verlag, Dordrecht, 2017. doi: 10.1007/978-94-024-1146-1. Google Scholar

[28]

T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, Berlin-Heidelberg, 1995. Google Scholar

[29]

P. Magal and S. G. Ruan, Theory and Applications of Abstract Semilinear Cauchy Problems, Applied Mathematical Sciences. Vol. 201. Springer, Switzerland, 2018. doi: 10.1007/978-3-030-01506-0. Google Scholar

[30]

P. Magal and S. G. Ruan, Center manifolds for semilinear equations with non-dense domain and applications to Hopf bifurcation in age structured models, Mem. Amer. Math. Soc., 202 (2009), 71 pp. doi: 10.1090/S0065-9266-09-00568-7. Google Scholar

[31]

J. A. J. Metz and O. Diekmann, The Dynamics of Physiologically Structured Populations, Springer-Verlag, Berlin, 1986. doi: 10.1007/978-3-662-13159-6. Google Scholar

[32]

R. OlendorfF. H. RoddD. PunzalanA. E. HoudeC. HurtD. N. Reznick and K. A. Hughes, Frequency-dependent survival in natural guppy populations, Nature, 441 (2006), 633-636. doi: 10.1038/nature04646. Google Scholar

[33]

H. H. Schäfer, Banach Lattices and Positive Operators, Springer-Verlag, Berlin, 1974. Google Scholar

[34]

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

[35]

H. R. Thieme, Remarks on resolvent positive operators and their perturbation, Discrete Contin. Dynam. Systems, 4 (1998), 73-90. doi: 10.3934/dcds.1998.4.73. Google Scholar

[36]

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. Google Scholar

[37]

G. F. Webb, Theory of Nonlinear Age-Dependent Population Dynamics, Marcel Dekker, lnc., New York, 1985. Google Scholar

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