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  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, 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.   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

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

[1]

Xiyou Cheng, Zhitao Zhang. Structure of positive solutions to a class of Schrödinger systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020461

[2]

Felix Finster, Jürg Fröhlich, Marco Oppio, Claudio F. Paganini. Causal fermion systems and the ETH approach to quantum theory. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020451

[3]

Maoding Zhen, Binlin Zhang, Vicenţiu D. Rădulescu. Normalized solutions for nonlinear coupled fractional systems: Low and high perturbations in the attractive case. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020379

[4]

Meng Chen, Yong Hu, Matteo Penegini. On projective threefolds of general type with small positive geometric genus. Electronic Research Archive, , () : -. doi: 10.3934/era.2020117

[5]

Pierre-Etienne Druet. A theory of generalised solutions for ideal gas mixtures with Maxwell-Stefan diffusion. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020458

[6]

Juan Pablo Pinasco, Mauro Rodriguez Cartabia, Nicolas Saintier. Evolutionary game theory in mixed strategies: From microscopic interactions to kinetic equations. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2020051

[7]

Huiying Fan, Tao Ma. Parabolic equations involving Laguerre operators and weighted mixed-norm estimates. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5487-5508. doi: 10.3934/cpaa.2020249

[8]

Anton A. Kutsenko. Isomorphism between one-Dimensional and multidimensional finite difference operators. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020270

[9]

Yuxia Guo, Shaolong Peng. A direct method of moving planes for fully nonlinear nonlocal operators and applications. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020462

[10]

Zedong Yang, Guotao Wang, Ravi P. Agarwal, Haiyong Xu. Existence and nonexistence of entire positive radial solutions for a class of Schrödinger elliptic systems involving a nonlinear operator. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020436

[11]

Marco Ghimenti, Anna Maria Micheletti. Compactness results for linearly perturbed Yamabe problem on manifolds with boundary. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020453

[12]

Claudianor O. Alves, Rodrigo C. M. Nemer, Sergio H. Monari Soares. The use of the Morse theory to estimate the number of nontrivial solutions of a nonlinear Schrödinger equation with a magnetic field. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020276

[13]

Wenmeng Geng, Kai Tao. Large deviation theorems for dirichlet determinants of analytic quasi-periodic jacobi operators with Brjuno-Rüssmann frequency. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5305-5335. doi: 10.3934/cpaa.2020240

[14]

Anna Canale, Francesco Pappalardo, Ciro Tarantino. Weighted multipolar Hardy inequalities and evolution problems with Kolmogorov operators perturbed by singular potentials. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020274

[15]

Antoine Benoit. Weak well-posedness of hyperbolic boundary value problems in a strip: when instabilities do not reflect the geometry. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5475-5486. doi: 10.3934/cpaa.2020248

[16]

Mengni Li. Global regularity for a class of Monge-Ampère type equations with nonzero boundary conditions. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020267

[17]

Shenglan Xie, Maoan Han, Peng Zhu. A posteriori error estimate of weak Galerkin fem for second order elliptic problem with mixed boundary condition. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020340

[18]

Mehdi Badsi. Collisional sheath solutions of a bi-species Vlasov-Poisson-Boltzmann boundary value problem. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2020052

[19]

Mokhtar Bouloudene, Manar A. Alqudah, Fahd Jarad, Yassine Adjabi, Thabet Abdeljawad. Nonlinear singular $ p $ -Laplacian boundary value problems in the frame of conformable derivative. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020442

2019 Impact Factor: 1.27

Metrics

  • PDF downloads (82)
  • HTML views (229)
  • Cited by (0)

Other articles
by authors

[Back to Top]