On a Free Boundary Problem for a Two-Species Weak Competition System

Springer Science and Business Media LLC - Tập 24 - Trang 873-895 - 2012
Jong-Shenq Guo1, Chang-Hong Wu1
1Department of Mathematics, Tamkang University, New Taipei City, Taiwan

Tóm tắt

We study a Lotka–Volterra type weak competition model with a free boundary in a one-dimensional habitat. The main objective is to understand the asymptotic behavior of two competing species spreading via a free boundary. We also provide some sufficient conditions for spreading success and spreading failure, respectively. Finally, when spreading successfully, we provide an estimate to show that the spreading speed (if exists) cannot be faster than the minimal speed of traveling wavefront solutions for the competition model on the whole real line without a free boundary.

Tài liệu tham khảo

Aronson, D.G.: The asymptotic speed of propagation of a simple epidemic. In: Nonlinear Diffusion (NSF-CBMS Regional Conference on Nonlinear Diffusion Equations, University of Houston, Houston, TX, 1976). Research Notes in Mathematics, vol. 14, pp. 1–23. Pitman, London (1977)

Aronson, D.G., Weinberger, H.F.: Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation. In: Partial Differential Equations and Related Topics (Program, Tulane University, New Orleans, LA, 1974). Lecture Notes in Mathematics, vol. 446, pp. 5–49. Springer, Berlin (1975)

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

Bunting, G., Du, Y., Krakowski, K.: Spreading speed revisited: analysis of a free boundary model. Preprint (2011)

Cantrell R.S., Cosner C.: Spatial Ecology via Reaction-Diffusion Equations. Wiley, Chichester (2003)

Chen X.F., Friedman A.: A free boundary problem arising in a model of wound healing. SIAM J. Math. Anal. 32, 778–800 (2000)

Chen X.F., Friedman A.: A free boundary problem for an elliptic-hyperbolic system: an application to tumor growth. SIAM J. Math. Anal. 35, 974–986 (2003)

Conley C., Gardner R.: An application of the generalized Morse index to traveling wave solutions of a competitive reaction diffusion model. Indiana Univ. Math. J. 33, 319–343 (1984)

Du Y., Guo Z.M.: Spreading–vanishing dichotomy in a diffusive logistic model with a free boundary II. J. Differ. Equ. 250, 4336–4366 (2011)

Du Y., Guo Z.M.: The Stefan problem for the Fisher-KPP equation. J. Differ. Equ. 253, 996–1035 (2012)

Du Y., Lin Z.G.: Spreading–vanishing dichotomy in the diffusive logistic model with a free boundary. SIAM J. Math. Anal. 42, 377–405 (2010)

Du, Y., Lou, B.: Spreading and vanishing in nonlinear diffusion problems with free boundaries. Preprint (2011)

Du Y., Ma L.: Logistic type equations on \({\mathbb{R}^N}\) by a squeezing method involving boundary blow-up solutions. J. Lond. Math. Soc. 64, 107–124 (2001)

Du, Y., Guo, Z.M., Peng, R.: A diffusive logistic model with a free boundary in time-periodic environment. Preprint (2011)

Gardner R.A.: Existence and stability of travelling wave solutions of competition models: a degree theoretic approach. J. Differ. Equ. 44, 343–364 (1982)

Guo J.S., Liang X.: The minimal speed of traveling fronts for the Lotka–Volterra competition system. J. Dyn. Diff. Equat. 23, 353–363 (2011)

Hilhorst D., Iida M., Mimura M., Ninomiya H.: A competition-diffusion system approximation to the classical two-phase Stefan problem. Jpn. J. Ind. Appl. Math. 18(2), 161–180 (2001)

Hilhorst D., Mimura M., Schatzle R.: Vanishing latent heat limit in a Stefan-like problem arising in biology. Nonlinear Anal. Real World Appl. 4, 261–285 (2003)

Hosono, Y.: Singular Perturbation Analysis of Travelling Waves for Diffusive Lotka–Volterra Competitive Models. Numerical and Applied Mathematics, Part II (Paris, 1988), pp. 687–692. Baltzer, Basel (1989)

Hosono Y.: The minimal speed for a diffusive Lotka–Volterra model. Bull. Math. Biol. 60, 435–448 (1998)

Huang W.: Problem on minimum wave speed for a Lotka–Volterra reaction-diffusion competition model. J. Dyn. Diff. Equat. 22, 285–297 (2010)

Kan-on Y.: Parameter dependence of propagation speed of travelling waves for competition-diffusion equations. SIAM J. Math. Anal. 26, 340–363 (1995)

Kan-on Y.: Fisher wave fronts for the Lotka–Volterra competition model with diffusion. Nonlinear Anal. 28, 145–164 (1997)

Kolmogorov A.N., Petrovsky I.G., Piskunov N.S.: Étude de l’équation de la diffusion avec croissance de la quantité de matière et son application à un probléme biologique. Bull. Univ. Moskov. Ser. Int. Sect. A 1, 1–25 (1937)

Lewis M.A., Li B., Weinberger H.F.: 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)

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., Zhao X.Q.: Spreading speeds and traveling waves for abstract monostable evolution systems. J. Funct. Anal. 259, 857–903 (2010)

Lin Z.G.: A free boundary problem for a predator–prey model. Nonlinearity 20, 1883–1892 (2007)

Mimura M., Yamada Y., Yotsutani S.: A free boundary problem in ecology. Jpn. J. Appl. Math. 2, 151–186 (1985)

Mimura M., Yamada Y., Yotsutani S.: Stability analysis for free boundary problems in ecology. Hiroshima Math. J. 16, 477–498 (1986)

Mimura M., Yamada Y., Yotsutani S.: Free boundary problems for some reaction-diffusion equations. Hiroshima Math. J. 17, 241–280 (1987)

Murray J.D., Sperb R.P.: Minimum domains for spatial patterns in a class of reaction diffusion equations. J. Math. Biol. 18, 169–184 (1983)

Okubo A., Maini P.K., Williamson M.H., Murray J.D.: On the spatial spread of the grey squirrel in Britain. Proc. R. Soc. Lond. B 238, 113–125 (1989)

Peng, R., Zhao, X.-Q.: The diffusive logistic model with a free boundary and seasonal succession. Discret. Contin. Dyn. Syst. (Ser. A), in press (2012)

Ricci R., Tarzia D.A.: Asymptotic behavior of the solutions of the dead-core problem. Nonlinear Anal. 13, 405–411 (1989)

Tang M.M., Fife P.C.: Propagating fronts for competing species equations with diffusion. Arch. Ration. Mech. Anal. 73, 69–77 (1980)

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

Zhao X.Q.: Spatial dynamics of some evolution systems in biology. In: Du, Y., Ishii, H., Lin, W.Y. (eds) Recent Progress on Reaction-Diffusion Systems and Viscosity Solutions, World Scientific, Singapore (2009)

Thông tin
Thông tin xuất bản

Springer Science and Business Media LLC

Tập 24

873-895

Thông tin tác giả