The diffusive competition model witha free boundary: Invasion of a superior or inferior competitor

Yihong Du; Zhigui Lin

doi:10.3934/dcdsb.2014.19.3105

Article Contents

2014, Volume 19, Issue 10: 3105-3132. Doi: 10.3934/dcdsb.2014.19.3105

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The diffusive competition model with a free boundary: Invasion of a superior or inferior competitor

Yihong Du^1, and
Zhigui Lin^2,

1.
School of Science and Technology, University of New England, Armidale, NSW 2351
2.
School of Mathematical Science, Yangzhou University, Yangzhou 225002

Received: February 28, 2013

Revised: April 30, 2013

Published: November 2014

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Abstract

In this paper we consider the diffusive competition model consisting of an invasive species with density $u$ and a native species with density $v$, in a radially symmetric setting with free boundary. We assume that $v$ undergoes diffusion and growth in $\mathbb{R}^N$, and $u$ exists initially in a ball ${r < h(0)}$, but invades into the environment with spreading front ${r = h(t)}$, with $h(t)$ evolving according to the free boundary condition $h'(t) = -\mu u_r(t, h(t))$, where $\mu>0$ is a given constant and $u(t,h(t))=0$. Thus the population range of $u$ is the expanding ball ${r < h(t)}$, while that for $v$ is $\mathbb{R}^N$. In the case that $u$ is a superior competitor (determined by the reaction terms), we show that a spreading-vanishing dichotomy holds, namely, as $t\to\infty$, either $h(t)\to\infty$ and $(u,v)\to (u^*,0)$, or $\lim_{t\to\infty} h(t)<\infty$ and $(u,v)\to (0,v^*)$, where $(u^*,0)$ and $(0, v^*)$ are the semitrivial steady-states of the system. Moreover, when spreading of $u$ happens, some rough estimates of the spreading speed are also given. When $u$ is an inferior competitor, we show that $(u,v)\to (0,v^*)$ as $t\to\infty$, so the invasive species $u$ always vanishes in the long run.

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Mathematics Subject Classification: Primary: 35K20, 35R35, 35J60; Secondary: 92B05.

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References

[1]	G. Bunting, Y. Du and K. Krakowski, Spreading speed revisited: Analysis of a free boundary model, Netw. Heterog. Media, 7 (2012), 583-603.doi: 10.3934/nhm.2012.7.583.
[2]	X. F. Chen and A. Friedman, A free boundary problem arising in a model of wound healing, SIAM J. Math. Anal., 32 (2000), 778-800.doi: 10.1137/S0036141099351693.
[3]	R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations, John Wiley& Sons Ltd, 2003.doi: 10.1002/0470871296.
[4]	Y. Du and Z. M. Guo, Spreading-vanishing dichotomy in the diffusive logistic model with a free boundary II, J. Diff. Eqns., 250 (2011), 4336-4366.doi: 10.1016/j.jde.2011.02.011.
[5]	Y. Du and Z. G. Lin, Spreading-vanishing dichotomy in the diffusive logistic model with a free boundary, SIAM J. Math. Anal., 42 (2010), 377-405.doi: 10.1137/090771089.
[6]	Y. Du and L. Ma, Logistic type equations on $\mathbbR^N$ by a squeezing method involving boundary blow-up solutions, J. London Math. Soc., 64 (2001), 107-124.doi: 10.1017/S0024610701002289.
[7]	J.-S. Guo and C.-H. Wu, On a free boundary problem for a two-species weak competition system, J. Dyn. Diff. Equat., 24 (2012), 873-895.doi: 10.1007/s10884-012-9267-0.
[8]	Y. Kan-On, Fisher wave fronts for the Lotka-Volterra competition model with diffusion, Nonl. Anal. TMA, 28 (1997), 145-164.doi: 10.1016/0362-546X(95)00142-I.
[9]	K. Kishimoto and H. F. Weinberger, The spatial homogeneity of stable equilibria of some reaction-diffusion systems on convex domains, J. Diff. Eqns., 58 (1985), 15-21.doi: 10.1016/0022-0396(85)90020-8.
[10]	O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, Amer. Math. Soc., Providence, RI, 1968.
[11]	G. M. Lieberman, Second Order Parabolic Differential Equations, World Scientific, Singapore, 1996.doi: 10.1142/3302.
[12]	Z. G. Lin, A free boundary problem for a predator-prey model, Nonlinearity, 20 (2007), 1883-1892.doi: 10.1088/0951-7715/20/8/004.
[13]	Y. Morita and K. Tachibana, An entire solution to the Lotka-Volterra competition-diffusion equations, SIAM J. Math. Anal., 40 (2009), 2217-2240.doi: 10.1137/080723715.
[14]	C. V. Pao, Nonliear Parabolic and Elliptic Equations, Plenum Press, New York, 1992.
[15]	H. L. Smith, Monotone Dynamical Systems, American Math. Soc., Providence, 1995.
[16]	M. X. Wang, On some free boundary problems of the prey-predator model, J. Diff. Eqns., 256 (2014), 3365-3394.doi: 10.1016/j.jde.2014.02.013.
[17]	J. F. Zhao and M. X. Wang, A free boundary problem for a predator-prey model with double free boundaries, preprint.