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.
Similar content being viewed by others
References
Aronson, D.G., Weinberger, H.F.: Multidimensional nonlinear diffusion arising in population genetics. Adv. Math. 30, 33–76 (1978)
Berestycki, H., Hamel, F.: Reaction–Diffusion Equations and Propagation Phenomena. Applied Mathematical Sciences. Springer, Berlin (2007)
Berestycki, H., Hamel, F., Nadin, G.: Asymptotic spreading in heterogeneous diffusive media. J. Funct. Anal. 255, 2146–2189 (2008)
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)
Ducrot, A.: Convergence to generalized transition waves for some Holling–Tanner prey–predator reaction–diffusion system. J. Math. Pures Appl. 100, 1–15 (2013)
Ducrot, A.: Spatial propagation for a two components reaction–diffusion system arising in population dynamics. J. Differ. Eq. 260, 8316–8357 (2016)
Dunbar, S.R.: Travelling wave solutions of diffusive Lotka–Volterra equations. J. Math. Biol. 17(1), 11–32 (1983)
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)
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)
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)
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)
Fisher, R.A.: The wave of advantageous genes. Ann. Eugen. 7, 355–369 (1937)
Gardner, R.: Existence of travelling wave solutions of predator–prey systems via the connection index. SIAM J. Appl. Math. 44(1), 56–79 (1984)
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)
Holling, C.S.: Some characteristics of simple types of predation and parasitism. Can. Entomol. 91, 385–398 (1959)
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)
Huang, W.: Traveling wave solutions for a class of predator–prey systems. J. Dyn. Differ. Equ. 23, 633–644 (2012)
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)
Lewis, M., Li, B., Weinberger, H.: Spreading speed and linear determinacy for two-species competition models. J. Math. Biol. 45, 219–233 (2002)
Li, B., Weinberger, H.F., Lewis, M.A.: Spreading speeds as slowest wave speeds for cooperative systems. Math. Biosci. 196, 82–98 (2005)
Li, H., Xiao, H.: Traveling wave solutions for diffusive predator–prey type systems with nonlinear density dependence. Comput. Math. Appl. 74, 2221–2230 (2017)
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)
Liang, X., Yi, Y., Zhao, X.-Q.: Spreading speeds and traveling waves for periodic evolution systems. J. Differ. Eq. 231, 57–77 (2006)
Magal, P., Zhao, X.-Q.: Global attractors and steady states for uniformly persistent dynamical systems. SIAM J. Math. Anal. 37, 251–275 (2005)
May, R.M.: Stability and Complexity in Model Ecosystems. Princeton University Press, Princeton (1974)
Oaten, A., Murdoch, W.W.: Functional response and stability in predator–prey system. Am. Natur. 109, 289–298 (1975)
Owen, M.R., Lewis, M.A.: How predation can slow, stop or reverse a prey invasion. Bull. Math. Biol. 01, 1–35 (2000)
Smith, H.L., Thieme, H.R.: Dynamical Systems and Population Persistence. American Mathematical Society, Providence (2011)
Thieme, H.R.: Density-dependent regulation of spatially distributed populations and their asymptotic speed of spread. J. Math. Biol. 8, 173–187 (1979)
Volpert, A., Volpert, V., Volpert, V.: Travelling Wave Solutions of Parabolic Systems, Translations of Mathematical Monographs, vol. 140. AMS, Providence (1994)
Wang, H., Castillo-Chavez, C.: Spreading speeds and traveling waves for non-cooperative integro-difference systems. DCDS-B 17, 2243–2266 (2012)
Weinberger, H.: On spreading speed and travelling waves for growth and migration models in a periodic habitat. J. Math. Biol. 45, 511–548 (2002)
Weinberger, H., Kawasaki, K., Shigesada, N.: Spreading speeds for a partially cooperative \(2-\)species reaction–diffusion model. DCDS-A 23, 1087–1098 (2009)
Author information
Authors and Affiliations
Corresponding author
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
About this article
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
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00526-019-1576-2