Competitive systems with migration and the Poincaré-Bendixson theorem for a 4-dimensional case
Authors:
Jifa Jiang and Xing Liang
Journal:
Quart. Appl. Math. 64 (2006), 483-498
MSC (2000):
Primary 34C12, 37N25; Secondary 92D40
DOI:
https://doi.org/10.1090/S0033-569X-06-01016-0
Published electronically:
June 14, 2006
MathSciNet review:
2259050
Full-text PDF Free Access
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Abstract: In this paper, dynamics of the $n$-species competitive system with migration is studied. It is proved that if the Jacobian matrix of the system is irreducible at every point in Int $\mathbb {R}^{2n}_+$, then there is a defined countable family of invariant $(2n-1)$-cells which attract all nonconvergent persistent trajectories. Moreover, it is proved that the Poincaré-Bendixson theorem holds for $2$-species competitive systems with migration.
- Morris W. Hirsch, Systems of differential equations which are competitive or cooperative. I. Limit sets, SIAM J. Math. Anal. 13 (1982), no. 2, 167–179. MR 647119, DOI https://doi.org/10.1137/0513013
- Morris W. Hirsch, Systems of differential equations that are competitive or cooperative. II. Convergence almost everywhere, SIAM J. Math. Anal. 16 (1985), no. 3, 423–439. MR 783970, DOI https://doi.org/10.1137/0516030
- Morris W. Hirsch, Systems of differential equations which are competitive or cooperative. III. Competing species, Nonlinearity 1 (1988), no. 1, 51–71. MR 928948
- S. B. Hsu, H. L. Smith, and Paul Waltman, Competitive exclusion and coexistence for competitive systems on ordered Banach spaces, Trans. Amer. Math. Soc. 348 (1996), no. 10, 4083–4094. MR 1373638, DOI https://doi.org/10.1090/S0002-9947-96-01724-2
Kamke E. Kamke, Zur Theorie der Systeme Gewoknlicher Differentialiechungen, II, Acta Math. 58(1932), 57-85.
- Xing Liang and Jifa Jiang, On the finite-dimensional dynamical systems with limited competition, Trans. Amer. Math. Soc. 354 (2002), no. 9, 3535–3554. MR 1911510, DOI https://doi.org/10.1090/S0002-9947-02-03032-5
- Xing Liang and Jifa Jiang, The classification of the dynamical behavior of 3-dimensional type $K$ monotone Lotka-Volterra systems, Nonlinear Anal. 51 (2002), no. 5, Ser. A: Theory Methods, 749–763. MR 1921374, DOI https://doi.org/10.1016/S0362-546X%2801%2900860-4
- Xing Liang and Jifa Jiang, Discrete infinite-dimensional type-$K$ monotone dynamical systems and time-periodic reaction-diffusion systems, J. Differential Equations 189 (2003), no. 1, 318–354. MR 1968324, DOI https://doi.org/10.1016/S0022-0396%2802%2900062-1
- Xing Liang and Jifa Jiang, The dynamical behaviour of type-K competitive Kolmogorov systems and its application to three-dimensional type-K competitive Lotka-Volterra systems, Nonlinearity 16 (2003), no. 3, 785–801. MR 1975782, DOI https://doi.org/10.1088/0951-7715/16/3/301
- J. D. Murray, Mathematical biology, Biomathematics, vol. 19, Springer-Verlag, Berlin, 1989. MR 1007836
- S. Smale, On the differential equations of species in competition, J. Math. Biol. 3 (1976), no. 1, 5–7. MR 406579, DOI https://doi.org/10.1007/BF00307854
- Hal L. Smith, Monotone dynamical systems, Mathematical Surveys and Monographs, vol. 41, American Mathematical Society, Providence, RI, 1995. An introduction to the theory of competitive and cooperative systems. MR 1319817
