Dynamics of strongly competing systems with many species
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- by E. N. Dancer, Kelei Wang and Zhitao Zhang PDF
- Trans. Amer. Math. Soc. 364 (2012), 961-1005 Request permission
Abstract:
In this paper, we prove that the solution of the Lotka-Volterra competing species system with strong competition converges to a stationary point under some natural conditions. We also study the moving boundary problem of the singular limit equation, which plays an important role in our proof.References
- Luis A. Caffarelli and Xavier Cabré, Fully nonlinear elliptic equations, American Mathematical Society Colloquium Publications, vol. 43, American Mathematical Society, Providence, RI, 1995. MR 1351007, DOI 10.1090/coll/043
- L. A. Caffarelli, A. L. Karakhanyan, and Fang-Hua Lin, The geometry of solutions to a segregation problem for nondivergence systems, J. Fixed Point Theory Appl. 5 (2009), no. 2, 319–351. MR 2529504, DOI 10.1007/s11784-009-0110-0
- L. A. Cafferelli and Fang Hua Lin, An optimal partition problem for eigenvalues, J. Sci. Comput. 31 (2007), no. 1-2, 5–18. MR 2304268, DOI 10.1007/s10915-006-9114-8
- L. A. Caffarelli and Fang-Hua Lin, Singularly perturbed elliptic systems and multi-valued harmonic functions with free boundaries, J. Amer. Math. Soc. 21 (2008), no. 3, 847–862. MR 2393430, DOI 10.1090/S0894-0347-08-00593-6
- Luis Caffarelli and Fanghua Lin, Nonlocal heat flows preserving the $L^2$ energy, Discrete Contin. Dyn. Syst. 23 (2009), no. 1-2, 49–64. MR 2449068, DOI 10.3934/dcds.2009.23.49
- Xu-Yan Chen, A strong unique continuation theorem for parabolic equations, Math. Ann. 311 (1998), no. 4, 603–630. MR 1637972, DOI 10.1007/s002080050202
- M. Conti, S. Terracini, and G. Verzini, Nehari’s problem and competing species systems, Ann. Inst. H. Poincaré C Anal. Non Linéaire 19 (2002), no. 6, 871–888 (English, with English and French summaries). MR 1939088, DOI 10.1016/S0294-1449(02)00104-X
- M. Conti, S. Terracini, and G. Verzini, An optimal partition problem related to nonlinear eigenvalues, J. Funct. Anal. 198 (2003), no. 1, 160–196. MR 1962357, DOI 10.1016/S0022-1236(02)00105-2
- Monica Conti, Susanna Terracini, and Gianmaria Verzini, A variational problem for the spatial segregation of reaction-diffusion systems, Indiana Univ. Math. J. 54 (2005), no. 3, 779–815. MR 2151234, DOI 10.1512/iumj.2005.54.2506
- Monica Conti, Susanna Terracini, and G. Verzini, Asymptotic estimates for the spatial segregation of competitive systems, Adv. Math. 195 (2005), no. 2, 524–560. MR 2146353, DOI 10.1016/j.aim.2004.08.006
- Monica Conti, Susanna Terracini, and Gianmaria Verzini, Uniqueness and least energy property for solutions to strongly competing systems, Interfaces Free Bound. 8 (2006), no. 4, 437–446. MR 2283921, DOI 10.4171/IFB/150
- Norman Dancer, Competing species systems with diffusion and large interactions, Rend. Sem. Mat. Fis. Milano 65 (1995), 23–33 (1997). MR 1459414, DOI 10.1007/BF02925250
- E. N. Dancer and Yi Hong Du, Competing species equations with diffusion, large interactions, and jumping nonlinearities, J. Differential Equations 114 (1994), no. 2, 434–475. MR 1303035, DOI 10.1006/jdeq.1994.1156
- E. N. Dancer and Yi Hong Du, Positive solutions for a three-species competition system with diffusion. I. General existence results, Nonlinear Anal. 24 (1995), no. 3, 337–357. MR 1312772, DOI 10.1016/0362-546X(94)E0063-M
- E. N. Dancer and Yi Hong Du, Positive solutions for a three-species competition system with diffusion. II. The case of equal birth rates, Nonlinear Anal. 24 (1995), no. 3, 359–373. MR 1312773, DOI 10.1016/0362-546X(94)E0064-N
