Skip to main content
SpringerLink
Log in

Criteria for Pointwise Growth and Their Role in Invasion Processes

  • Published:
Journal of Nonlinear Science Aims and scope Submit manuscript

Abstract

This article is concerned with pointwise growth and spreading speeds in systems of parabolic partial differential equations. Several criteria exist for quantifying pointwise growth rates. These include the location in the complex plane of singularities of the pointwise Green’s function and pinched double roots of the dispersion relation. The primary aim of this work is to establish some rigorous properties related to these criteria and the relationships between them. In the process, we discover that these concepts are not equivalent and point to some interesting consequences for nonlinear front invasion problems. Among the more striking is the fact that pointwise growth does not depend continuously on system parameters. Other results include a determination of the circumstances under which pointwise growth on the real line implies pointwise growth on a semi-infinite interval. As a final application, we consider invasion fronts in an infinite cylinder and show that the linear prediction always favors the formation of stripes in the leading edge.

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

Similar content being viewed by others

Notes

  1. That is, with a sufficiently strong exponential rate; see Sandstede and Scheel (2004) for cases where convergence is too weak.

  2. The situation is analogous to supercritical and subcritical bifurcation scenarios: the linearization often gives good predictions in supercritical bifurcations, but one does not expect accurate linear predictions in subcritical bifurcations.

References

  • Abraham, R., Robbin, J.: Transversal Mappings and Flows. W. A. Benjamin, New York (1967)

    MATH  Google Scholar 

  • Alexander, J., Gardner, R., Jones, R.C.: A topological invariant arising in the stability analysis of travelling waves. J. Reine Angew. Math. 410, 167–212 (1990)

    Google Scholar 

  • Aronson, D., Weinberger, H.: Multidimensional nonlinear diffusion arising in population genetics. Adv. Math. 30, 33–76 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  • Archer, A., Robbins, M., Thiele, U., Knobloch, E.: Solidification fronts in supercooled liquids: How rapid fronts can lead to disordered glassy solids. Phys. Rev. E 86, 031603 (2012)

    Article  Google Scholar 

  • Bers, A.N.: Space-time evolution of plasma instabilities-absolute and convective. In: Rosenbluth, M.N., Sagdeev, R.Z. (eds.) Handbook of Plasma Physics. North-Holland, Amsterdam (1983)

    Google Scholar 

  • Bose, K., Cox, T., Silvestri, S., Varin, P.: Invasion fronts and pattern formation in a model of chemotaxis in one and two dimensions. SIAM Undergrad. Res. Online. 6, 228–245 (2013)

    Article  Google Scholar 

  • Brevdo, L.: A dynamical system approach to the absolute instability of spatially developing localized open flows and media. R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 458, 1375–1397 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  • Brevdo, L., Bridges, T.J.: Absolute and convective instabilities of spatially periodic flows. Phil. Trans. R. Soc. London A 354, 1027–1064 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  • Briggs, R.J.: Electron-stream interaction with plasmas. MIT Press, Cambridge (1964)

    Google Scholar 

  • Collet, P., Eckmann, J.-P.: Instabilities and Fronts in Extended Systems. Princeton University Press, Princeton, (1990)

  • Csahók, Z., Misbah, C.: On the invasion of an unstable structureless state by a stable hexagonal pattern. Europhy. Lett. 47, 331–337 (1999)

    Article  Google Scholar 

  • Dee, G., Langer, J.S.: Propagating pattern selection. Phys. Rev. Lett. 50, 383–386 (1983)

    Article  Google Scholar 

  • Fiedler, B., Scheel, A.: Spatio-Temporal Dynamics of Reaction-Diffusion Patterns. In: Kirkilionis, M., Krmker, S., Rannacher, R., Tomi, F. (Eds.) Trends in Nonlinear Analysis, Springer-Verlag, Berlin, (2003) (145 pages)

