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December  2014, 19(10): 3105-3132. doi: 10.3934/dcdsb.2014.19.3105

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  March 2013 Revised  May 2013 Published  October 2014

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.
Keywords: free boundary, spreading-vanishing dichotomy, invasive population., Diffusive competition model.
Mathematics Subject Classification: Primary: 35K20, 35R35, 35J60; Secondary: 92B0.
Citation: Yihong Du, Zhigui Lin. The diffusive competition model with a free boundary: Invasion of a superior or inferior competitor. Discrete & Continuous Dynamical Systems - B, 2014, 19 (10) : 3105-3132. doi: 10.3934/dcdsb.2014.19.3105
References:
[1]

G. Bunting, Y. Du and K. Krakowski, Spreading speed revisited: Analysis of a free boundary model,, Netw. Heterog. Media, 7 (2012), 583.  doi: 10.3934/nhm.2012.7.583.  Google Scholar

[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.  doi: 10.1137/S0036141099351693.  Google Scholar

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R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations,, John Wiley& Sons Ltd, (2003).  doi: 10.1002/0470871296.  Google Scholar

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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.  doi: 10.1016/j.jde.2011.02.011.  Google Scholar

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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.  doi: 10.1137/090771089.  Google Scholar

[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.  doi: 10.1017/S0024610701002289.  Google Scholar

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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.  doi: 10.1007/s10884-012-9267-0.  Google Scholar

[8]

Y. Kan-On, Fisher wave fronts for the Lotka-Volterra competition model with diffusion,, Nonl. Anal. TMA, 28 (1997), 145.  doi: 10.1016/0362-546X(95)00142-I.  Google Scholar

[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.  doi: 10.1016/0022-0396(85)90020-8.  Google Scholar

[10]

O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type,, Amer. Math. Soc., (1968).   Google Scholar

[11]

G. M. Lieberman, Second Order Parabolic Differential Equations,, World Scientific, (1996).  doi: 10.1142/3302.  Google Scholar

[12]

Z. G. Lin, A free boundary problem for a predator-prey model,, Nonlinearity, 20 (2007), 1883.  doi: 10.1088/0951-7715/20/8/004.  Google Scholar

[13]

Y. Morita and K. Tachibana, An entire solution to the Lotka-Volterra competition-diffusion equations,, SIAM J. Math. Anal., 40 (2009), 2217.  doi: 10.1137/080723715.  Google Scholar

[14]

C. V. Pao, Nonliear Parabolic and Elliptic Equations,, Plenum Press, (1992).   Google Scholar

[15]

H. L. Smith, Monotone Dynamical Systems,, American Math. Soc., (1995).   Google Scholar

[16]

M. X. Wang, On some free boundary problems of the prey-predator model,, J. Diff. Eqns., 256 (2014), 3365.  doi: 10.1016/j.jde.2014.02.013.  Google Scholar

[17]

J. F. Zhao and M. X. Wang, A free boundary problem for a predator-prey model with double free boundaries,, preprint., ().   Google Scholar

show all references

References:
[1]

G. Bunting, Y. Du and K. Krakowski, Spreading speed revisited: Analysis of a free boundary model,, Netw. Heterog. Media, 7 (2012), 583.  doi: 10.3934/nhm.2012.7.583.  Google Scholar

[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.  doi: 10.1137/S0036141099351693.  Google Scholar

[3]

R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations,, John Wiley& Sons Ltd, (2003).  doi: 10.1002/0470871296.  Google Scholar

[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.  doi: 10.1016/j.jde.2011.02.011.  Google Scholar

[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.  doi: 10.1137/090771089.  Google Scholar

[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.  doi: 10.1017/S0024610701002289.  Google Scholar

[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.  doi: 10.1007/s10884-012-9267-0.  Google Scholar

[8]

Y. Kan-On, Fisher wave fronts for the Lotka-Volterra competition model with diffusion,, Nonl. Anal. TMA, 28 (1997), 145.  doi: 10.1016/0362-546X(95)00142-I.  Google Scholar

[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.  doi: 10.1016/0022-0396(85)90020-8.  Google Scholar

[10]

O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type,, Amer. Math. Soc., (1968).   Google Scholar

[11]

G. M. Lieberman, Second Order Parabolic Differential Equations,, World Scientific, (1996).  doi: 10.1142/3302.  Google Scholar

[12]

Z. G. Lin, A free boundary problem for a predator-prey model,, Nonlinearity, 20 (2007), 1883.  doi: 10.1088/0951-7715/20/8/004.  Google Scholar

[13]

Y. Morita and K. Tachibana, An entire solution to the Lotka-Volterra competition-diffusion equations,, SIAM J. Math. Anal., 40 (2009), 2217.  doi: 10.1137/080723715.  Google Scholar

[14]

C. V. Pao, Nonliear Parabolic and Elliptic Equations,, Plenum Press, (1992).   Google Scholar

[15]

H. L. Smith, Monotone Dynamical Systems,, American Math. Soc., (1995).   Google Scholar

[16]

M. X. Wang, On some free boundary problems of the prey-predator model,, J. Diff. Eqns., 256 (2014), 3365.  doi: 10.1016/j.jde.2014.02.013.  Google Scholar

[17]

J. F. Zhao and M. X. Wang, A free boundary problem for a predator-prey model with double free boundaries,, preprint., ().   Google Scholar

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