Some results based on maximal regularity regarding population models with age and spatial structure

  • Christoph Walker


We review some results on abstract linear and nonlinear population models with age and spatial structure. The results are mainly based on the assumption of maximal \(L_p\)-regularity of the spatial dispersion term. In particular, this property allows us to characterize completely the generator of the underlying linear semigroup and to give a simple proof of asynchronous exponential growth of the semigroup. Moreover, maximal regularity is also a powerful tool in order to establish the existence of nontrivial positive equilibrium solutions to nonlinear equations by fixed point arguments or bifurcation techniques. We illustrate the results with examples.


Population models Age and spatial structure Maximal regularity Bifurcation theory 

Mathematics Subject Classification

35K59 35B32 47D06 92D25 47H07 


  1. 1.
    Amann, H.: Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces. SIAM Rev. 18, 620–709 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Amann, H.: Dual semigroups and second-order linear elliptic boundary value problems. Israel J. Math. 45, 225–254 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Amann, H.: Linear and quasilinear parabolic problems, vol. I. Monographs in Mathematics, vol. 89, Birkhäuser Boston Inc., Boston, MA (1995)Google Scholar
  4. 4.
    Anita, L.-I., Anita, S.: Asymptotic behavior of the solutions to semi-linear age-dependent population dynamics with diffusion and periodic vital rates. Math. Popul. Stud. 15, 114–121 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Ayati, B.: A structured-population model of Proteus mirabilis swarm-colony development. J. Math. Biol. 52, 93–114 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Ayati, B., Webb, G., Anderson, R.: Computational methods and results for structured multiscale models of tumor invasion. SIAM J. Multsc. Mod. Simul. 5, 1–20 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Bátkai, A., Fijavž, M.K., Rhandi, A.: Positive operator semigroups. Oper. Theor. Adv. Appl. 257, Birkhäuser Cham (2017)Google Scholar
  8. 8.
    Busenberg, S., Langlais, M.: Global behaviour in age structured S.I.S. models with seasonal periodicities and vertical transmission. J. Math. Anal. Appl. 213, 511–533 (2007)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Crandall, M.G., Rabinowitz, P.H.: Bifurcation from simple eigenvalues. J. Funct. Anal. 8, 321–340 (1971)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Cushing, J.M.: Existence and stability of equilibria in age-structured population dynamics. J. Math. Biol. 20, 259–276 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Cushing, J.M.: Global branches of equilibrium solutions of the McKendrick equations for age structured population growth. Comp. Math. Appl. 11, 175–188 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Cushing, J.: Equilibria in structured populations. J. Math. Biol. 23, 15–39 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Daners, D., Koch-Medina, P.: Abstract evolution equations, periodic problems, and applications. Pitman Res. Notes Math. Ser., 279, Longman, Harlow (1992)Google Scholar
  14. 14.
    Delgado, M., Molina-Becerra, M., Suárez, A.: Nonlinear age-dependent diffusive equations: a bifurcation approach. J. Diff. Equ. 244, 2133–2155 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Delgado, M., Molina-Becerra, M., Suárez, A.: A nonlinear age-dependent model with spatial diffusion. J. Math. Anal. Appl. 313, 366–380 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Delgado, M., Molina-Becerra, M., Suárez, A.: The sub-supersolution method for an evolutionary reaction-diffusion age-dependent problem. Diff. Integr. Equ. 18, 155–168 (2005)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Delgado, M., Suárez, A.: Age-dependent diffusive Lotka–Volterra type systems. Math. Comput. Model. 45, 668–680 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Dore, G.: Maximal regularity in \(L^p\) spaces for an abstract Cauchy problem. Adv. Diff. Equ. 5, 293–322 (2000)zbMATHGoogle Scholar
  19. 19.
    Ducrot, A., Magal, P.: Travelling wave solutions for an infection-age structured model with diffusion. Proc. Roy. Soc. Edinburgh Sect. A 139, 459–482 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Dyson, J., Sanchez, E., Villella-Bressan, R., Webb, G.F.: An age and spatially structured model of tumor invasion with haptotaxis. Discrete Contin. Dyn. Syst. Ser. B 8, 45–60 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Dyson, J., Villella-Bressan, R., Webb, G.F.: An age and spatially structured model of tumor invasion with haptotaxis. II. Math. Popul. Stud. 15, 73–95 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Engel, K.-J.: Nagel, R.: One-parameter semigroups for linear evolution equations. Springer New York Inc. (2000)Google Scholar
