Different asymptotic spreading speeds induced by advection in a diffusion problem with free boundaries

Gu, Hong; Lin, Zhigui; Lou, Bendong

doi:10.1090/S0002-9939-2014-12214-3

Different asymptotic spreading speeds induced by advection in a diffusion problem with free boundaries
HTML articles powered by AMS MathViewer

by Hong Gu, Zhigui Lin and Bendong Lou PDF

Proc. Amer. Math. Soc. 143 (2015), 1109-1117 Request permission

Abstract:

In this paper, we consider a Fisher-KPP equation with an advection term and two free boundaries, which models the behavior of an invasive species in one dimension space. When spreading happens (that is, the solution converges to a positive constant), we use phase plane analysis and upper/lower solutions to prove that the rightward and leftward asymptotic spreading speeds exist and both are positive constants. Moreover, one of them is bigger and the other is smaller than the spreading speed in the corresponding problem without advection term.

References

Isabel E. Averill, The effect of intermediate advection on two competing species, ProQuest LLC, Ann Arbor, MI, 2011. Thesis (Ph.D.)–The Ohio State University. MR 3004385
D. G. Aronson and H. F. Weinberger, Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation, Partial differential equations and related topics (Program, Tulane Univ., New Orleans, La., 1974) Lecture Notes in Math., Vol. 446, Springer, Berlin, 1975, pp. 5–49. MR 0427837
D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusion arising in population genetics, Adv. in Math. 30 (1978), no. 1, 33–76. MR 511740, DOI 10.1016/0001-8708(78)90130-5
Yihong Du and Zongming Guo, Spreading-vanishing dichotomy in a diffusive logistic model with a free boundary, II, J. Differential Equations 250 (2011), no. 12, 4336–4366. MR 2793257, DOI 10.1016/j.jde.2011.02.011
Yihong Du and Zongming Guo, The Stefan problem for the Fisher-KPP equation, J. Differential Equations 253 (2012), no. 3, 996–1035. MR 2922661, DOI 10.1016/j.jde.2012.04.014
Yihong Du and Zhigui Lin, Spreading-vanishing dichotomy in the diffusive logistic model with a free boundary, SIAM J. Math. Anal. 42 (2010), no. 1, 377–405. MR 2607347, DOI 10.1137/090771089
Yihong Du and Bendong Lou, Spreading and vanishing in nonlinear diffusion problems with free boundaries, J. Eur. Math. Soc. (to appear) (arXiv:1301.5373)
Hong Gu, Zhigui Lin, and Bendong Lou, Long time behavior of solutions of a diffusion-advection logistic model with free boundaries, Appl. Math. Lett. 37 (2014), 49–53. MR 3231725, DOI 10.1016/j.aml.2014.05.015
Norberto Aníbal Maidana and Hyun Mo Yang, Spatial spreading of West Nile Virus described by traveling waves, J. Theoret. Biol. 258 (2009), no. 3, 403–417. MR 2973250, DOI 10.1016/j.jtbi.2008.12.032
I. G. Petrovski, Ordinary differential equations, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1966. Revised English edition. Translated from the Russian and edited by Richard A. Silverman. MR 0193298
Aizik I. Volpert, Vitaly A. Volpert, and Vladimir A. Volpert, Traveling wave solutions of parabolic systems, Translations of Mathematical Monographs, vol. 140, American Mathematical Society, Providence, RI, 1994. Translated from the Russian manuscript by James F. Heyda. MR 1297766, DOI 10.1090/mmono/140