Hostname: page-component-586b7cd67f-rdxmf Total loading time: 0 Render date: 2024-11-22T13:24:48.294Z Has data issue: false hasContentIssue false
  • English
  • Français

Heterogeneity and strong competition in ecology

Published online by Cambridge University Press:  25 June 2018

H. HUTRIDURGA  and
C. VENKATARAMAN
H. HUTRIDURGA
Affiliation:
Department of Mathematics, Imperial College London, London, SW7 2AZ, UK email: [email protected]
C. VENKATARAMAN
Affiliation:
Department of Mathematics, School of Mathematical and Physical Sciences, University of Sussex, Brighton, BN1 9QH, UK email: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We study a competition-diffusion model while performing simultaneous homogenization and strong competition limits. The limit problem is shown to be a Stefan-type evolution equation with effective coefficients. We also perform some numerical simulations in one and two spatial dimensions that suggest that oscillations in motilities are detrimental to invasion behaviour of a species.

Keywords

HomogenizationEquations in media with periodic structuresAsymptotic analysisStefan problemsFinite element methodsEcology and invasion
Type
Papers
Information
European Journal of Applied Mathematics , Volume 30 , Issue 4 , August 2019 , pp. 682 - 706
Copyright
Copyright © Cambridge University Press 2018 

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.)

Footnotes

†The research of the first author was supported by the EPSRC programme grant ‘Mathematical fundamentals of Metamaterials for multiscale Physics and Mechanic’ (EP/L024926/1). The second author would like to thank the Isaac Newton Institute for Mathematical Sciences for support and hospitality during the program ‘Coupling geometric PDEs with physics for cell morphology, motility and pattern formation’ where this project was initiated.

References

[1] Allaire, G. (1992) Homogenization and two-scale convergence. SIAM J. Math. Anal. 23 (6), 1482–1518.Google Scholar
[2] Allaire, G. (2002) Shape Optimization by the Homogenization Method, Applied Mathematical Sciences, Vol. 146, Springer, New York.Google Scholar
[3] Bardos, C. & Hutridurga, H. (2016) Simultaneous diffusion and homogenization asymptotic for the linear Boltzmann equation. Asymptotic Anal. 100 (1–2), 111–130.Google Scholar
[4] Brezis, H. (2010) Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer, New York.Google Scholar
[5] Cioranescu, D. & Donato, P. (1999) An Introduction to Homogenization, Oxford Lecture Series in Mathematics and Its Applications. Oxford University Press, Oxford.Google Scholar
[6] Crank, J. (1975) Free and Moving Boundary Problems, 2nd ed, Oxford University Press, New York.Google Scholar
[7] Dancer, E. N., Hilhorst, D., Mimura, M. & Peletier, L. A. (1999) Spatial segregation limit of a competition–diffusion system. Eur. J. Appl. Math. 10 (2), 97–115.Google Scholar
[8] Girardin, L. & Nadin, G. (2015) Travelling waves for diffusive and strongly competitive systems: Relative motility and invasion speed. Eur. J. Appl. Math. 26 (04), 521–534.Google Scholar
[9] Hilhorst, D., Iida, M., Mimura, M. & Ninomiya, H. (2001) A competition-diffusion system approximation to the classical two-phase Stefan problem. Japan J. Indust. Appl. Math. 18 (2), 161–180.Google Scholar
[10] Hilhorst, D., Mimura, M. & Schätzle, R. (2003) Vanishing latent heat limit in a Stefan-like problem arising in biology. Nonlinear Anal.: Real World Appl. 4 (2), 261–285.Google Scholar
[11] Javierre, E., Vuik, C., Vermolen, F. J. & Van der Zwaag, S. (2006) A comparison of numerical models for one-dimensional Stefan problems. J. Comput. Appl. Math. 192 (2), 445–459.Google Scholar
[12] Kim, I. C. & Mellet, A. (2008) Homogenization of a Hele–Shaw problem in periodic and random media. Arch. Rat. Mech. Anal. 194 (2), 507–530.Google Scholar
[13] Kim, I. C. & Mellet, A. (2010) Homogenization of one-phase Stefan-type problems in periodic and random media. Trans. Amer. Math. Soc. 362 (08), 4161–4190.Google Scholar
[14] Lakkis, O., Madzvamuse, A. & Venkataraman, C. (2013) Implicit–explicit timestepping with finite element approximation of reaction–diffusion systems on evolving domains. SIAM J. Numer. Anal. 51 (4), 2309–2330.Google Scholar
[15] Lukkassen, D., Nguetseng, G. & Wall, P. (2002) Two-scale convergence. Int. J. Pure Appl. Math. 2 (1), 33–81.Google Scholar
[16] Nguetseng, G. (1989) A general convergence result for a functional related to the theory of homogenization. SIAM J. Math. Anal. 20 (3), 608–623.Google Scholar
[17] Pierre, M. (2010) Global existence in reaction-diffusion systems with control of mass: A survey. Milan J. Math. 78 (2), 417–455.Google Scholar
[18] Pavliotis, G. A. & Stuart, A. M. (2008) Multiscale Methods: Averaging and Homogenization, Texts in Applied Mathematics, Vol. 53, Springer, New York.Google Scholar
[19] Rodrigues, J.-F. (1982) Free boundary convergence in the homogenization of the one phase Stefan problem. Trans. Amer. Math. Soc. 274 (1), 297.Google Scholar
[20] Simon, J. (1986) Compact sets in the space Lp(0,T;B). Ann. Mat. Pur. Appl. 146 (1), 65–96.Google Scholar
[21] Schmidt, A. & Siebert, K. G. (2005) Design of Adaptive Finite Element Software: The Finite Element Toolbox ALBERTA, Springer-Verlag, Berlin.Google Scholar
[22] Visintin, A. (Jan. 2007) Homogenization of a doubly nonlinear Stefan-type problem. SIAM J. Math. Anal. 39 (3), 987–1017.Google Scholar