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Spreading Speeds and Traveling Waves for Non-cooperative Reaction–Diffusion Systems

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Abstract

Much has been studied on the spreading speed and traveling wave solutions for cooperative reaction–diffusion systems. In this paper, we shall establish the spreading speed for a large class of non-cooperative reaction–diffusion systems and characterize the spreading speed as the slowest speed of a family of non-constant traveling wave solutions. Our results are applied to a partially cooperative system describing interactions between ungulates and grass.

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References

  • Aronson, D.G., Weinberger, H.F.: Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation. In: Goldstein, J.A. (ed.) Partial Differential Equations and Related Topics. Lecture Notes in Mathematics, vol. 446, pp. 5–49. Springer, Berlin (1975)

    Chapter  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  • Boumenir, A., Nguyen, V.: Perron theorem in the monotone iteration method for traveling waves in delayed reaction–diffusion equations. J. Differ. Equ. 244, 1551–1570 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  • Brown, K., Carr, J.: Deterministic epidemic waves of critical velocity. Math. Proc. Camb. Philos. Soc. 81, 431–433 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  • Cohen, J.: Convexity of the dominant eigenvalue of an essentially non-negative matrix. Proc. Am. Math. Soc. 81, 657–658 (1981)

    MATH  Google Scholar 

  • Crooks, E.C.M.: On the Vol’pert theory of traveling-wave solutions for parabolic systems. Nonlinear Anal. 26, 1621–1642 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  • Dale, P.D., Maini, P.K., Sherratt, J.A.: Mathematical modelling of corneal epithelial wound healing. Math. Biosci. 124, 127–147 (1994)

    Article  MATH  Google Scholar 

  • Diekmann, O.: Thresholds and travelling waves for the geographical spread of an infection. J. Math. Biol. 6, 109–130 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  • Fang, J., Zhao, X.: Monotone wavefronts for partially degenerate reaction–diffusion systems. J. Dyn. Differ. Equ. 21, 663–680 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  • Fife, P.: Mathematical Aspects of Reacting and Diffusing Systems. Lecture Notes in Biomathematics, vol. 28. Springer, Berlin (1979)

    MATH  Google Scholar 

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

    Article  Google Scholar 

  • Hadeler, K., Rothe, F.: Travelling fronts in nonlinear diffusion equations. J. Math. Biol. 2, 251–263 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  • Horn, R., Johnson, C.: Matrix Analysis. Cambridge University Press, Cambridge (1985)

    MATH  Google Scholar 

  • Hsu, S., Zhao, X.: Spreading speeds and traveling waves for nonmonotone integrodifference equations. SIAM J. Math. Anal. 40, 776–789 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  • Kolmogorov, A., Petrovsky, I., Piscounov, N.: Etude de l’équation de la diffusion avec croissance de la quantité de matière et son application a un problème biologique. Bull. Mosc. Univ. Math. Mech. 1(6), 1–26 (1937)

    Google Scholar 

  • 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  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  • Li, B., Lewis, M., Weinberger, H.: Existence of traveling waves for integral recursions with nonmonotone growth functions. J. Math. Biol. 58, 323–338 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  • Lui, R.: Biological growth and spread modeled by systems of recursions. I. Mathematical theory. Math. Biosci. 93(2), 269–295 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  • Ma, S.: Traveling wavefronts for delayed reaction–diffusion systems via a fixed point theorem. J. Differ. Equ. 171, 294–314 (2001)

    Article  MATH  Google Scholar 

  • Ma, S.: Traveling waves for non-local delayed diffusion equations via auxiliary equation. J. Differ. Equ. 237, 259–277 (2007)

    Article  MATH  Google Scholar 

  • Protter, M., Weinberger, H.: Maximum principles in differential equations. Springer, New York (1984)

    Book  MATH  Google Scholar 

  • Rass, L., Radcliffe, J.: Spatial Deterministic Epidemics. American Mathematical Society, Providence (2003)

    MATH  Google Scholar 

  • Sherratt, J., Murray, J.D.: Models of epidermal wound healing. Proc. R. Soc. Lond. B 241, 29–36 (1990)

    Article  Google Scholar 

  • Sherratt, J., Murray, J.: Mathematical analysis of a basic model for epidermal wound healing. J. Math. Biol. 29, 389–404 (1991)

    Article  MATH  Google Scholar 

  • Smoller, J.: Shock Waves and Reaction–Diffusion Equations. Springer, New York (1994)

    MATH  Google Scholar 

  • 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  MATH  Google Scholar 

  • Volpert, A.I., Volpert, V.A., Volpert, V.A.: Traveling Wave Solutions of Parabolic Systems. Transl. Math. Monogr., vol. 140. American Mathematical Society, Providence (1994)

    Google Scholar 

  • Wang, H.: On the existence of traveling waves for delayed reaction–diffusion equations. J. Differ. Equ. 247, 887–905 (2009)

    Article  MATH  Google Scholar 

  • Wang, H.: Spreading speeds and traveling waves for a model of epidermal wound healing. arXiv:1007.1442v1 (2010)

  • Wang, H., Castillo-Chavez, C.: Spreading speeds and traveling waves for non-cooperative integro-difference systems. arXiv:1003.1600v1 (2010)

  • Weinberger, H.F.: Asymptotic behavior of a model in population genetics. In: Chadam, J.M. (ed.) Nonlinear Partial Differential Equations and Applications. Lecture Notes in Mathematics, vol. 648, pp. 47–96. Springer, Berlin (1978)

    Chapter  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  • Weinberger, H.F., Lewis, M.A., Li, B.: Analysis of linear determinacy for spread in cooperative models. J. Math. Biol. 45, 183–218 (2002)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  • Weinberger, H.F., Kawasaki, K., Shigesada, N.: Spreading speeds for a partially cooperative 2-species reaction–diffusion model. Discrete Contin. Dyn. Syst. 23, 1087–1098 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  • Weng, P., Zhao, X.: Spreading speed and traveling waves for a multi-type SIS epidemic model. J. Differ. Equ. 229, 270–296 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  • Wu, J., Zou, X.: Traveling wave fronts of reaction diffusion systems with delay. J. Dyn. Differ. Equ. 13, 651–687 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  • Wu, J., Zou, X.: Erratum to “Traveling wave fronts of reaction–diffusion systems with delays” [J. Dyn. Differ. Equ. 13, 651, 687 (2001)]. J. Dyn. Differ. Equ. 20, 531–533 (2008)

    Article  MathSciNet  MATH  Google Scholar 

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Correspondence to Haiyan Wang.

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Communicated by P. Newton.

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Wang, H. Spreading Speeds and Traveling Waves for Non-cooperative Reaction–Diffusion Systems. J Nonlinear Sci 21, 747–783 (2011). https://doi.org/10.1007/s00332-011-9099-9

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