-
Previous Article
Mean field equations on tori: Existence and uniqueness of evenly symmetric blow-up solutions
- DCDS Home
- This Issue
-
Next Article
Evolution equations involving nonlinear truncated Laplacian operators
Global dynamics of competition models with nonsymmetric nonlocal dispersals when one diffusion rate is small
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 |
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 [
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[26] |
J. G. Skellam,
Random dispersal in theoretical populations, Biometrika, 38 (1951), 196-218.
doi: 10.1093/biomet/38.1-2.196. |
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[26] |
J. G. Skellam,
Random dispersal in theoretical populations, Biometrika, 38 (1951), 196-218.
doi: 10.1093/biomet/38.1-2.196. |
[1] |
Guo-Bao Zhang, Ruyun Ma, Xue-Shi Li. Traveling waves of a Lotka-Volterra strong competition system with nonlocal dispersal. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 587-608. doi: 10.3934/dcdsb.2018035 |
[2] |
Shanshan Chen, Junping Shi, Guohong Zhang. Spatial pattern formation in activator-inhibitor models with nonlocal dispersal. Discrete & Continuous Dynamical Systems - B, 2021, 26 (4) : 1843-1866. doi: 10.3934/dcdsb.2020042 |
[3] |
Rafael Luís, Sandra Mendonça. A note on global stability in the periodic logistic map. Discrete & Continuous Dynamical Systems - B, 2020, 25 (11) : 4211-4220. doi: 10.3934/dcdsb.2020094 |
[4] |
Lakmi Niwanthi Wadippuli, Ivan Gudoshnikov, Oleg Makarenkov. Global asymptotic stability of nonconvex sweeping processes. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1129-1139. doi: 10.3934/dcdsb.2019212 |
[5] |
Fernando P. da Costa, João T. Pinto, Rafael Sasportes. On the convergence to critical scaling profiles in submonolayer deposition models. Kinetic & Related Models, 2018, 11 (6) : 1359-1376. doi: 10.3934/krm.2018053 |
[6] |
Chin-Chin Wu. Existence of traveling wavefront for discrete bistable competition model. Discrete & Continuous Dynamical Systems - B, 2011, 16 (3) : 973-984. doi: 10.3934/dcdsb.2011.16.973 |
[7] |
Jon Aaronson, Dalia Terhesiu. Local limit theorems for suspended semiflows. Discrete & Continuous Dynamical Systems - A, 2020, 40 (12) : 6575-6609. doi: 10.3934/dcds.2020294 |
[8] |
Dan Wei, Shangjiang Guo. Qualitative analysis of a Lotka-Volterra competition-diffusion-advection system. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2599-2623. doi: 10.3934/dcdsb.2020197 |
[9] |
J. Frédéric Bonnans, Justina Gianatti, Francisco J. Silva. On the convergence of the Sakawa-Shindo algorithm in stochastic control. Mathematical Control & Related Fields, 2016, 6 (3) : 391-406. doi: 10.3934/mcrf.2016008 |
[10] |
Alberto Bressan, Carlotta Donadello. On the convergence of viscous approximations after shock interactions. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 29-48. doi: 10.3934/dcds.2009.23.29 |
[11] |
Caifang Wang, Tie Zhou. The order of convergence for Landweber Scheme with $\alpha,\beta$-rule. Inverse Problems & Imaging, 2012, 6 (1) : 133-146. doi: 10.3934/ipi.2012.6.133 |
[12] |
Jonathan DeWitt. Local Lyapunov spectrum rigidity of nilmanifold automorphisms. Journal of Modern Dynamics, 2021, 17: 65-109. doi: 10.3934/jmd.2021003 |
[13] |
Michael Grinfeld, Amy Novick-Cohen. Some remarks on stability for a phase field model with memory. Discrete & Continuous Dynamical Systems - A, 2006, 15 (4) : 1089-1117. doi: 10.3934/dcds.2006.15.1089 |
[14] |
Guangying Lv, Jinlong Wei, Guang-an Zou. Noise and stability in reaction-diffusion equations. Mathematical Control & Related Fields, 2021 doi: 10.3934/mcrf.2021005 |
[15] |
Jianping Gao, Shangjiang Guo, Wenxian Shen. Persistence and time periodic positive solutions of doubly nonlocal Fisher-KPP equations in time periodic and space heterogeneous media. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2645-2676. doi: 10.3934/dcdsb.2020199 |
[16] |
Cicely K. Macnamara, Mark A. J. Chaplain. Spatio-temporal models of synthetic genetic oscillators. Mathematical Biosciences & Engineering, 2017, 14 (1) : 249-262. doi: 10.3934/mbe.2017016 |
[17] |
Jian Yang, Bendong Lou. Traveling wave solutions of competitive models with free boundaries. Discrete & Continuous Dynamical Systems - B, 2014, 19 (3) : 817-826. doi: 10.3934/dcdsb.2014.19.817 |
[18] |
Carlos Gutierrez, Nguyen Van Chau. A remark on an eigenvalue condition for the global injectivity of differentiable maps of $R^2$. Discrete & Continuous Dynamical Systems - A, 2007, 17 (2) : 397-402. doi: 10.3934/dcds.2007.17.397 |
[19] |
Seung-Yeal Ha, Shi Jin. Local sensitivity analysis for the Cucker-Smale model with random inputs. Kinetic & Related Models, 2018, 11 (4) : 859-889. doi: 10.3934/krm.2018034 |
[20] |
Bernold Fiedler, Carlos Rocha, Matthias Wolfrum. Sturm global attractors for $S^1$-equivariant parabolic equations. Networks & Heterogeneous Media, 2012, 7 (4) : 617-659. doi: 10.3934/nhm.2012.7.617 |
2019 Impact Factor: 1.338
Tools
Metrics
Other articles
by authors
[Back to Top]