Skip to main content
SpringerLink
Log in

On the modelling of spatially heterogeneous nonlocal diffusion: deciding factors and preferential position of individuals

  • Published:
Journal of Mathematical Biology Aims and scope Submit manuscript

Abstract

We develop general heterogeneous nonlocal diffusion models and investigate their connection to local diffusion models by taking a singular limit of focusing kernels. We reveal the link between the two groups of diffusion equations which include both spatial heterogeneity and anisotropy. In particular, we introduce the notion of deciding factors which single out a nonlocal diffusion model and typically consist of the total jump rate and the average jump length. In this framework, we also discuss the dependence of the profile of the steady state solutions on these deciding factors, thus shedding light on the preferential position of individuals.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8

Similar content being viewed by others

References

  • Alfaro M, Carles R (2014) Explicit solutions for replicator–mutator equations: extinction versus acceleration. SIAM J Appl Math 74:1919–1934

    Article  MathSciNet  Google Scholar 

  • Alfaro M, Coville J (2017) Propagation phenomena in monostable integro-differential equations: Acceleration or not? J Differ Equ 263:5727–5758

    Article  MathSciNet  Google Scholar 

  • Alfaro M, Gabriel P, Kavian O (2021) Confining integro-differential equations originating from evolutionary biology: ground states and long time dynamics (submitted)

  • Andreu-Vaillo F, Mazón JM, Rossi JD, Toledo-Melero JJ (2010) Nonlocal diffusion problems, vol 165. Mathematical Surveys and Monographs, American Mathematical Society, Providence. RI; Real Sociedad Matemática Española, Madrid

  • Applebaum D (2009) Lévy Processes and Stochastic Calculus, 2nd edn. Cambridge Studies in Advanced Mathematics, Cambridge University Press

    Book  Google Scholar 

  • Averill I, Lam K-Y, Lou Y (2017) The role of advection in a two-species competition model: a bifurcation approach, Mem. Amer. Math. Soc., 245, pp v+117

  • Bartlett MS, Porporato A (2018) State-dependent jump processes: Itô-stratonovich interpretations, potential, and transient solutions. Phys Rev E 98:052132

    Article  MathSciNet  Google Scholar 

  • Bürger R (1986) On the maintenance of genetic variation: global analysis of Kimura’s continuum-of-alleles model. J Math Biol 24:341–351

    Article  MathSciNet  Google Scholar 

  • Bürger R (1988) Perturbations of positive semigroups and applications to population genetics. Math Z 197:259–272

    Article  MathSciNet  Google Scholar 

  • Bürger R (2000) The mathematical theory of selection, recombination, and mutation, Wiley Series in Mathematical and Computational Biology. John Wiley & Sons Ltd, Chichester

    Google Scholar 

  • Cantrell RS, Cosner C, Lou Y (2006) Movement toward better environments and the evolution of rapid diffusion. Math Biosci 204:199–214

    Article  MathSciNet  Google Scholar 

  • Cantrell RS, Cosner C, Lou Y (2009) Evolution of dispersal in heterogeneous landscapes. Spatial ecology. Chapman Hall/CRC Press, Boca Raton

    MATH  Google Scholar 

  • Cantrell RS, Cosner C, Lou Y (2010) Evolution of dispersal and the ideal free distribution. Math Biosci Eng 7:17–36

    Article  MathSciNet  Google Scholar 

  • Cantrell RS, Cosner C, Lou Y, Ryan D (2012) Evolutionary stability of ideal free dispersal strategies: a nonlocal dispersal model. Can Appl Math Q 20:15–38

    MathSciNet  MATH  Google Scholar 

  • Chapman S (1928) On the brownian displacements and thermal diffusion of grains suspended in a non-uniform fluid, Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character, 119, 34–54

  • Chasseigne E, Chaves M, Rossi JD (2006) Asymptotic behavior for nonlocal diffusion equations. J Math Pures Appl (9) 86:271–291

    Article  MathSciNet  Google Scholar 

  • Choi S-H, Kim Y-J (2015) Chemotactic traveling waves by the metric of food. SIAM J Appl Math 75:2268–2289

    Article  MathSciNet  Google Scholar 

  • Clark JS (1998) Why trees migrate so fast: confronting theory with dispersal biology and the paleorecord. Am Natl 152:204–224

    Article  Google Scholar 

  • Cortázar C, Coville J, Elgueta M, Martínez S (2007) A nonlocal inhomogeneous dispersal process. J Differ Equ 241:332–358

    Article  MathSciNet  Google Scholar 

  • Cortázar C, Elgueta M, García-Melián J, Martínez S (2009) Existence and asymptotic behavior of solutions to some inhomogeneous nonlocal diffusion problems. SIAM J Math Anal 41:2136–2164

    Article  MathSciNet  Google Scholar 

  • Cortázar C, Elgueta M, García-Melián J, Martínez S (2015) Finite mass solutions for a nonlocal inhomogeneous dispersal equation. Discrete Contin Dyn Syst 35:1409–1419

    Article  MathSciNet  Google Scholar 

  • Cortázar C, Elgueta M, García-Melián J, Martínez S (2016) An inhomogeneous nonlocal diffusion problem with unbounded steps. J Evol Equ 16:209–232

    Article  MathSciNet  Google Scholar 

  • Cosner C, Dávila J, Martínez S (2012) Evolutionary stability of ideal free nonlocal dispersal. J Biol Dyn 6:395–405

    Article  MathSciNet  Google Scholar 

  • Coville J, Dupaigne L (2007) On a non-local equation arising in population dynamics. Proc R Soc Edinb Sect A 137:727–755