  • Foard, E., Wagner, A.J.: Survey of morphologies in the wake of an enslaved phase-separation front in two dimensions. Phys. Rev. E 85, 011501 (2012)

    Article  Google Scholar 

  • Freidlin, M.: Coupled reaction-diffusion equations. Ann. Probab. 19, 2957 (1991)

    Article  MathSciNet  Google Scholar 

  • Friedrich, R., Radons, G., Ditzinger, T., Henning, A.: Ripple Formation through an Interface Instability from Moving Growth and Erosion Sources. Phys. Rev. Lett. 85, 4884 (2000)

    Article  Google Scholar 

  • Fulton, W.: Algebraic Curves. Benjamin, New York (1969)

    MATH  Google Scholar 

  • Goh, R., Mesuro, S., Scheel, A.: Coherent structures in reaction-diffusion models for precipitation, In: Precipitation Patterns in Reaction-Diffusion Systems, pp. 73–93. Research Signpost, Trivandrum, (2010)

  • Goh, R., Scheel, A., Triggered fronts in the complex Ginzburg-Landau equation. J. Nonl. Sci. 24(1), 117–144 (in press).

  • Haragus, M., Schneider, G.: Bifurcating fronts for the Taylor-Couette problem in infinite cylinders. Z. Angew. Math. Phys. 50, 120–151 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  • Hari, A., Nepomnyashchy, A.A.: Nonpotential effects in dynamics of fronts between convection patterns. Phys. Rev. E 61, 4835–4847 (2000)

    Article  MathSciNet  Google Scholar 

  • Holmes, E., Lewis, M., Banks, J., Veit, R.: Partial differential equations in ecology: spatial interactions and population dynamics. Ecology 75, 17–29 (1994)

    Article  Google Scholar 

  • Holzer, M.: Anomalous spreading in a system of coupled Fisher-KPP equations. Physica D. 270, 1–10 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  • Holzer, M., Scheel, A.: A slow pushed front in a Lotka-Volterra competition model. Nonlinearity 25, 2151–2179 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  • Holzer, M., Scheel, A.: Accelerated fronts in a two stage invasion process. SIAM J. Math. Anal. 46, 397–427 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  • Kapitula, T., Promislow, K.: Spectral and Dynamical Stability of Nonlinear Waves. Springer, New York, (2013)

  • Kapitula, T., Sandstede, B.: Edge bifurcations for near integrable systems via Evans function techniques. SIAM J. Math. Anal. 33, 1117–1143 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  • Kotzagiannidis, M., Peterson, J., Redford, J., Scheel, A., Wu, Q.: Stable pattern selection through invasion fronts in closed two-species reaction-diffusion systems. In: Ogawa, T., Ueda, K. (eds.) Far-From-Equilibrium Dynamics, pp 79–93. RIMS Kokyuroku Bessatsu B31, Kyoto, (2012)

  • Krantz, S.G.: Function Theory of Several Complex Variables. AMS Chelsea Publishing, Providence (1992)

    MATH  Google Scholar 

  • Krekhov, A.: Formation of regular structures in the process of phase separation. Phys. Rev. E 79(3):035302R (2009)

    Google Scholar 

  • Liang, X., Lin, X., Matano, H.: A variational problem associated with the minimal speed of travelling waves for spatially periodic reaction-diffusion equations. Trans. Amer. Math. Soc. 362, 5605–5633 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  • Lunardi, A.: Analytic semigroups and optimal regularity in parabolic problems. In: Progress in Nonlinear Differential Equations and their Applications, pp. 16. Birkhuser Verlag, Basel, (1995)

  • Matano, H.: Traveling waves in spatially inhomogeneous diffusive media: the non-periodic case (2009, preprint).

