Skip to main content
SpringerLink
Log in

Spreading speeds for multidimensional reaction–diffusion systems of the prey–predator type

  • Published:
Calculus of Variations and Partial Differential Equations Aims and scope Submit manuscript

Abstract

We investigate spreading properties of solutions of a large class of two-component reaction–diffusion systems, including prey–predator systems as a special case. By spreading properties we mean the long time behaviour of solution fronts that start from localized (i.e. compactly supported) initial data. Though there are results in the literature on the existence of travelling waves for such systems, very little has been known—at least theoretically—about the spreading phenomena exhibited by solutions with compactly supported initial data. The main difficulty comes from the fact that the comparison principle does not hold for such systems. Furthermore, the techniques that are known for travelling waves such as fixed point theorems and phase portrait analysis do not apply to spreading fronts. In this paper, we first prove that spreading occurs with definite spreading speeds. Intriguingly, two separate fronts of different speeds may appear in one solution—one for the prey and the other for the predator—in some situations.

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

Similar content being viewed by others

References

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

    Article  MathSciNet  Google Scholar 

  2. Berestycki, H., Hamel, F.: Reaction–Diffusion Equations and Propagation Phenomena. Applied Mathematical Sciences. Springer, Berlin (2007)

    MATH  Google Scholar 

  3. Berestycki, H., Hamel, F., Nadin, G.: Asymptotic spreading in heterogeneous diffusive media. J. Funct. Anal. 255, 2146–2189 (2008)

    Article  MathSciNet  Google Scholar 

  4. Cheng, K.-S., Hsu, S.-B., Lin, S.-S.: Some results on global stability of a predator–prey system. J. Math. Biol. 12, 115–126 (1981)

    Article  MathSciNet  Google Scholar 

  5. Ducrot, A.: Convergence to generalized transition waves for some Holling–Tanner prey–predator reaction–diffusion system. J. Math. Pures Appl. 100, 1–15 (2013)

    Article  MathSciNet  Google Scholar 

  6. Ducrot, A.: Spatial propagation for a two components reaction–diffusion system arising in population dynamics. J. Differ. Eq. 260, 8316–8357 (2016)

    Article  MathSciNet  Google Scholar 

  7. Dunbar, S.R.: Travelling wave solutions of diffusive Lotka–Volterra equations. J. Math. Biol. 17(1), 11–32 (1983)

    Article  MathSciNet  Google Scholar 

  8. Dunbar, S.R.: Traveling waves in diffusive predator–prey equations: periodic orbits and point-to-periodic heteroclinic orbits. SIAM J. Appl. Math. 46(6), 1057–1078 (1986)

    Article  MathSciNet  Google Scholar 

  9. Ducrot, A., Giletti, T., Matano, H.: Existence and convergence to a propagating terrace in one-dimensional reaction–diffusion equations. Trans. Am. Math. Soc. 366, 5541–5566 (2014)

    Article  MathSciNet  Google Scholar 

  10. Fang, J., Zhao, X.-Q.: Existence and uniqueness of traveling waves for non-monotone integral equations with applications. J. Differ. Eq. 248, 2199–2226 (2010)

    Article  MathSciNet  Google Scholar 

  11. Fife, P.C., McLeod, J. B.: A phase plane discussion of convergence to travelling fronts for nonlinear diffusion. Arch. Ration. Mech. Anal. 75(4), 281–314 (1980/81)

    Article  MathSciNet  Google Scholar 

  12. Fisher, R.A.: The wave of advantageous genes. Ann. Eugen. 7, 355–369 (1937)

    Article  Google Scholar 

  13. Gardner, R.: Existence of travelling wave solutions of predator–prey systems via the connection index. SIAM J. Appl. Math. 44(1), 56–79 (1984)

    Article  MathSciNet  Google Scholar 

  14. Gardner, R., Jones, C.K.R.T.: Stability of travelling wave solutions of diffusive predator–prey systems. Trans. Am. Math. Soc. 327(2), 465–524 (1991)

    Article  MathSciNet  Google Scholar 

  15. Holling, C.S.: Some characteristics of simple types of predation and parasitism. Can. Entomol. 91, 385–398 (1959)

    Article  Google Scholar 

  16. Huang, J.H., Lu, G., Ruan, S.: Existence of traveling wave solutions in a diffusive predator–prey model. J. Math. Biol. 46, 132–152 (2003)