    Article  MathSciNet  Google Scholar 

  • dos Santos BC, Oliva SM, Rossi JD (2021) A local/nonlocal diffusion model. Appl Anal 1–34

  • Fick AV (1855) On liquid diffusion. Lond Edinb Dublin Philos Mag J Sci 10:30–39

    Article  Google Scholar 

  • Fisher RA (1937) The wave of advance of advantageous genes. Ann Eugen 7:355–369

    Article  Google Scholar 

  • Fleming WH (1979) Equilibrium distributions of continuous polygenic traits. SIAM J Appl Math 36:148–168

    Article  MathSciNet  Google Scholar 

  • Garnier J (2011) Accelerating solutions in integro-differential equations. SIAM J Math Anal 43:1955–1974

    Article  MathSciNet  Google Scholar 

  • Gil M-E, Hamel F, Martin G, Roques L (2017) Mathematical properties of a class of integro-differential models from population genetics. SIAM J Appl Math 77:1536–1561

    Article  MathSciNet  Google Scholar 

  • Gil M-E, Hamel F, Martin G, Roques L (2019) Dynamics of fitness distributions in the presence of a phenotypic optimum: an integro-differential approach. Nonlinearity 32:3485–3522

    Article  MathSciNet  Google Scholar 

  • Hutson V, Martinez S, Mischaikow K, Vickers GT (2003) The evolution of dispersal. J Math Biol 47:483–517

    Article  MathSciNet  Google Scholar 

  • Kim Y-J, Seo H (2021) Model for heterogeneous diffusion. SIAM J Appl Math 81:335–354

    Article  MathSciNet  Google Scholar 

  • Kimura M (1965) A stochastic model concerning the maintenance of genetic variability in quantitative characters. Proc Natl Acad Sci USA 54:731–736

    Article  Google Scholar 

  • Kolmogorov AN, Petrovsky IG, Piskunov NS (1937) Etude de l’équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique, Bull. Univ. Etat Moscou, Sér. Inter. A 1, pp 1–26

  • Lande R (1975) The maintenance of genetic variability by mutation in a polygenic character with linked loci. Genet Res 26:221–235

    Article  Google Scholar 

  • Medlock J, Kot M (2003) Spreading disease: integro-differential equations old and new. Math Biosci 184:201–222

    Article  MathSciNet  Google Scholar 

  • Molino A, Rossi JD (2016) Nonlocal diffusion problems that approximate a parabolic equation with spatial dependence. Z Angew Math Phys 67, Art. 41, 14

  • Molino A, Rossi JD (2019) Nonlocal approximations to Fokker–Planck equations. Funkcial Ekvac 62:35–60

    Article  MathSciNet  Google Scholar 

  • Murray J (2001) Mathematical biology II: spatial models and biomedical applications. Springer, New York

    Google Scholar 

  • Novak S (2014) Habitat heterogeneities versus spatial type frequency variances as driving forces of dispersal evolution. Ecol Evol 4:4589–4597

    Article  Google Scholar 

  • Potts JR, Schlägel UE (2020) Parametrizing diffusion-taxis equations from animal movement trajectories using step selection analysis. Methods Ecol Evol 11:1092–1105

    Article  Google Scholar 

  • Shigesada N, Kawasaki K (1997) Biological invasions: theory and practice. Oxford University Press, UK

    Google Scholar 

  • Skellam JG (1951) Random dispersal in theoretical populations. Biometrika 38:196–218

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  • Turchin P (1998) Quantitative analysis of movement: measuring and modeling population redistribution in animals and plants. Sinauer Associates

  • Wereide MT (1914) La diffusion d’une solution dont la concentration et la température sont variables. In Annales de Physique, vol. 9, EDP Sciences, 67–83

Download references

Acknowledgements

The authors are grateful to the anonymous referees whose precise comments have improved the presentation of the results. Matthieu Alfaro and Gwenaël Peltier are supported by the ANR project DEEV ANR-20-CE40-0011-01. Thomas Giletti is supported by the ANR Project JCJC ANR-21-CE40-0008. Hyowon Seo is supported by NRF of Korea (no. 2020R1I1A1A01069585). The authors also acknowledge support from CNRS International Research Network ReaDiNet.

Author information

Authors and Affiliations

  1. Laboratoire de Mathématiques Raphaël Salem, CNRS, Université de Rouen Normandie, Saint-Etienne-du-Rouvray, France

    Matthieu Alfaro

  2. BioSP, INRAE, 84914, Avignon, France

    Matthieu Alfaro

  3. IECL, Université de Lorraine, B.P. 70239, 54506, Vandoeuvre-lès-Nancy Cedex, France

    Thomas Giletti

  4. Department of Mathematical Sciences, KAIST, 291 Daehak-ro, Yuseong-gu, Daejeon, 34141, Korea

    Yong-Jung Kim

  5. Laboratoire J.-L. Lions, CNRS, Inria, Université de Paris-Cité, Sorbonne Université, 75005, Paris, France

    Gwenaël Peltier

  6. Department of mathematics, Kunsan National University, 558 Daehak-ro, Gunsan-si, Jeollabuk-do, 54150, South Korea

    Hyowon Seo

Authors
  1. Matthieu Alfaro
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Thomas Giletti
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yong-Jung Kim
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Gwenaël Peltier
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Hyowon Seo
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Matthieu Alfaro.

Additional information

Publisher's Note

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

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Alfaro, M., Giletti, T., Kim, YJ. et al. On the modelling of spatially heterogeneous nonlocal diffusion: deciding factors and preferential position of individuals. J. Math. Biol. 84, 38 (2022). https://doi.org/10.1007/s00285-022-01738-y

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00285-022-01738-y

Keywords

Mathematics Subject Classification

Search

Navigation