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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
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show all references

References:
[1]

Netw. Heterog. Media, 7 (2012), 583-603. doi: 10.3934/nhm.2012.7.583.  Google Scholar

[2]

SIAM J. Math. Anal., 32 (2000), 778-800. doi: 10.1137/S0036141099351693.  Google Scholar

[3]

John Wiley& Sons Ltd, 2003. doi: 10.1002/0470871296.  Google Scholar

[4]

J. Diff. Eqns., 250 (2011), 4336-4366. doi: 10.1016/j.jde.2011.02.011.  Google Scholar

[5]

SIAM J. Math. Anal., 42 (2010), 377-405. doi: 10.1137/090771089.  Google Scholar

[6]

J. London Math. Soc., 64 (2001), 107-124. doi: 10.1017/S0024610701002289.  Google Scholar

[7]

J. Dyn. Diff. Equat., 24 (2012), 873-895. doi: 10.1007/s10884-012-9267-0.  Google Scholar

[8]

Nonl. Anal. TMA, 28 (1997), 145-164. doi: 10.1016/0362-546X(95)00142-I.  Google Scholar

[9]

J. Diff. Eqns., 58 (1985), 15-21. doi: 10.1016/0022-0396(85)90020-8.  Google Scholar

[10]

Amer. Math. Soc., Providence, RI, 1968.  Google Scholar

[11]

World Scientific, Singapore, 1996. doi: 10.1142/3302.  Google Scholar

[12]

Nonlinearity, 20 (2007), 1883-1892. doi: 10.1088/0951-7715/20/8/004.  Google Scholar

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SIAM J. Math. Anal., 40 (2009), 2217-2240. doi: 10.1137/080723715.  Google Scholar

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Plenum Press, New York, 1992.  Google Scholar

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American Math. Soc., Providence, 1995.  Google Scholar

[16]

J. Diff. Eqns., 256 (2014), 3365-3394. 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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