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June  2020, 40(6): 3075-3092. doi: 10.3934/dcds.2020035

Global dynamics of competition models with nonsymmetric nonlocal dispersals when one diffusion rate is small

Xueli Bai 1, and  Fang Li 2,,

1. 

Department of Applied Mathematics, Northwestern Polytechnical University, 127 Youyi Road(West), Beilin 710072, Xi'an, China
2. 

School of Mathematics, Sun Yat-sen University, No. 135, Xingang Xi Road, Guangzhou 510275, China

* Corresponding author: Fang Li

Received  February 2019 Revised  April 2019 Published  October 2019

Fund Project: The first author is supported by Shaanxi NSF (No. S2017-ZRJJ-MS-0104), Special financial aid to post-doctor research fellow (No. 2017T100768), Shaanxi Postdoctoral Science Foundation (2017BSHTDZZ16) and Alexander von Humboldt Foundation. The second author is supported by NSF of China (No. 11431005, 11971498) and the Fundamental Research Funds for the Central Universities

In this paper, we study the global dynamics of a general $ 2\times 2 $ competition models with nonsymmetric nonlocal dispersal operators. Our results indicate that local stability implies global stability provided that one of the diffusion rates is sufficiently small. This paper extends the work in [3], where Lotka-Volterra competition models with symmetric nonlocal operators are considered, to more general competition models with nonsymmetric operators.

Keywords: Nonlocal dispersal, competition models, local stability, global convergence.
Mathematics Subject Classification: Primary: 45G15, 45M05; Secondary: 45M10, 45M20.
Citation: Xueli Bai, Fang Li. Global dynamics of competition models with nonsymmetric nonlocal dispersals when one diffusion rate is small. Discrete & Continuous Dynamical Systems, 2020, 40 (6) : 3075-3092. doi: 10.3934/dcds.2020035
References:
[1]

L. J. S. Allen, E. J. Allen and S. Ponweera, A mathematical model for weed dispersal and control, Bull. Math. Biol., 58 (1996), 815-834.  doi: 10.1007/BF02459485.  Google Scholar

[2]

X. Bai and F. Li, Global dynamics of a competition model with nonlocal dispersal II: The full system, J. Differential Equations, 258 (2015), 2655-2685.  doi: 10.1016/j.jde.2014.12.014.  Google Scholar

[3]

X. Bai and F. Li, Classification of global dynamics of competition models with nonlocal dispersals I: Symmetric kernels, Calc. Var. Partial Differential Equations, 57 (2018), Art. 144, 35 pp. doi: 10.1007/s00526-018-1419-6.  Google Scholar

[4]

M. L. Cain, B. G. Milligan and A. E. Strand, Long-distance seed dispersal in plant populations, Am. J. Bot., 87 (2000), 1217-1227.  doi: 10.2307/2656714.  Google Scholar

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R. S. Cantrell and C. Cosner, On the effects of spatial heterogeneity on the persistence of interacting species, J. Math. Biol., 37 (1998), 103-145.  doi: 10.1007/s002850050122.  Google Scholar

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R. S. Cantrell and C. Cosner, Spatial Ecology Via Reaction-Diffusion Equations, Wiley Series in Mathematical and Computational Biology. John Wiley and Sons, Ltd., Chichester, 2003. doi: 10.1002/0470871296.  Google Scholar

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J. S. Clark, Why trees migrate so fast: Confronting theory with dispersal biology and the paleorecord, Am. Nat., 152 (1998), 204-224.  doi: 10.1086/286162.  Google Scholar

[8]

J. S. Clark, C. Fastie, G. Hurtt, S. T. Jackson, C. Johnson, G. A. King, M. Lewis, J. Lynch, S. Pacala, C. Prentice, E. W. Schupp, T. III. Webb and P. Wyckoff, Reid's paradox of rapid plant migration, BioScience, 48 (1998), 13-24.  doi: 10.2307/1313224.  Google Scholar

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X. He and W.-M. Ni, Global dynamics of the Lotka-Volterra competition-diffusion system: Diffusion and spatial heterogeneity, I, Comm. Pure Appl. Math., 69 (2016), 981-1014.  doi: 10.1002/cpa.21596.  Google Scholar

[10]

G. Hetzer, T. Nguyen and W. Shen, Coexistence and extinction in the Volterra-Lotka competition model with nonlocal dispersal, Commun. Pure Appl. Anal., 11 (2012), 1699-1722.  doi: 10.3934/cpaa.2012.11.1699.  Google Scholar

[11]

V. Huston, S. Martinez, K. Miscaikow and G. T. Vichers, The evolution of dispersal, J. Math. Biol., 47 (2003), 483-517.  doi: 10.1007/s00285-003-0210-1.  Google Scholar

[12]

M. Kot, M. A. Lewis and P. van den Driessche, Dispersal data and the spread of invading organisms, Ecology, 77 (1996), 2027-2042.  doi: 10.2307/2265698.  Google Scholar

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K.-Y. Lam and W.-M. Ni, Uniqueness and complete dynamics in the heterogeneous competition-diffusion systems, SIAM J. Appl. Math., 72 (2012), 1695-1712.  doi: 10.1137/120869481.  Google Scholar

