-
Previous Article
Lessons in uncertainty quantification for turbulent dynamical systems
- DCDS Home
- This Issue
-
Next Article
Conservation laws in mathematical biology
Dynamics of a three species competition model
1. | Department of Mathematics, Mathematical Bioscience Institute, Ohio State University, Columbus, Ohio 43210, United States |
2. | Department of Mathematics and Statistics, York University, Toronto, M3J 1P3, Canada |
References:
[1] |
I. Averill, Y. Lou and D. Munther, On several conjectures from evolution of dispersal,, J. Biol. Dyn., ().
|
[2] |
F. Belgacem, "Elliptic Boundary Value Problems with Indefinite Weights: Variational Formulations of the Principal Eigenvalue and Applications," Pitman Res. Notes Math. Ser., 368, Longman Sci, 1997. |
[3] |
F. Belgacem and C. Cosner, The effects of dispersal along environmental gradients on the dynamics of populations in heterogeneous environment, Canadian Appl. Math. Quarterly, 3 (1995), 379-397. |
[4] |
N. P. Bhatia and G. P. Szegö, "Stability Theory of Dynamical Systems," Springer-Verlag, New York, 1970. |
[5] |
R. S. Cantrell and C. Cosner, Practical persistence in ecological models via comparison methods, Proc. Roy. Soc. Edinb. A, 126 (1996), 247-272.
doi: 10.1017/S0308210500022721. |
[6] |
R. S. Cantrell and C. Cosner, "Spatial Ecology via Reaction-Diffusion Equations," Wiley Series in Mathematical and Computational Biology, John Wiley & Sons, Ltd., Chichester, 2003. |
[7] |
R. S. Cantrell, C. Cosner and Y. Lou, Advection mediated coexistence of competing species, Proc. Roy. Soc. Edinb. A, 137 (2007), 497-518. |
[8] |
R. S. Cantrell, C. Cosner and Y. Lou, Evolution of dispersal and the ideal free distribution, Math. Bios. Eng., 7 (2010), 17-36.
doi: 10.3934/mbe.2010.7.17. |
[9] | |
[10] |
X. F. Chen and Y. Lou, Principal eigenvalue and eigenfunction of an elliptic operator with large convection and its application to a competition model, Indiana Univ. Math. J., 57 (2008), 627-658.
doi: 10.1512/iumj.2008.57.3204. |
[11] |
M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. Funct. Anal., 8 (1971), 321-340.
doi: 10.1016/0022-1236(71)90015-2. |
[12] |
J. Dockery, V. Hutson, K. Mischaikow and M. Pernarowski, The evolution of slow dispersal rates: A reaction-diffusion model, J. Math. Biol., 37 (1998), 61-83.
doi: 10.1007/s002850050120. |
[13] |
S. D. Fretwell and H. L. Lucas, On territorial behavior and other factors influencing habitat selection in birds: Theoretical development, Acta Biotheor., 19 (1970), 16-36.
doi: 10.1007/BF01601953. |
[14] |
A. Friedman, "Partial Differential Equations of Parabolic Type," Prentice-Hall, 1964. |
[15] |
R. Gejji, Y. Lou, D. Munther and J. Peyton, Evolutionary convergence to ideal free dispersal strategies and coexistence, Bull. Math. Biol., 74 (2012), 257-299.
doi: 10.1007/s11538-011-9662-4. |
[16] |
D. Gilbarg and N. Trudinger, "Elliptic Partial Differential Equation of Second Order," 2nd edition, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 224, Springer-Verlag, Berlin, 1983. |
[17] |
J. K. Hale, Dynamical systems and stability, J. Math. Anal. Appl., 26 (1969), 39-59.
doi: 10.1016/0022-247X(69)90175-9. |
[18] |
G. H. Hardy, J. E. Littlewood and G. Pólya, "Inequalities," University Press, Cambridge, UK, 1952. |
[19] |
A. Hastings, Can spatial variation alone lead to selection for dispersal?, Theor. Pop. Biol., 24 (1983), 244-251.
doi: 10.1016/0040-5809(83)90027-8. |
[20] |
P. Hess, "Periodic-Parabolic Boundary Value Problems and Positivity," Pitman Res. Notes Math. Ser., 247, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1991. |
[21] |
J. Huska, Harnack inequality and exponential separation for oblique derivative problems on Lipschitz domains, Journal of Differential Equations, 226 (2006), 541-557. |
[22] |
K. Y. Lam, Concentration phenomena of a semilinear elliptic equation with large advection in an ecological model, J. Differential Equations, 250 (2011), 161-181. |
[23] |
K. Y. Lam, Limiting profiles of semilinear elliptic equations with large advection in population dynamics, II,,, SIAM J. Math Anal., ().