- Hal L. Smith, Competing subcommunities of mutualists and a generalized Kamke theorem, SIAM J. Appl. Math. 46 (1986), no. 5, 856–874. MR 858998, DOI https://doi.org/10.1137/0146052
- H. L. Smith and H. R. Thieme, Stable coexistence and bi-stability for competitive systems on ordered Banach spaces, J. Differential Equations 176 (2001), no. 1, 195–222. MR 1861187, DOI https://doi.org/10.1006/jdeq.2001.3981
- Peter Takáč, Convergence to equilibrium on invariant $d$-hypersurfaces for strongly increasing discrete-time semigroups, J. Math. Anal. Appl. 148 (1990), no. 1, 223–244. MR 1052057, DOI https://doi.org/10.1016/0022-247X%2890%2990040-M
- Yasuhiro Takeuchi, Diffusion-mediated persistence in two-species competition Lotka-Volterra model, Math. Biosci. 95 (1989), no. 1, 65–83. MR 1001292, DOI https://doi.org/10.1016/0025-5564%2889%2990052-7
- Yasuhiro Takeuchi and Zheng Yi Lu, Permanence and global stability for competitive Lotka-Volterra diffusion systems, Nonlinear Anal. 24 (1995), no. 1, 91–104. MR 1308472, DOI https://doi.org/10.1016/0362-546X%2894%29E0024-B
- Tu Caifeng and Jiang Jifa, The coexistence of a community of species with limited competition, J. Math. Anal. Appl. 217 (1998), no. 1, 233–245. MR 1492087, DOI https://doi.org/10.1006/jmaa.1997.5711
- Caifeng Tu and Jifa Jiang, The necessary and sufficient conditions for the global stability of type-$K$ Lotka-Volterra system, Proc. Amer. Math. Soc. 127 (1999), no. 11, 3181–3186. MR 1628420, DOI https://doi.org/10.1090/S0002-9939-99-05077-7
- Caifeng Tu and Jifa Jiang, Global stability and permanence for a class of type $K$ monotone systems, SIAM J. Math. Anal. 30 (1999), no. 2, 360–378. MR 1664764, DOI https://doi.org/10.1137/S0036141097325290
- Yi Wang and Jifa Jiang, The long-run behavior of periodic competitive Kolmogorov systems, Nonlinear Anal. Real World Appl. 3 (2002), no. 4, 471–485. MR 1930616, DOI https://doi.org/10.1016/S1468-1218%2801%2900034-7
- Yi Wang and Jifa Jiang, Uniqueness and attractivity of the carrying simplex for discrete-time competitive dynamical systems, J. Differential Equations 186 (2002), no. 2, 611–632. MR 1942224, DOI https://doi.org/10.1016/S0022-0396%2802%2900025-6
- M. L. Zeeman, Hopf bifurcations in competitive three-dimensional Lotka-Volterra systems, Dynam. Stability Systems 8 (1993), no. 3, 189–217. MR 1246002, DOI https://doi.org/10.1080/02681119308806158
H1 M. Hirsch, Systems of differential equations which are competitive or cooperative. I: Limit sets, SIAM J. Math. Anal., 13 (1982), 167–179.
H2 M. Hirsch, Systems of differential equations which are competitive or cooperative. II: Convergence almost everywhere, SIAM J. Math. Anal., 16 (1985), 423–439.
hirsch1 M. Hirsch, Systems of differential equations which are competitive or cooperative. III: Competing species, Nonlinearity, 1 (1988), 51-71.
Hsu S. B. Hsu, H. L. Smith, and P. Waltman, Competitive exclusion and coexistence for competitive systems on ordered Banach spaces, Trans. Amer. Math. Soc., 348(1996), 4083–4094.
Kamke E. Kamke, Zur Theorie der Systeme Gewoknlicher Differentialiechungen, II, Acta Math. 58(1932), 57-85.
liang X. Liang and J. Jiang, On the finite dimensional dynamical systems with limited competition, Trans. Amer. Math. Soc., 354 (2002), 3535-3554.
typek X. Liang and J. Jiang, The classification of the dynamical behavior of 3-dimensional type-K competitive Lotka-Volterra systems, Nonlinear Analysis, TMA, 51:5(2002), 749-763.