  • Pelinovsky, D., Scheel, A.: Stability analysis of stationary light transmission in nonlinear photonic structures. J. Nonl. Sci. 13, 347–396 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  • Pismen, L.M., Nepomnyashchy, A.A.: Propagation of the hexagonal pattern. Europhys. Lett. 27, 433436 (1994)

    Google Scholar 

  • Rademacher, J.D.M., Sandstede, B., Scheel, A.: Computing absolute and essential spectra using continuation. Physica D 229, 166–183 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  • Sandstede, B.: Stability of Travelling Waves. In: Handbook of dynamical systems, Vol. 2, pp. 983–1055, North-Holland, Amsterdam, (2002)

  • Sandstede, B., Scheel, A.: Absolute and convective instabilities of waves on unbounded and large bounded domains. Physica D 145, 233–277 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  • Sandstede, B., Scheel, A.: Evans function and blow-up methods in critical eigenvalue problems. Discr. Cont. Dyn. Sys. 10, 941–964 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  • Sandstede, B., Scheel, A.: Relative morse indices, fredholm indices, and group velocities. Discr. Cont. Dyn. Sys. 20, 139–158 (2008)

    MATH  MathSciNet  Google Scholar 

  • Scheel, A.: Radially symmetric patterns of reaction-diffusion systems. Mem. Amer. Math. Soc. 165(786):86 (2003)

    Google Scholar 

  • Scheel, A.: Spinodal decomposition and coarsening fronts in the Cahn-Hilliard equation (2012, preprint).

  • Shen, W.: Dynamical systems and traveling waves in almost periodic structures. J. Differ. Equ. 169, 493–548 (2001)

    Article  MATH  Google Scholar 

  • Shen, W.: Traveling waves in diffusive random media. J. Dyn. Diff. Eq. 16, 1011–1060 (2004)

    Article  MATH  Google Scholar 

  • Shubin, M.A.: On holomorphic families of subspaces of a banach space. Integr. Equ. Operat. Theory 2, 407–420 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  • van Saarloos, W.: Front propagation into unstable states. Phys. Rep. 386, 29–222 (2003)

    Google Scholar 

  • van Saarloos, W., Hohenberg, P.: Fronts, pulses, sources and sinks in generalized complex ginzburg-landau equations. Physica D: Nonlinear Phenom. 56, 303–367 (1992)

    Article  MATH  Google Scholar 

  • Suslov, S.: Numerical aspects of searching convective/absolute instability transition. J. Comp. Phys. 212, 188–217 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  • Weinberger, H.: Long-time behavior of a class of biological models. SIAM J. Math. Anal. 13, 353–396 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  • Weinberger, H., Lewis, M., Li, B.: Anomalous spreading speeds of cooperative recursion systems. J. Math. Biol. 55, 207–222 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  • Zumbrun, K., Howard, P.: Pointwise semigroup methods and stability of viscous shock waves. Indiana Univ. Math. J. 47, 741–871 (1998)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Acknowledgments

The authors acknowledge partial support through the National Science Foundation (NSF) [DMS-1004517 (M.H.), DMS-0806614 (A.S.), and DMS-1311740 (A.S.)]. This research was initiated during an NSF-sponsored REU program in the summer of 2012 (Bose et al. 2013). We are grateful to Koushiki Bose, Tyler Cox, Stefano Silvestri, and Patrick Varin for working out some of the examples in this article. We also thank Ryan Goh for many stimulating discussions related to the material in Sects. 4 and 5.

Author information

Author notes

  1. Matt Holzer

    Present address: Department of Mathematics, George Mason University, Fairfax, VA, 22030, USA

Authors and Affiliations

  1. School of Mathematics, University of Minnesota, 127 Vincent Hall, 206 Church St SE, Minneapolis, MN, 55455, USA

    Matt Holzer & Arnd Scheel

Authors
  1. Matt Holzer
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Arnd Scheel
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Matt Holzer.

Additional information

Communicated by Govind Menon.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Holzer, M., Scheel, A. Criteria for Pointwise Growth and Their Role in Invasion Processes. J Nonlinear Sci 24, 661–709 (2014). https://doi.org/10.1007/s00332-014-9202-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00332-014-9202-0

Keywords

Search

Navigation