|
[24] |
K. Y. Lam and W. M. Ni, Limiting profiles of semilinear elliptic equations with large advection in population dynamics, Discrete Contin. Dynam. Syst., 28 (2010), 1051-1067.
doi: 10.3934/dcds.2010.28.1051. |
[25] |
Y. Lou, On the effects of migration and spatial heterogeneity on single and multiple species, Journal of Differential Equations, 223 (2006), 400-426. |
[26] |
W.-M. Ni, "The Mathematics of Diffusion," CBMS-NSF Regional Conference Series in Applied Mathematics, 82, SIAM, 2011. |
[27] |
A. Okubo and S. A. Levin, "Diffusion and Ecological Problems: Modern Perspectives, Interdisciplinary Applied Mathematics," Vol. 14, 2nd edition, Springer, Berlin, 2001. |
[28] |
M. H. Protter and H. F. Weinberger, "Maximum Principles in Differential Equations," 2nd edition, Springer-Verlag, Berlin, 1984. |
[29] |
R. Redlinger, Über die $C^2$-kompaktheit der bahn der lösungen semilinearer parabolischer systeme, Proc. Roy. Soc. Edinb. A, 93 (1983), 99-103.
doi: 10.1017/S0308210500031693. |
[30] |
N. Shigesada and K. Kawasaki, "Biological Invasions: Theory and Practice," Oxford Series in Ecology and Evolution, Oxford University Press, Oxford, New York, Tokyo, 1997. |
[31] |
H. Smith, "Monotone Dynamical Systems. An Introduction to the Theory of Competitive and Cooperative Systems," Mathematical Surveys and Monographs, 41, American Mathematical Society, Providence, RI, 1995. |
[32] |
P. Turchin, "Qualitative Analysis of Movement," Sinauer Press, Sunderland, MA, 1998. |
[33] |
E. Zeidler, "Nonlinear Functional Analysis and its Applications. I. Fixed Point Theorems," Springer-Verlag, New York, 1985. |
show all references
References:
[1] |
I. Averill, Y. Lou and D. Munther, On several conjectures from evolution of dispersal,, J. Biol. Dyn., ().
|
[2] |
F. Belgacem, "Elliptic Boundary Value Problems with Indefinite Weights: Variational Formulations of the Principal Eigenvalue and Applications," Pitman Res. Notes Math. Ser., 368, Longman Sci, 1997. |
[3] |
F. Belgacem and C. Cosner, The effects of dispersal along environmental gradients on the dynamics of populations in heterogeneous environment, Canadian Appl. Math. Quarterly, 3 (1995), 379-397. |
[4] |
N. P. Bhatia and G. P. Szegö, "Stability Theory of Dynamical Systems," Springer-Verlag, New York, 1970. |
[5] |
R. S. Cantrell and C. Cosner, Practical persistence in ecological models via comparison methods, Proc. Roy. Soc. Edinb. A, 126 (1996), 247-272.
doi: 10.1017/S0308210500022721. |
[6] |
R. S. Cantrell and C. Cosner, "Spatial Ecology via Reaction-Diffusion Equations," Wiley Series in Mathematical and Computational Biology, John Wiley & Sons, Ltd., Chichester, 2003. |
[7] |
R. S. Cantrell, C. Cosner and Y. Lou, Advection mediated coexistence of competing species, Proc. Roy. Soc. Edinb. A, 137 (2007), 497-518. |
[8] |
R. S. Cantrell, C. Cosner and Y. Lou, Evolution of dispersal and the ideal free distribution, Math. Bios. Eng., 7 (2010), 17-36.
doi: 10.3934/mbe.2010.7.17. |
[9] | |
[10] |
X. F. Chen and Y. Lou, Principal eigenvalue and eigenfunction of an elliptic operator with large convection and its application to a competition model, Indiana Univ. Math. J., 57 (2008), 627-658.
doi: 10.1512/iumj.2008.57.3204. |
[11] |
M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. Funct. Anal., 8 (1971), 321-340.