LJ1 X. Liang and J. Jiang, Discrete infinite dimensional type-K monotone dynamical systems and time periodic reaction-diffusion systems, J. Differential Equation, 189:1(2003), 318-354.
LX1 X. Liang and J. Jiang, The dynamical behavior of type-K competitive Kolmogorov systems and its applications to 3-dimensional type-K competitive Lotka-Volterra systems, Nonlinearity, 16:3(2003), 785-801.
murray J. Murray, Mathematical Biology, 1989, Springer-Verlag, Berlin.
smale S. Smale, On the differential equations species in competition, J. Math. Biol., 3(1976), 5–7.
Smi1 H. L. Smith, “Monotone Dynamical Systems, an Introduction to the Theory of Competitive and Cooperative Systems,” Mathematical Surveys and Monographs, vol. 41(1995), Amer. Math. Soc., Providence, RI.
smith1 H. L. Smith, Competing subcommunities of mutualists and a generalized Kamke theorem, SIAM J. Appl. Math., 46(1986), 856–874.
smithand H. L. Smith and H. R. Thieme, Stable coexistence and bi-stability for competitive systems on ordered Banach spaces, J. Diff. Eqns., 176 (2001), no. 1, 195–222.
Ta2 P. Takáč, Convergence to equilibrium on invariant $d$-hypersurfaces for strongly increasing discrete-time semigroups, J. Math. Anal. Appl. 148 (1990), 223-244.
Tak1Y. Takeuchi, Diffusion-mediated persistence in two-species competition Lotka-Volterra model, Math. Biosciences 95(1989), 65-83.
Tak2Y. Takeuchi and Z. Lu, Permanence and global stability for competitive Lotka-Volterra diffusion systems, Nonlinear Analysis, TMA, 24(1995), 91-104.
TuJiang1 C. F. Tu and J. F. Jiang, The coexistence of a community of species with limited competition, J. Math. Anal. Appl., 217(1998), 553–571.
TuJiang3 C. F. Tu and J. F. Jiang, The necessary and sufficient conditions for the global stability of type-K Lotka-Volterra systems, Proc. Amer. Math. Soc., 127(1999), 3181–3186.
TuJiang2 C. F. Tu and J. F. Jiang, Global stability and permanence for a class of type-K monotone systems, SIAM J. Math. Anal., 30(1999), 360–378.
WJ2 Y. Wang and J. Jiang, The long-run behavior of periodic competitive Kolmogorov systems, Nonlinear Anal. Real World Appl., 3(2002), 471–485.
WJ3 Y. Wang and J. Jiang, Uniqueness and attractivity of the carrying simples for the discrete-time competitive dynamical systems, J. Differential Equations, 186(2002), 611–632.
zeeman M. L. Zeeman, Hopf bifurations in competitive three-dimensional Lotka-Volterra systems, Dynamics and Stability of Systems, 3(1993), 190–217.
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Additional Information
Jifa Jiang
Affiliation:
Department of Applied Mathematics, Tongji University, Shanghai 200092, People’s Republic of China
Email:
jiangjf@mail.tongji.edu.cn
Xing Liang
Affiliation:
Department of Mathematics, University of Science and Technology of China, Hefei, Anhui 230026, People’s Republic of China
Email:
xliang@ustc.edu.cn
Keywords:
Competitive systems with migration,
Poincaré-Bendixson theorem
Received by editor(s):
October 19, 2005
Published electronically:
June 14, 2006
Additional Notes:
Research of the first author supported by the National Natural Science Foundation of China.
Research of the second author supported by the National Natural Science Foundation of China grant 10401032.
Article copyright:
© Copyright 2006
Brown University