Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-27T12:08:01.054Z Has data issue: false hasContentIssue false
  • English
  • Français

The effects of diffusion on the principal eigenvalue for age-structured models with random diffusion

Part of: Parabolic equations and systems General theory of linear operators

Published online by Cambridge University Press:  23 March 2021

Hao Kang
Hao Kang*
Affiliation:
Department of Mathematics, University of Miami, Coral Gables, FL 33146, USA (haokang@math.miami.edu)
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper, we study the principal spectral theory of age-structured models with random diffusion. First, we provide an equivalent characteristic for the principal eigenvalue, the strong maximum principle and a positive strict super-solution. Then, we use the result to investigate the effects of diffusion rate on the principal eigenvalue. Finally, we study how the principal eigenvalue affects the global dynamics of the KPP model and verify that the principal eigenvalue being zero is a critical value.

Keywords

Age structureglobal dynamicsprincipal eigenvaluerandom diffusionstrong maximum principle

MSC classification

Primary: 35K57: Reaction-diffusion equations
Secondary: 92D25: Population dynamics (general) 47A10: Spectrum, resolvent
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 152 , Issue 1 , February 2022 , pp. 258 - 280
Check for updates
Copyright
Copyright © The Author(s), 2021. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Chan, W. L. and Guo, B. Z.. On the semigroups of age-size dependent population dynamics with spatial diffusion. Manuscr. Math. 66 (1990), 161–181.CrossRefGoogle Scholar
Chekroun, A. and Kuniya, T.. Global threshold dynamics of an infection age-structured SIR epidemic model with diffusion under the Dirichlet boundary condition. J. Differ. Equ. 269 (2020), 117–148.CrossRefGoogle Scholar
Chen, X. and Lou, Y.. Effects of diffusion and advection on the smallest eigenvalue of an elliptic operator and their applications. Indiana Univ. Math. J. 61 (2012), 45–80.CrossRefGoogle Scholar
Coville, J.. On a simple criterion for the existence of a principal eigenfunction of some nonlocal operators. J. Differ. Equ. 249 (2010), 2921–2953.CrossRefGoogle Scholar
Crandall, M. G. and Rabinowitz, P. H.. Bifurcation from simple eigenvalues. J. Funct. Anal. 8 (1971), 321–340.CrossRefGoogle Scholar
Daners, D. and Medina, K. P.. Abstract evolution equations, periodic problems and applications (London: Longman, 1992).Google Scholar
Delgado, M., Molina-Becerra, M. and Suárez, A.. A nonlinear age-dependent model with spatial diffusion. J. Math. Anal. Appl. 313 (2006), 366–380.CrossRefGoogle Scholar
Delgado, M., Molina-Becerra, M. and Suárez, A.. Nonlinear age-dependent diffusive equations: a bifurcation approach. J. Differ. Equ. 244 (2008), 2133–2155.CrossRefGoogle Scholar
Ducrot, A.. Travelling wave solutions for a scalar age-structured equation. Discrete Contin. Dyn. Syst. B 7 (2007), 251–273.Google Scholar
Ducrot, A. and Magal, P.. Travelling wave solutions for an infection-age structured model with diffusion. Proc. R. Soc. Edinburgh Sect. A 139 (2009), 459–482.CrossRefGoogle Scholar
Ducrot, A. and Magal, P.. Travelling wave solutions for an infection-age structured epidemic model with external supplies. Nonlinearity 24 (2011), 2891–2911.CrossRefGoogle Scholar
Ducrot, A., Magal, P. and Ruan, S.. Travelling wave solutions in multigroup age-structured epidemic models. Arch. Ration. Mech. Anal. 195 (2010), 311–331.CrossRefGoogle Scholar
Freedman, H. I. and Zhao, X.-Q.. Global asymptotics in some quasimonotone reaction-diffusion systems with delays. J. Differ. Equ. 137 (1997), 340–362.CrossRefGoogle Scholar
García-Melián, J. and Rossi, J. D.. On the principal eigenvalue of some nonlocal diffusion problems. J. Differ. Equ. 246 (2009), 21–38.CrossRefGoogle Scholar
Guo, B. Z. and Chan, W. L.. On the semigroup for age dependent population dynamics with spatial diffusion. J. Math. Anal. Appl. 184 (1994), 190–199.CrossRefGoogle Scholar
Gurtin, M. E.. A system of equations for age-dependent population diffusion. J. Theor. Biol. 40 (1973), 389–392.CrossRefGoogle ScholarPubMed
Gurtin, M. E. and MacCamy, R. C. Non-linear age-dependent population dynamics. Arch. Ration. Mech. Anal. 54 (1974), 281–300.CrossRefGoogle Scholar
Gurtin, M. E. and MacCamy, R. C.. Diffusion models for age-structured populations. Math. Biosci. 54 (1981), 49–59.CrossRefGoogle Scholar
Hutson, V., Martinez, S., Mischaikow, K. and Vickers, G. T.. The evolution of dispersal. J. Math. Biol. 47 (2003), 483–517.CrossRefGoogle ScholarPubMed
Kang, H. and Ruan, S.. Age-structured models with nonlocal diffusion principal spectral theory, limiting properties and global dynamics. submitted.Google Scholar