doi: 10.1016/0022-1236(71)90015-2. |
[12] |
J. Dockery, V. Hutson, K. Mischaikow and M. Pernarowski, The evolution of slow dispersal rates: A reaction-diffusion model, J. Math. Biol., 37 (1998), 61-83.
doi: 10.1007/s002850050120. |
[13] |
S. D. Fretwell and H. L. Lucas, On territorial behavior and other factors influencing habitat selection in birds: Theoretical development, Acta Biotheor., 19 (1970), 16-36.
doi: 10.1007/BF01601953. |
[14] |
A. Friedman, "Partial Differential Equations of Parabolic Type," Prentice-Hall, 1964. |
[15] |
R. Gejji, Y. Lou, D. Munther and J. Peyton, Evolutionary convergence to ideal free dispersal strategies and coexistence, Bull. Math. Biol., 74 (2012), 257-299.
doi: 10.1007/s11538-011-9662-4. |
[16] |
D. Gilbarg and N. Trudinger, "Elliptic Partial Differential Equation of Second Order," 2nd edition, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 224, Springer-Verlag, Berlin, 1983. |
[17] |
J. K. Hale, Dynamical systems and stability, J. Math. Anal. Appl., 26 (1969), 39-59.
doi: 10.1016/0022-247X(69)90175-9. |
[18] |
G. H. Hardy, J. E. Littlewood and G. Pólya, "Inequalities," University Press, Cambridge, UK, 1952. |
[19] |
A. Hastings, Can spatial variation alone lead to selection for dispersal?, Theor. Pop. Biol., 24 (1983), 244-251.
doi: 10.1016/0040-5809(83)90027-8. |
[20] |
P. Hess, "Periodic-Parabolic Boundary Value Problems and Positivity," Pitman Res. Notes Math. Ser., 247, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1991. |
[21] |
J. Huska, Harnack inequality and exponential separation for oblique derivative problems on Lipschitz domains, Journal of Differential Equations, 226 (2006), 541-557. |
[22] |
K. Y. Lam, Concentration phenomena of a semilinear elliptic equation with large advection in an ecological model, J. Differential Equations, 250 (2011), 161-181. |
[23] |
K. Y. Lam, Limiting profiles of semilinear elliptic equations with large advection in population dynamics, II,,, SIAM J. Math Anal., ().
|
[24] |
K. Y. Lam and W. M. Ni, Limiting profiles of semilinear elliptic equations with large advection in population dynamics, Discrete Contin. Dynam. Syst., 28 (2010), 1051-1067.
doi: 10.3934/dcds.2010.28.1051. |
[25] |
Y. Lou, On the effects of migration and spatial heterogeneity on single and multiple species, Journal of Differential Equations, 223 (2006), 400-426. |
[26] |
W.-M. Ni, "The Mathematics of Diffusion," CBMS-NSF Regional Conference Series in Applied Mathematics, 82, SIAM, 2011. |
[27] |
A. Okubo and S. A. Levin, "Diffusion and Ecological Problems: Modern Perspectives, Interdisciplinary Applied Mathematics," Vol. 14, 2nd edition, Springer, Berlin, 2001. |
[28] |
M. H. Protter and H. F. Weinberger, "Maximum Principles in Differential Equations," 2nd edition, Springer-Verlag, Berlin, 1984. |
[29] |
R. Redlinger, Über die $C^2$-kompaktheit der bahn der lösungen semilinearer parabolischer systeme, Proc. Roy. Soc. Edinb. A, 93 (1983), 99-103.