[14]

C. T. Lee, M. F. Hoopes, J. Diehl, W. Gilliland, G. Huxel, E. V. Leaver, K. McCann, J. Umbanhowar and A. Mogilner, Non-local concepts and models in biology, J. Theor. Biol., 210 (2001), 201-219.  doi: 10.1006/jtbi.2000.2287.  Google Scholar

[15]

F. Li, J. Coville and X. Wang, On eigenvalue problems arising from nonlocal diffusion models, Discrete Contin. Dyn. Syst., 37 (2017), 879-903.  doi: 10.3934/dcds.2017036.  Google Scholar

[16]

F. Li, Y. Lou and Y. Wang, Global dynamics of a competition model with non-local dispersal I: The shadow system, J. Math. Anal. Appl., 412 (2014), 485-497.  doi: 10.1016/j.jmaa.2013.10.071.  Google Scholar

[17]

F. Li, L. Wang and Y. Wang, On the effects of migration and inter-specific competitions in steady state of some Lotka-Volterra model, Discrete Contin. Dyn. Syst. Ser. B, 15 (2011), 669-686.  doi: 10.3934/dcdsb.2011.15.669.  Google Scholar

[18]

Y. Lou, On the effects of migration and spatial heterogeneity on single and multiple species, J. Differential Equations, 223 (2006), 400-426.  doi: 10.1016/j.jde.2005.05.010.  Google Scholar

[19]

F. Lutscher, E. Pachepsky and M. A. Lewis, The effect of dispersal patterns on stream populations, SIAM Rev, 47 (2005), 749-772.  doi: 10.1137/050636152.  Google Scholar

[20]

J. Medlock and M. Kot, Spreading disease: Integro-differential equations old and new, Math. Biosci., 184 (2003), 201-222.  doi: 10.1016/S0025-5564(03)00041-5.  Google Scholar

[21]

F. J. R. Meysman, B. P. Boudreau and J. J. Middelburg, Relations between local, nonlocal, discrete and continuous models of bioturbation, J. Marine Research, 61 (2003), 391-410.  doi: 10.1357/002224003322201241.  Google Scholar

[22]

A. Mogilner and L. Edelstein-Keshet, A non-local model for a swarm, J. Math. Biol., 38 (1999), 534-570.  doi: 10.1007/s002850050158.  Google Scholar

[23]

A. Okubo and S. A. Levin, Diffusion and Ecological Problems: Modern Perspectives, Second edition, Interdisciplinary Applied Mathematics, 14, Springer-Verlag, New York, 2001. doi: 10.1007/978-1-4757-4978-6.  Google Scholar

[24]

H. G. Othmer, S. R. Dunbar and W. Alt, Models of dispersal in biological systems, J. Math. Biol., 26 (1988), 263-298.  doi: 10.1007/BF00277392.  Google Scholar

[25]

F. M. Schurr, O. Steinitz and R. Nathan, Plant fecundity and seed dispersal in spatially heterogeneous environments: Models, mechanisms and estimation, J. Ecol., 96 (2008), 628-641.  doi: 10.1111/j.1365-2745.2008.01371.x.  Google Scholar

[26]

J. G. Skellam, Random dispersal in theoretical populations, Biometrika, 38 (1951), 196-218.  doi: 10.1093/biomet/38.1-2.196.  Google Scholar

show all references

References:
[1]

L. J. S. Allen, E. J. Allen and S. Ponweera, A mathematical model for weed dispersal and control, Bull. Math. Biol., 58 (1996), 815-834.  doi: 10.1007/BF02459485.  Google Scholar

[2]

X. Bai and F. Li, Global dynamics of a competition model with nonlocal dispersal II: The full system, J. Differential Equations, 258 (2015), 2655-2685.  doi: 10.1016/j.jde.2014.12.014.  Google Scholar

[3]

X. Bai and F. Li, Classification of global dynamics of competition models with nonlocal dispersals I: Symmetric kernels, Calc. Var. Partial Differential Equations, 57 (2018), Art. 144, 35 pp. doi: 10.1007/s00526-018-1419-6.  Google Scholar

[4]

M. L. Cain, B. G. Milligan and A. E. Strand, Long-distance seed dispersal in plant populations, Am. J. Bot., 87 (2000), 1217-1227.  doi: 10.2307/2656714.  Google Scholar

[5]

R. S. Cantrell and C. Cosner, On the effects of spatial heterogeneity on the persistence of interacting species, J. Math. Biol., 37 (1998), 103-145.  doi: 10.1007/s002850050122.  Google Scholar

[6]

R. S. Cantrell and C. Cosner, Spatial Ecology Via Reaction-Diffusion Equations, Wiley Series in Mathematical and Computational Biology. John Wiley and Sons, Ltd., Chichester, 2003. doi: 10.1002/0470871296.  Google Scholar

[7]

J. S. Clark, Why trees migrate so fast: Confronting theory with dispersal biology and the paleorecord, Am. Nat., 152 (1998), 204-224.  doi: 10.1086/286162.  Google Scholar