  23. 23.
    Engel, K.-J., Nagel, R.: A short course on operator semigroups. Springer, New York (2006)zbMATHGoogle Scholar
  24. 24.
    Esipov, S.E., Shapiro, J.A.: Kinetic model of Proteus mirabilis swarm colony development. J. Math. Biol. 36, 249–268 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Feller, W.: On the integral equation of renewal theory. Ann. Math. Stat. 12, 243–267 (1941)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Fitzgibbon, W., Parrott, M., Webb, G.: Diffusion epidemic models with incubation and crisscross dynamics. Math. Biosci. 128, 131–155 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Guo, B.Z., Chan, W.L.: On the semigroup for age dependent population dynamics with spatial diffusion. J. Math. Anal. Appl. 184(1), 190–199 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Gurtin, M.E.: A system of equations for age-dependent population diffusion. J. Thero. Biol. 40, 389–392 (1973)CrossRefGoogle Scholar
  29. 29.
    Gurtin, M.E., MacCamy, R.C.: Nonlinear age-dependent population dynamics. Arch. Rat. Mech. Anal. 54, 281–300 (1974)CrossRefzbMATHGoogle Scholar
  30. 30.
    Gurtin, M.E., MacCamy, R.C.: Diffusion models for age-structured populations. Math. Biosci. 54, 49–59 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Gyllenberg, M., Webb, G.F.: Asynchronous exponential growth of semigroups of nonlinear operators. J. Math. Anal. Appl. 167, 443–467 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Huyer, W.: Semigroup formulation and approximation of a linear age-dependent population problem with spatial diffusion. Semigroup Forum 49, 99–114 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Iannelli, M., Martcheva, M., Milner, F.A.: Gender-structured population modeling. Mathematical Methods, Numerics, and Simulations. SIAM Frontiers in Applied Mathematics, Philadelphia (2005)Google Scholar
  34. 34.
    Kato, T.: Integration of the equation of evolution in Banach spaces. J. Math. Soc. Japan 5, 208–234 (1953)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Kunya, T., Oizumi, R.: Existence result for an age-structured SIS epidemic model with spatial diffusion. Nonlinear Anal. Real World Appl. 23, 196–208 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Langlais, M.: Large time behavior in a nonlinear age-dependent population dynamics problem with spatial diffusion. J. Math. Biol. 26, 319–346 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Laurençot, Ph, Walker, Ch.: An age and spatially structured population model for Proteus mirabilis swarm-colony development. Math. Mod. Nat. Phen. 7, 49–77 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Magal, P., Thieme, H.R.: Eventual compactness for semiflows generated by nonlinear age-structured models. Commun. Pure Appl. Anal. 3, 695–727 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    McKendrick, A.G.: Applications of mathematics to medical problems. Proc. Edin. Math. Soc. 44, 435–438 (1926)Google Scholar
  40. 40.
    Pazy, A.: Semigroups of linear operators and applications to partial differential equations. Appl. Math. Sci. 44, Springer (1983)Google Scholar
  41. 41.
    Pejsachowicz, J., Rabier, P.J.: Degree theory for \(C^1\) Fredholm mappings of index 0. J. Anal. Math. 76, 289–319 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Prüß, J.: On the qualitative behaviour of populations with age-specific interactions. Comp. Math. Appl. 9, 327–339 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Prüß, J.: Maximal regularity for evolution equations in L p-spaces. Conf. Sem. Mat. Univ. Bari 285, 1–39 (2003)Google Scholar
  44. 44.
    Rabinowitz, P.H.: Some global results for nonlinear eigenvalue problems. J. Funct. Anal. 7, 487–513 (1971)MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Restrepo, G.: Differentiable norms in Banach spaces. Bull. Am. Math. Soc. 70, 413–414 (1964)MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    Rhandi, A.: Positivity and stability for a population equation with diffusion on \(L^1\). Positivity 2(2), 101–113 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  47. 47.