- E. N. Dancer and Yihong Du, On a free boundary problem arising from population biology, Indiana Univ. Math. J. 52 (2003), no. 1, 51–67. MR 1970020, DOI 10.1512/iumj.2003.52.2196
- E. N. Dancer and Z. M. Guo, Uniqueness and stability for solutions of competing species equations with large interactions, Comm. Appl. Nonlinear Anal. 1 (1994), no. 2, 19–45. MR 1280113
- E. N. Dancer and Zhitao Zhang, Dynamics of Lotka-Volterra competition systems with large interaction, J. Differential Equations 182 (2002), no. 2, 470–489. MR 1900331, DOI 10.1006/jdeq.2001.4102
- David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, Classics in Mathematics, Springer-Verlag, Berlin, 2001. Reprint of the 1998 edition. MR 1814364
- Mikhail Gromov and Richard Schoen, Harmonic maps into singular spaces and $p$-adic superrigidity for lattices in groups of rank one, Inst. Hautes Études Sci. Publ. Math. 76 (1992), 165–246. MR 1215595
- Qing Han, F. Lin, Nodal Sets of Solutions of Elliptic Differential Equations, books available on Han’s homepage.
- O. A. Ladyženskaja, V. A. Solonnikov, and N. N. Ural′ceva, Linear and quasilinear equations of parabolic type, Translations of Mathematical Monographs, Vol. 23, American Mathematical Society, Providence, R.I., 1968 (Russian). Translated from the Russian by S. Smith. MR 0241822
- Anthony W. Leung, Systems of nonlinear partial differential equations, Mathematics and its Applications, Kluwer Academic Publishers, Dordrecht, 1989. Applications to biology and engineering. MR 1621827, DOI 10.1007/978-94-015-3937-1
- Fanghua Lin and Xiaoping Yang, Geometric measure theory—an introduction, Advanced Mathematics (Beijing/Boston), vol. 1, Science Press Beijing, Beijing; International Press, Boston, MA, 2002. MR 2030862
- Gary M. Lieberman, Second order parabolic differential equations, World Scientific Publishing Co., Inc., River Edge, NJ, 1996. MR 1465184, DOI 10.1142/3302
- Chi-Cheung Poon, Unique continuation for parabolic equations, Comm. Partial Differential Equations 21 (1996), no. 3-4, 521–539. MR 1387458, DOI 10.1080/03605309608821195
- S. Smale, On the differential equations of species in competition, J. Math. Biol. 3 (1976), no. 1, 5–7. MR 406579, DOI 10.1007/BF00307854
- Kelei Wang and Zhitao Zhang, Some new results in competing systems with many species, Ann. Inst. H. Poincaré C Anal. Non Linéaire 27 (2010), no. 2, 739–761. MR 2595199, DOI 10.1016/j.anihpc.2009.11.004
Additional Information
- E. N. Dancer
- Affiliation: School of Mathematics and Statistics, University of Sydney, NSW 2006 Australia
- Email: normd@maths.usyd.edu.au
- Kelei Wang
- Affiliation: Academy of Mathematics and Systems Science, The Chinese Academy of Sciences, Beijing 100190, People’s Republic of China
- Address at time of publication: School of Mathematics and Statistics, University of Sydney, NSW 2006 Australia
- Email: wangkelei05@mails.gucas.ac.cn, kelei@maths.usyd.edu.au
- Zhitao Zhang
- Affiliation: Academy of Mathematics and Systems Science, The Chinese Academy of Sciences, Beijing 100190, People’s Republic of China
- Email: zzt@math.ac.cn
- Received by editor(s): April 8, 2010
- Received by editor(s) in revised form: September 3, 2010
- Published electronically: September 15, 2011
- Additional Notes: This work was supported by the Australian Research Council and the National Natural Science Foundation of China (10831005, 10971046)
- © Copyright 2011 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 364 (2012), 961-1005
- MSC (2010): Primary 35B40, 35R35, 35K57, 92D25
- DOI: https://doi.org/10.1090/S0002-9947-2011-05488-7
- MathSciNet review: 2846360