[8]

J. S. Clark, C. Fastie, G. Hurtt, S. T. Jackson, C. Johnson, G. A. King, M. Lewis, J. Lynch, S. Pacala, C. Prentice, E. W. Schupp, T. III. Webb and P. Wyckoff, Reid's paradox of rapid plant migration, BioScience, 48 (1998), 13-24.  doi: 10.2307/1313224.  Google Scholar

[9]

X. He and W.-M. Ni, Global dynamics of the Lotka-Volterra competition-diffusion system: Diffusion and spatial heterogeneity, I, Comm. Pure Appl. Math., 69 (2016), 981-1014.  doi: 10.1002/cpa.21596.  Google Scholar

[10]

G. Hetzer, T. Nguyen and W. Shen, Coexistence and extinction in the Volterra-Lotka competition model with nonlocal dispersal, Commun. Pure Appl. Anal., 11 (2012), 1699-1722.  doi: 10.3934/cpaa.2012.11.1699.  Google Scholar

[11]

V. Huston, S. Martinez, K. Miscaikow and G. T. Vichers, The evolution of dispersal, J. Math. Biol., 47 (2003), 483-517.  doi: 10.1007/s00285-003-0210-1.  Google Scholar

[12]

M. Kot, M. A. Lewis and P. van den Driessche, Dispersal data and the spread of invading organisms, Ecology, 77 (1996), 2027-2042.  doi: 10.2307/2265698.  Google Scholar

[13]

K.-Y. Lam and W.-M. Ni, Uniqueness and complete dynamics in the heterogeneous competition-diffusion systems, SIAM J. Appl. Math., 72 (2012), 1695-1712.  doi: 10.1137/120869481.  Google Scholar

[14]

C. T. Lee, M. F. Hoopes, J. Diehl, W. Gilliland, G. Huxel, E. V. Leaver, K. McCann, J. Umbanhowar and A. Mogilner, Non-local concepts and models in biology, J. Theor. Biol., 210 (2001), 201-219.  doi: 10.1006/jtbi.2000.2287.  Google Scholar

[15]

F. Li, J. Coville and X. Wang, On eigenvalue problems arising from nonlocal diffusion models, Discrete Contin. Dyn. Syst., 37 (2017), 879-903.  doi: 10.3934/dcds.2017036.  Google Scholar

[16]

F. Li, Y. Lou and Y. Wang, Global dynamics of a competition model with non-local dispersal I: The shadow system, J. Math. Anal. Appl., 412 (2014), 485-497.  doi: 10.1016/j.jmaa.2013.10.071.  Google Scholar

[17]

F. Li, L. Wang and Y. Wang, On the effects of migration and inter-specific competitions in steady state of some Lotka-Volterra model, Discrete Contin. Dyn. Syst. Ser. B, 15 (2011), 669-686.  doi: 10.3934/dcdsb.2011.15.669.  Google Scholar

[18]

Y. Lou, On the effects of migration and spatial heterogeneity on single and multiple species, J. Differential Equations, 223 (2006), 400-426.  doi: 10.1016/j.jde.2005.05.010.  Google Scholar

[19]

F. Lutscher, E. Pachepsky and M. A. Lewis, The effect of dispersal patterns on stream populations, SIAM Rev, 47 (2005), 749-772.  doi: 10.1137/050636152.  Google Scholar

[20]

J. Medlock and M. Kot, Spreading disease: Integro-differential equations old and new, Math. Biosci., 184 (2003), 201-222.  doi: 10.1016/S0025-5564(03)00041-5.  Google Scholar

[21]

F. J. R. Meysman, B. P. Boudreau and J. J. Middelburg, Relations between local, nonlocal, discrete and continuous models of bioturbation, J. Marine Research, 61 (2003), 391-410.  doi: 10.1357/002224003322201241.  Google Scholar

[22]

A. Mogilner and L. Edelstein-Keshet, A non-local model for a swarm, J. Math. Biol., 38 (1999), 534-570.  doi: 10.1007/s002850050158.  Google Scholar

[23]

A. Okubo and S. A. Levin, Diffusion and Ecological Problems: Modern Perspectives, Second edition, Interdisciplinary Applied Mathematics, 14, Springer-Verlag, New York, 2001. doi: 10.1007/978-1-4757-4978-6.  Google Scholar

[24]

H. G. Othmer, S. R. Dunbar and W. Alt, Models of dispersal in biological systems, J. Math. Biol., 26 (1988), 263-298.  doi: 10.1007/BF00277392.  Google Scholar

[25]

F. M. Schurr, O. Steinitz and R. Nathan, Plant fecundity and seed dispersal in spatially heterogeneous environments: Models, mechanisms and estimation, J. Ecol., 96 (2008), 628-641.  doi: 10.1111/j.1365-2745.2008.01371.x.  Google Scholar

[26]

J. G. Skellam, Random dispersal in theoretical populations, Biometrika, 38 (1951), 196-218.  doi: 10.1093/biomet/38.1-2.196.  Google Scholar

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