Kang, H., Ruan, S. and Yu, X.. Age-structured population dynamics with nonlocal diffusion. J. Dyn. Differ. Equ. (2020). https://doi.org/10.1007/s10884-020-09860-5.CrossRefGoogle ScholarPubMed
Kato, T.. Perturbation theory for linear operators, vol. 132 (Springer-Verlag Berlin Heidelberg, 2013).Google Scholar
Lam, K.-Y. and Lou, Y.. Asymptotic behavior of the principal eigenvalue for cooperative elliptic systems and applications. J. Dynam. Differ. Equ. 28 (2016), 29–48.CrossRefGoogle Scholar
Langlais, M.. A nonlinear problem in age-dependent population diffusion. SIAM J. Math. Anal. 16 (1985), 510–529.CrossRefGoogle Scholar
Langlais, M.. Large time behavior in a nonlinear age-dependent population dynamics problem with spatial diffusion. J. Math. Biol. 26 (1988), 319–346.CrossRefGoogle Scholar
Liang, X., Zhang, L. and Zhao, X.-Q.. The principal eigenvalue for periodic nonlocal dispersal systems with time delay. J. Differ. Equ. 266 (2019), 2100–2124.CrossRefGoogle Scholar
Liu, S., Lou, Y., Peng, R. and Zhou, M.. Monotonicity of the principal eigenvalue for a linear time-periodic parabolic operator. Proc. Am. Math. Soc. 147 (2019), 5291–5302.CrossRefGoogle Scholar
Marek, I.. Frobenius theory of positive operators: comparison theorems and applications. SIAM J. Appl. Math. 19 (1970), 607–628.CrossRefGoogle Scholar
Medlock, J. and Kot, M.. Spreading disease: integro-differential equations old and new. Math. Biosci. 184 (2003), 201–222.CrossRefGoogle ScholarPubMed
Murray, J. D.. Mathematical biology: I. An introduction, vol. 17 (Springer-Verlag New York, 2007).Google Scholar
Peng, R. and Zhao, X.-Q.. Effects of diffusion and advection on the principal eigenvalue of a periodic-parabolic problem with applications. Calc. Var. Partial Differ. Equ. 54 (2015), 1611–1642.CrossRefGoogle Scholar
Rabinowitz, P. H.. Some global results for nonlinear eigenvalue problems. J. Funct. Anal. 7 (1971), 487–513.CrossRefGoogle Scholar
Rheinboldt, W. C.. Local mapping relations and global implicit function theorems. Trans. Am. Math. Soc. 138 (1969), 183–198.CrossRefGoogle Scholar
Sandberg, I.. Global implicit function theorems. IEEE Trans. Circuits Syst. 28 (1981), 145–149.CrossRefGoogle Scholar
Sawashima, I.. On spectral properties of some positive operators. Nat. Sci. Rep., Ochanomizu Univ. 15 (1964), 53–64.Google Scholar
Shen, Z. and Vo, H.-H.. Nonlocal dispersal equations in time-periodic media: principal spectral theory, limiting properties and long-time dynamics. J. Differ. Equ. 267 (2019), 1423–1466.CrossRefGoogle Scholar
Thieme, H. R.. Remarks on resolvent positive operators and their perturbation. Discrete Contin. Dyn. Syst. 4 (1998), 73–90.CrossRefGoogle Scholar
Thieme, H. R.. Spectral bound and reproduction number for infinite-dimensional population structure and time heterogeneity. SIAM J. Appl. Math. 70 (2009), 188–211.CrossRefGoogle Scholar
Walker, C.. Positive equilibrium solutions for age-and spatially-structured population models. SIAM J. Math. Anal. 41 (2009), 1366–1387.CrossRefGoogle Scholar
Walker, C.. Age-dependent equations with non-linear diffusion. Discrete Contin. Dyn. Syst. 26 (2010), 691–712.CrossRefGoogle Scholar
Walker, C.. Global bifurcation of positive equilibria in nonlinear population models. J. Differ. Equ. 248 (2010), 1756–1776.CrossRefGoogle Scholar
Walker, C.. Bifurcation of positive equilibria in nonlinear structured population models with varying mortality rates. Ann. Mat. Pura Appl. 190 (2011), 1–19.CrossRefGoogle Scholar
Walker, C.. On nonlocal parabolic steady-state equations of cooperative or competing systems. Nonlinear Anal. Real World Appl. 12 (2011), 3552–3571.CrossRefGoogle Scholar
Walker, C.. On positive solutions of some system of reaction-diffusion equations with nonlocal initial conditions. J. Reine Angew. Math. 2011 (2011), 149–179.CrossRefGoogle Scholar
Walker, C.. A note on a nonlocal nonlinear reaction–diffusion model. Appl. Math. Lett. 25 (2012), 1772–1777.CrossRefGoogle Scholar
Walker, C.. Global continua of positive solutions for some quasilinear parabolic equation with a nonlocal initial condition. J. Dyn. Differ. Equ. 25 (2013), 159–172.CrossRefGoogle Scholar
Walker, C.. Some remarks on the asymptotic behavior of the semigroup associated with age-structured diffusive populations. Monatsh. Math. 170 (2013), 481–501.CrossRefGoogle Scholar
Webb, G. F.. An age-dependent epidemic model with spatial diffusion. Arch. Ration. Mech. Anal. 75 (1980), 91–102.CrossRefGoogle Scholar
Webb, G. F.. Diffusive age-dependent population models and an application to genetics. Math. Biosci. 61 (1982), 1–16.CrossRefGoogle Scholar
Webb, G. F.. Theory of nonlinear age-dependent population dynamics (New York: Marcel Dekker, 1984).Google Scholar