# American Institute of Mathematical Sciences

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

 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.
Citation: Yihong Du, Zhigui Lin. The diffusive competition model with a free boundary: Invasion of a superior or inferior competitor. Discrete and 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), 583-603. 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 $\mathbb{R}^N2$ 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.

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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 $\mathbb{R}^N2$ 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.
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