    Article  MathSciNet  Google Scholar 

  17. Huang, W.: Traveling wave solutions for a class of predator–prey systems. J. Dyn. Differ. Equ. 23, 633–644 (2012)

    Article  MathSciNet  Google Scholar 

  18. Kolmogorov, A.N., Petrovsky, I.G., Piskunov, N.S.: Etude de l’équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique. Bulletin de l’Université d’Etat de Moscou, Série Internationale A 1, 1–26 (1937)

    Google Scholar 

  19. Lewis, M., Li, B., Weinberger, H.: Spreading speed and linear determinacy for two-species competition models. J. Math. Biol. 45, 219–233 (2002)

    Article  MathSciNet  Google Scholar 

  20. Li, B., Weinberger, H.F., Lewis, M.A.: Spreading speeds as slowest wave speeds for cooperative systems. Math. Biosci. 196, 82–98 (2005)

    Article  MathSciNet  Google Scholar 

  21. Li, H., Xiao, H.: Traveling wave solutions for diffusive predator–prey type systems with nonlinear density dependence. Comput. Math. Appl. 74, 2221–2230 (2017)

    Article  MathSciNet  Google Scholar 

  22. Liang, X., Zhao, X.-Q.: Asymptotic speeds of spread and traveling waves for monotone semiflows with applications. Commun. Pure Appl. Math. 60, 1–40 (2007)

    Article  MathSciNet  Google Scholar 

  23. Liang, X., Yi, Y., Zhao, X.-Q.: Spreading speeds and traveling waves for periodic evolution systems. J. Differ. Eq. 231, 57–77 (2006)

    Article  MathSciNet  Google Scholar 

  24. Magal, P., Zhao, X.-Q.: Global attractors and steady states for uniformly persistent dynamical systems. SIAM J. Math. Anal. 37, 251–275 (2005)

    Article  MathSciNet  Google Scholar 

  25. May, R.M.: Stability and Complexity in Model Ecosystems. Princeton University Press, Princeton (1974)

    Google Scholar 

  26. Oaten, A., Murdoch, W.W.: Functional response and stability in predator–prey system. Am. Natur. 109, 289–298 (1975)

    Article  Google Scholar 

  27. Owen, M.R., Lewis, M.A.: How predation can slow, stop or reverse a prey invasion. Bull. Math. Biol. 01, 1–35 (2000)

    MATH  Google Scholar 

  28. Smith, H.L., Thieme, H.R.: Dynamical Systems and Population Persistence. American Mathematical Society, Providence (2011)

    MATH  Google Scholar 

  29. Thieme, H.R.: Density-dependent regulation of spatially distributed populations and their asymptotic speed of spread. J. Math. Biol. 8, 173–187 (1979)

    Article  MathSciNet  Google Scholar 

  30. Volpert, A., Volpert, V., Volpert, V.: Travelling Wave Solutions of Parabolic Systems, Translations of Mathematical Monographs, vol. 140. AMS, Providence (1994)

    Book  Google Scholar 

  31. Wang, H., Castillo-Chavez, C.: Spreading speeds and traveling waves for non-cooperative integro-difference systems. DCDS-B 17, 2243–2266 (2012)

    Article  MathSciNet  Google Scholar 

  32. Weinberger, H.: On spreading speed and travelling waves for growth and migration models in a periodic habitat. J. Math. Biol. 45, 511–548 (2002)

    Article  MathSciNet  Google Scholar 

  33. Weinberger, H., Kawasaki, K., Shigesada, N.: Spreading speeds for a partially cooperative \(2-\)species reaction–diffusion model. DCDS-A 23, 1087–1098 (2009)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. UNIHAVRE, LMAH, FR-CNRS-3335, ISCN, Normandie University, 76600, Le Havre, France

    Arnaud Ducrot

  2. Institut Elie Cartan Lorraine, UMR 7502, University of Lorraine, 54506, Vandoeuvre-lès-Nancy, France

    Thomas Giletti

  3. Meiji Institute for Advanced Study of Mathematical Sciences, Meiji University, 4-21-1 Nakano, Tokyo, 164-8525, Japan

    Hiroshi Matano

Authors
  1. Arnaud Ducrot
    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. Hiroshi Matano
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Arnaud Ducrot.

Additional information

Communicated by P. Rabinowitz.

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

Ducrot, A., Giletti, T. & Matano, H. Spreading speeds for multidimensional reaction–diffusion systems of the prey–predator type. Calc. Var. 58, 137 (2019). https://doi.org/10.1007/s00526-019-1576-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00526-019-1576-2

Mathematics Subject Classification

Search

Navigation