    Rhandi, A., Schnaubelt., R.: Asymptotic behaviour of a non-autonomous population equation with diffusion in \(L^1\). Disc. Contin. Dyn. Systems 5(3), 663–683 (1999)CrossRefzbMATHGoogle Scholar
  48. 48.
    Sharpe, F.R., Lotka, A.J.: A problem in age distributions. Phil. Mag. 21, 98–130 (1911)CrossRefzbMATHGoogle Scholar
  49. 49.
    Shi, J., Wang, X.: On global bifurcation for quasilinear elliptic systems on bounded domains. J. Diff. Equ. 246(7), 2788–2812 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  50. 50.
    Simon, J.: Compact sets in the space \(L^p(0,T;B)\). Ann. Mat. Pura Appl. 146(4), 65–96 (1987)Google Scholar
  51. 51.
    Thieme, H.R.: Positive perturbation of operator semigroups: growth bounds, essential compactness, and asynchronous exponential growth. Discrete Contin. Dyn. Syst. 4(4), 735–764 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  52. 52.
    Thieme, H.R.: Mathematics in Population Biology. Princeton series in theoretical and computational biology. Princeton University Press, Princeton (2003)zbMATHGoogle Scholar
  53. 53.
    Triebel, H.: Interpolation theory, function spaces, differential operators, 2nd edn. Johann Ambrosius Barth, Heidelberg (1995)zbMATHGoogle Scholar
  54. 54.
    von Foerster, H.: Some remarks on changing populations. The Kinetics of Cellular Proliferation, Grune and Stratton, pp. 382–407 (1959)Google Scholar
  55. 55.
    Walker, Ch.: Global well-posedness of a haptotaxis model including age and spatial structure. Diff. Int. Eq. 20, 1053–1074 (2007)zbMATHGoogle Scholar
  56. 56.
    Walker, Ch.: Global existence for an age and spatially structured haptotaxis model with nonlinear age-boundary conditions. Europ. J. Appl. Math. 19, 113–147 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  57. 57.
    Walker, Ch.: Positive equilibrium solutions for age and spatially structured population models. SIAM J. Math. Anal. 41, 1366–1387 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  58. 58.
    Walker, Ch.: Global bifurcation of positive equilibria in nonlinear population models. J. Diff. Equ. 248, 1756–1776 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  59. 59.
    Walker, Ch.: Age-dependent equations with non-linear diffusion. Discrete Contin. Dyn. Syst. 26, 691–712 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  60. 60.
    Walker, Ch.: Bifurcation of positive equilibria in nonlinear structured population models with varying mortality rates. Ann. Mat. Pura Appl. 190, 1–19 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  61. 61.
    Walker, Ch.: On positive solutions of some system of reaction-diffusion equations with nonlocal initial conditions. J. Reine Angew. Math. 660, 149–179 (2011)MathSciNetzbMATHGoogle Scholar
  62. 62.
    Walker, Ch.: On nonlocal parabolic steady-state equations of cooperative or competing systems. Nonlinear Anal. Real World Appl. 12, 3552–3571 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  63. 63.
    Walker, Ch.: Positive solutions of some parabolic system with cross-diffusion and nonlocal initial conditions. NoDEA Nonlinear Diff. Equ. Appl. 19, 195–218 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  64. 64.
    Walker, Ch.: Some remarks on the asymptotic behavior of the semigroup associated with age-structured diffusive populations. Monatsh. Math. 170, 481–501 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  65. 65.
    Walker, Ch.: Global continua of positive solutions for some quasilinear parabolic equations with a nonlocal initial condition. J. Dyn. Diff. Equ. 25, 159–172 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  66. 66.
    Webb, G.F.: Theory of nonlinear age-dependent population dynamics. Monographs and Textbooks in Pure and Applied Mathematics, 89. Marcel Dekker, Inc., New York (1985)Google Scholar
  67. 67.
    Webb, G.F.: An age-dependent epidemic model with spatial diffusion. Arch. Rat. Mech. Anal. 75, 91–102 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  68. 68.
    Webb, G.F.: An operator-theoretic formulation of asynchronous exponential growth. Trans. Am. Math. Soc. 303(2), 751–763 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  69. 69.
    Webb, G.F.: Population models structured by age, size, and spatial position. Structured population models in biology and epidemiology, 149, Lecture Notes in Math, vol. 1936, : Math. Biosci. Subser., Springer, Berlin (2008)Google Scholar

Copyright information

© Orthogonal Publishing and Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institut für Angewandte MathematikLeibniz Universität HannoverHannoverGermany

Personalised recommendations