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March  2017, 22(2): 483-490. doi: 10.3934/dcdsb.2017023

Expanding speed of the habitat for a species in an advective environment

 1 School of Mathematical Sciences, Tongji University, Shanghai 200092, China 2 School of Applied Mathematics, Nanjing University of Finance & Economics, Nanjing 210023, China 3 Mathematics & Science College, Shanghai Normal University, Shanghai 200234, China

* Corresponding author

Received  March 2016 Revised  June 2016 Published  December 2016

Fund Project: This research was partly supported by NSFC (No. 11671262,11601225).

Recently, Gu et al. [7,8] studied a reaction-diffusion-advection equation $u_t =u_{xx} -β u_x + f(u)$ in $(g(t), h(t))$, where $g(t)$ and $h(t)$ are two free boundaries satisfying Stefan conditions, $f(u)$ is a Fisher-KPP type of nonlinearity. When $β ∈ [0,c_0)$, where $c_0 := 2\sqrt{f'(0)}$, they found that for a spreading solution $(u,g,h)$, $h(t)/t \to c^*_r (β)$ and $g(t)/t \to c^*_l (β)$ as $t \to ∞$, and $c^*_r (β) > c^*_r(0) = - c^*_l (0) > - c^*_l (β) >0$. In this paper we study the expanding speed $C^*(β) :=c^*_r(β) - c^*_l (β)$ of the habitat $(g(t), h(t))$, and show that $C^*(β)$ is strictly increasing in $β ∈ [0,c_0)$. When $β ∈ [c_0, β^*)$ for some $β^*>c_0$, [8] also found a virtual spreading phenomena: $h(t)/t \to c^*_r(β)$ as $t\to∞$, and a back forms in the solution which moves rightward with a speed $β - c_0$. Hence the expanding speed of the main habitat for such a solution is $C^*(β) := c^*_r(β) -[β -c_0]$. In this paper we show that $C^*(β)$ is strictly decreasing in $β∈ [c_0, β^*)$ with $C^*(β^* -0)=0$, and so there exists a unique $β_0∈ (c_0, β^*)$ such that the advection is favorable to the expanding speed of the habitat if and only if $β∈ (0,β_0)$.

Citation: Junfan Lu, Hong Gu, Bendong Lou. Expanding speed of the habitat for a species in an advective environment. Discrete and Continuous Dynamical Systems - B, 2017, 22 (2) : 483-490. doi: 10.3934/dcdsb.2017023
References:
 [1] I. E. Averill, The Effect of Intermediate Advection on Two Competing Species Doctor of Philosophy, Ohio State University, Mathematics, 2012. [2] Y. Du and Z. G. Lin, Spreading-vanishing dichtomy in the diffusive logistic model with a free boundary, SIAM J. Math. Anal., 45 (2013), 1995-1996.  doi: 10.1137/090771089. [3] Y. Du and B. Lou, Spreading and vanishing in nonlinear diffusion problems with free boundaries, J. Eur. Math. Soc., 17 (2015), 2673-2724.  doi: 10.4171/JEMS/568. [4] Y. Du, H. Matsuzawa and M. Zhou, Sharp estimate of the spreading speed determined by nonlinear free boundary problems, SIAM J. Math. Anal., 46 (2014), 375-396.  doi: 10.1137/130908063. [5] J. Ge, K. I. Kim, Z. G. Lin and H. Zhu, A SIS reaction-diffusion-advection model in a low-risk and high-risk domain, J. Differential Equations, 259 (2015), 5486-5509.  doi: 10.1016/j.jde.2015.06.035. [6] H. Gu, Z. Lin and B. Lou, Long time behavior of solutions of a diffusion-advection logistic model with free boundaries, Appl. Math. Lett., 37 (2014), 49-53.  doi: 10.1016/j.aml.2014.05.015. [7] H. Gu, Z. Lin and B. Lou, Different asymptotic spreading speeds induced by advection in a diffusion problem with free boundaries, Proc. Amer. Math. Soc., 143 (2015), 1109-1117.  doi: 10.1090/S0002-9939-2014-12214-3. [8] H. Gu, B. Lou and M. Zhou, Long time behavior of solutions of Fisher-KPP equation with advection and free boundaries, J. Funct. Anal., 269 (2015), 1714-1768.  doi: 10.1016/j.jfa.2015.07.002. [9] N. A. Maidana and H. Yang, Spatial spreading of West Nile virus described by traveling waves, J. Theoret. Biol., 258 (2009), 403-417.  doi: 10.1016/j.jtbi.2008.12.032. [10] Y. Zhao and M. Wang, A reaction-diffusion-advection equation with mixed and free boundary conditions, preprint, arXiv: 1312.7751.

show all references

References:
 [1] I. E. Averill, The Effect of Intermediate Advection on Two Competing Species Doctor of Philosophy, Ohio State University, Mathematics, 2012. [2] Y. Du and Z. G. Lin, Spreading-vanishing dichtomy in the diffusive logistic model with a free boundary, SIAM J. Math. Anal., 45 (2013), 1995-1996.  doi: 10.1137/090771089. [3] Y. Du and B. Lou, Spreading and vanishing in nonlinear diffusion problems with free boundaries, J. Eur. Math. Soc., 17 (2015), 2673-2724.  doi: 10.4171/JEMS/568. [4] Y. Du, H. Matsuzawa and M. Zhou, Sharp estimate of the spreading speed determined by nonlinear free boundary problems, SIAM J. Math. Anal., 46 (2014), 375-396.  doi: 10.1137/130908063. [5] J. Ge, K. I. Kim, Z. G. Lin and H. Zhu, A SIS reaction-diffusion-advection model in a low-risk and high-risk domain, J. Differential Equations, 259 (2015), 5486-5509.  doi: 10.1016/j.jde.2015.06.035. [6] H. Gu, Z. Lin and B. Lou, Long time behavior of solutions of a diffusion-advection logistic model with free boundaries, Appl. Math. Lett., 37 (2014), 49-53.  doi: 10.1016/j.aml.2014.05.015. [7] H. Gu, Z. Lin and B. Lou, Different asymptotic spreading speeds induced by advection in a diffusion problem with free boundaries, Proc. Amer. Math. Soc., 143 (2015), 1109-1117.  doi: 10.1090/S0002-9939-2014-12214-3. [8] H. Gu, B. Lou and M. Zhou, Long time behavior of solutions of Fisher-KPP equation with advection and free boundaries, J. Funct. Anal., 269 (2015), 1714-1768.  doi: 10.1016/j.jfa.2015.07.002. [9] N. A. Maidana and H. Yang, Spatial spreading of West Nile virus described by traveling waves, J. Theoret. Biol., 258 (2009), 403-417.  doi: 10.1016/j.jtbi.2008.12.032. [10] Y. Zhao and M. Wang, A reaction-diffusion-advection equation with mixed and free boundary conditions, preprint, arXiv: 1312.7751.
Graph of $C^*(\beta)$: a simulation result when $f(u)=u(1-u)$ and $\mu=1$
Graphs of $c_r^*(\beta)$ and $-c_l^*(\beta)$ when $f(u)=u(1-u)$ and $\mu=1$
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