Skip to main content
SpringerLink
Log in

A mixed hyperbolic-parabolic system to describe predator-prey dynamics

  • Published:
Bulletin of the Brazilian Mathematical Society, New Series Aims and scope Submit manuscript

Abstract

Following [2], a model aiming at the description of two competing populations is introduced. In particular, it is considered a nonlinear system consisting of a nonlocal conservation law for predators coupled with a parabolic equation for prey. The drift term in the equation for predators is in general a nonlocal and nonlinear function of the prey density: the movement of predators can hence be directed towards regions where the concentration of prey is higher. Lotka-Volterra type right hand sides describe the feeding. In [2] the resulting Cauchy problemis proved to be well posed in any space dimension with respect to the L 1 topology, and estimates on the growth of the solution in L 1 and L ∞ norm and on the time dependence are provided. Numerical integrations show a few qualitative features of the solutions.

This is a joint work with RinaldoM. Colombo.

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

Similar content being viewed by others

References

  1. H. Amann. Linear and quasilinear parabolic problems. Vol. I, volume 89 of “Monographs in Mathematics”. Birkhäuser Boston Inc., Boston, MA, 1995. Abstract linear theory.

    Book  Google Scholar 

  2. R. M. Colombo and E. Rossi. Hyperbolic predators vs. parabolic prey. Commun. Math. Sci., 13(2) (2015), 369–400.

    Article  MathSciNet  MATH  Google Scholar 

  3. C. M. Dafermos. Hyperbolic conservation laws in continuum physics, volume 325 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, third edition, (2010).

    Book  Google Scholar 

  4. A. Friedman. Partial differential equations of parabolic type. Prentice-Hall Inc., Englewood Cliffs, N. J., (1964).

    MATH  Google Scholar 

  5. S. N. Kružkov. First order quasilinear equations with several independent variables. Mat. Sb. (N. S.), 81(123) (1970), 228–255.

    MathSciNet  Google Scholar 

  6. M. Lécureux-Mercier. Improved stability estimates for general scalar conservation laws. J. Hyperbolic Differ. Equ., 8(4) (2011), 727–757.

    Article  MathSciNet  MATH  Google Scholar 

  7. M. Lécureux-Mercier. Improved stability estimates on general scalar balance laws. ArXiv e-prints, July (2013).

    MATH  Google Scholar 

  8. A. Lunardi. Analytic semigroups and optimal regularity in parabolic problems. Progress inNonlinearDifferential Equations and theirApplications, 16. Birkhäuser Verlag, Basel, (1995).

    Google Scholar 

  9. P. Quittner and P. Souplet. Superlinear parabolic problems. Birkhäuser Advanced Texts: Basler Lehrbücher. [BirkhäuserAdvanced Texts: Basel Textbooks]. Birkhäuser Verlag, Basel, 2007. Blow-up, global existence and steady states.

    Google Scholar 

  10. E. Rossi and V. Schleper. Convergence of a numerical scheme for a mixed hyperbolic-parabolic system in two space dimensions. ESAIM:M2AN, 2015. To appear.

    MATH  Google Scholar 

  11. D. Serre. Systems of conservation laws. 1&2. Cambridge University Press, Cambridge, 1999. Translated from the 1996 French original by I. N. Sneddon.

    Book  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Applications, University of Milano-Bicocca, Via R. Cozzi, 55, 20125, Milano, Italy

    Elena Rossi

Authors
  1. Elena Rossi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Elena Rossi.

Additional information

The author was supported by the PRIN 2012 project Nonlinear Hyperbolic Partial Differential Equations, Dispersive and Transport Equations: Theoretical and Applicative Aspects and by the GNAMPA 2014 project Conservation Laws in the Modeling of Collective Phenomena.

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Rossi, E. A mixed hyperbolic-parabolic system to describe predator-prey dynamics. Bull Braz Math Soc, New Series 47, 701–714 (2016). https://doi.org/10.1007/s00574-016-0179-1

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00574-016-0179-1

Keywords

Mathematical subject classification

Search

Navigation