December  2019, 24(12): 6675-6691. 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  December 2019 Early access  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 and Continuous Dynamical Systems - B, 2019, 24 (12) : 6675-6691. 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. 

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[13]

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

[14]

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

[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.

[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.

[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. 

[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.

[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.

[20]

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

[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.

[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.

[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.

[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.

[25]

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

[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.

[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.

[28]

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

[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.

[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.

[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.

[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.

[33]

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

[34]

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

[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.

[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.

[37]

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

show all references

References:
[1]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[13]

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

[14]

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

[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.

[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.

[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. 

[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.

[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.

[20]

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

[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.

[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.

[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.

[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.

[25]

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

[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.

[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.

[28]

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

[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.

[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.

[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.

[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.

[33]

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

[34]

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

[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.

[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.

[37]

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

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