Skip to main content
SpringerLink
Log in

Approximation of random diffusion by nonlocal diffusion in age-structured models

  • Published:
Zeitschrift für angewandte Mathematik und Physik Aims and scope Submit manuscript

Abstract

In this paper we study the approximation of random diffusion by nonlocal diffusion with properly rescaled kernels in age-structured models. First we show that solutions of age-structured models with nonlocal diffusion under Dirichlet and Neumann boundary conditions converge to solutions of the corresponding age-structured models with random diffusion under Dirichlet and Neumann boundary conditions, respectively. Then we prove that the principal eigenvalues of the nonlocal operators in age-structured models under Dirichlet and Neumann boundary conditions converge to the principal eigenvalues of the corresponding Laplace operators in age-structured models under Dirichlet and Neumann boundary conditions, respectively.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

References

  1. Andreu, F., Mazón, J.M., Rossi, J.D., Toledo, J.: A nonlocal p-Laplacian evolution equation with Neumann boundary conditions. J. Math. Pures Appl. 90(2), 201–227 (2008)

    Article  MathSciNet  Google Scholar 

  2. Andreu, F., Mazón, J.M., Rossi, J.D., Toledo, J.: The limit as \(p \rightarrow \infty \) in a nonlocal p-Laplacian evolution equation: a nonlocal approximation of a model for sandpiles. Calc. Var. Partial Differ. Equ. 35(3), 279–316 (2009)

    Article  MathSciNet  Google Scholar 

  3. Andreu, F., Mazón, J.M., Rossi, J.D., Toledo, J.: A nonlocal p-Laplacian evolution equation with nonhomogeneous Dirichlet boundary conditions. SIAM J. Math. Anal. 40(5), 1815–1851 (2009)

    Article  MathSciNet  Google Scholar 

  4. Andreu, F., Mazón, J.M., Rossi, J.D., Toledo, J.: Local and nonlocal weighted p-Laplacian evolution equations. Publ. Mat. 55(1), 27–66 (2011)

    Article  MathSciNet  Google Scholar 

  5. Andreu-Vaillo, F., Mazón, J.M., Rossi, J.D., Toledo-Melero, J.J.: Nonlocal Diffusion Problems. American Mathematical Society, Providence, RI (2010)

    Book  Google Scholar 

  6. Cao, J.-F., Du, Y., Li, F., Li, W.-T.: The dynamics of a Fisher-KPP nonlocal diffusion model with free boundaries. J. Funct. Anal. 277(8), 2772–2814 (2019)

    Article  MathSciNet  Google Scholar 

  7. Chasseigne, E., Chaves, M., Rossi, J.D.: Asymptotic behavior for nonlocal diffusion equations. J. Math. Pures Appl. 86(3), 271–291 (2006)

    Article  MathSciNet  Google Scholar 

  8. Cortazar, C., Elgueta, M., Rossi, J.D.: Nonlocal diffusion problems that approximate the heat equation with Dirichlet boundary conditions. Isr. J. Math. 170(1), 53–60 (2009)

    Article  MathSciNet  Google Scholar 

  9. Cortazar, C., Elgueta, M., Rossi, J.D., Wolanski, N.: How to approximate the heat equation with Neumann boundary conditions by nonlocal diffusion problems. Arch. Ration. Mech. Anal. 187(1), 137–156 (2008)

    Article  MathSciNet  Google Scholar 

  10. Delgado, M., Molina-Becerra, M., Suárez, A.: Nonlinear age-dependent diffusive equations: a bifurcation approach. J. Differ. Equ. 244(9), 2133–2155 (2008)

    Article  MathSciNet  Google Scholar 

  11. Du, Y., Ni, W.: Approximation of random diffusion equation by nonlocal diffusion equation in free boundary problems of one space dimension. arXiv preprint, arXiv:2003.05560 (2020)

  12. Ducrot, A.: Travelling wave solutions for a scalar age-structured equation. Discrete Contin. Dyn. Syst. Ser. B 7(2), 251–273 (2007)

    MathSciNet  MATH  Google Scholar 

  13. Ducrot, A., Magal, P.: Travelling wave solutions for an infection-age structured model with diffusion. Proc. R. Soc. Edinb. Sect. A 139(3), 459–482 (2009)

    Article  MathSciNet  Google Scholar 

  14. Ducrot, A., Magal, P.: Travelling wave solutions for an infection-age structured epidemic model with external supplies. Nonlinearity 24(10), 2891 (2011)

    Article  MathSciNet  Google Scholar 

  15. Ducrot, A., Magal, P., Ruan, S.: Travelling wave solutions in multigroup age-structured epidemic models. Arch. Ration. Mech. Anal. 195(1), 311–331 (2010)

    Article  MathSciNet  Google Scholar 

  16. Gripenberg, G., Londen, S.-O., Staffans, O.: Volterra Integral and Functional Equations, vol. 34. Cambridge University Press, Cambridge (1990)

    Book  Google Scholar 

  17. 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)

    Article  MathSciNet  Google Scholar 

  18. Iannelli, M.: Mathematical Theory of Age-Structured Population Dynamics. Giardini editori e stampatori, Pisa (1995)

    Google Scholar 

  19. Ignat, L.I., Rossi, J.D.: A nonlocal convection–diffusion equation. J. Funct. Anal. 251(2), 399–437 (2007)

    Article  MathSciNet  Google Scholar 

  20. Inaba, H.: Age-Structured Population Dynamics in Demography and Epidemiology. Springer, New York (2017)

    Book  Google Scholar 

  21. Kang, H., Ruan, S.: Age-structured models with nonlocal diffusion: principal spectral theory, limiting properties and global dynamics. Submitted

  22. Kang, H., Ruan, S.: Nonlinear age-structured population models with nonlocal diffusion and nonlocal boundary conditions. J. Differ. Equ. 278, 430–462 (2021)

    Article  MathSciNet  Google Scholar 

  23. Kang, H., Ruan, S., Yu, X.: Age-structured population dynamics with nonlocal diffusion. J. Dyn. Differ. Equ. (2020). https://doi.org/10.1007/s10884-020-09860-5

    Article  Google Scholar 

  24. Magal, P., Ruan, S.: Theory and Applications of Abstract Semilinear Cauchy Problems. Springer, New York (2018)

    Book  Google Scholar 

  25. Shen, W., Xie, X.: Approximations of random dispersal operators/equations by nonlocal dispersal operators/equations. J. Differ. Equ. 259(12), 7375–7405 (2015)

    Article  MathSciNet  Google Scholar 

  26. Sun, J.-W., Li, W.-T., Yang, F.-Y.: Approximate the Fokker–Planck equation by a class of nonlocal dispersal problems. Nonlinear Anal. Theory Methods Appl. 74(11), 3501–3509 (2011)

    Article  MathSciNet  Google Scholar 

  27. Webb, G.F.: Theory of Nonlinear Age-Dependent Population Dynamics. Marcel Dekker, New York (1984)

    Google Scholar 

Download references

Acknowledgements

We would like to thank the three anonymous reviewers for their helpful comments and valuable suggestions.

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Miami, Coral Gables, FL, 33146, USA

    Hao Kang & Shigui Ruan

Authors
  1. Hao Kang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Shigui Ruan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Shigui Ruan.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Research was partially supported by National Science Foundation (DMS-1853622)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kang, H., Ruan, S. Approximation of random diffusion by nonlocal diffusion in age-structured models. Z. Angew. Math. Phys. 72, 108 (2021). https://doi.org/10.1007/s00033-021-01538-2

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00033-021-01538-2

Keywords

Mathematics Subject Classification

Search

Navigation