doi: 10.1017/S0308210500031693. |
[30] |
N. Shigesada and K. Kawasaki, "Biological Invasions: Theory and Practice," Oxford Series in Ecology and Evolution, Oxford University Press, Oxford, New York, Tokyo, 1997. |
[31] |
H. Smith, "Monotone Dynamical Systems. An Introduction to the Theory of Competitive and Cooperative Systems," Mathematical Surveys and Monographs, 41, American Mathematical Society, Providence, RI, 1995. |
[32] |
P. Turchin, "Qualitative Analysis of Movement," Sinauer Press, Sunderland, MA, 1998. |
[33] |
E. Zeidler, "Nonlinear Functional Analysis and its Applications. I. Fixed Point Theorems," Springer-Verlag, New York, 1985. |
[1] |
Chris Cosner. Reaction-diffusion-advection models for the effects and evolution of dispersal. Discrete and Continuous Dynamical Systems, 2014, 34 (5) : 1701-1745. doi: 10.3934/dcds.2014.34.1701 |
[2] |
Yixiang Wu, Necibe Tuncer, Maia Martcheva. Coexistence and competitive exclusion in an SIS model with standard incidence and diffusion. Discrete and Continuous Dynamical Systems - B, 2017, 22 (3) : 1167-1187. doi: 10.3934/dcdsb.2017057 |
[3] |
Azmy S. Ackleh, Keng Deng, Yixiang Wu. Competitive exclusion and coexistence in a two-strain pathogen model with diffusion. Mathematical Biosciences & Engineering, 2016, 13 (1) : 1-18. doi: 10.3934/mbe.2016.13.1 |
[4] |
Xinfu Chen, King-Yeung Lam, Yuan Lou. Corrigendum: Dynamics of a reaction-diffusion-advection model for two competing species. Discrete and Continuous Dynamical Systems, 2014, 34 (11) : 4989-4995. doi: 10.3934/dcds.2014.34.4989 |
[5] |
Xinfu Chen, King-Yeung Lam, Yuan Lou. Dynamics of a reaction-diffusion-advection model for two competing species. Discrete and Continuous Dynamical Systems, 2012, 32 (11) : 3841-3859. doi: 10.3934/dcds.2012.32.3841 |
[6] |
Kwangjoong Kim, Wonhyung Choi, Inkyung Ahn. Reaction-advection-diffusion competition models under lethal boundary conditions. Discrete and Continuous Dynamical Systems - B, 2021 doi: 10.3934/dcdsb.2021250 |
[7] |
Hao Wang, Katherine Dunning, James J. Elser, Yang Kuang. Daphnia species invasion, competitive exclusion, and chaotic coexistence. Discrete and Continuous Dynamical Systems - B, 2009, 12 (2) : 481-493. doi: 10.3934/dcdsb.2009.12.481 |
[8] |
Alain Rapaport, Mario Veruete. A new proof of the competitive exclusion principle in the chemostat. Discrete and Continuous Dynamical Systems - B, 2019, 24 (8) : 3755-3764. doi: 10.3934/dcdsb.2018314 |
[9] |
Robert Stephen Cantrell, King-Yeung Lam. Competitive exclusion in phytoplankton communities in a eutrophic water column. Discrete and Continuous Dynamical Systems - B, 2021, 26 (4) : 1783-1795. doi: 10.3934/dcdsb.2020361 |
[10] |
M. R. S. Kulenović, Orlando Merino. Competitive-exclusion versus competitive-coexistence for systems in the plane. Discrete and Continuous Dynamical Systems - B, 2006, 6 (5) : 1141-1156. doi: 10.3934/dcdsb.2006.6.1141 |
[11] |
H. L. Smith, X. Q. Zhao. Competitive exclusion in a discrete-time, size-structured chemostat model. Discrete and Continuous Dynamical Systems - B, 2001, 1 (2) : 183-191. doi: 10.3934/dcdsb.2001.1.183 |
[12] |
Azmy S. Ackleh, Youssef M. Dib, S. R.-J. Jang. Competitive exclusion and coexistence in a nonlinear refuge-mediated selection model. Discrete and Continuous Dynamical Systems - B, 2007, 7 (4) : 683-698. doi: 10.3934/dcdsb.2007.7.683 |
[13] |
Dan Li, Hui Wan. Coexistence and exclusion of competitive Kolmogorov systems with semi-Markovian switching. Discrete and Continuous Dynamical Systems, 2021, 41 (9) : 4145-4183. doi: 10.3934/dcds.2021032 |
[14] |
Kokum R. De Silva, Tuoc V. Phan, Suzanne Lenhart. Advection control in parabolic PDE systems for competitive populations. Discrete and Continuous Dynamical Systems - B, 2017, 22 (3) : 1049-1072. doi: 10.3934/dcdsb.2017052 |
[15] |
Zhen-Hui Bu, Zhi-Cheng Wang. Curved fronts of monostable reaction-advection-diffusion equations in space-time periodic media. Communications on Pure and Applied Analysis, 2016, 15 (1) : 139-160. doi: 10.3934/cpaa.2016.15.139 |
[16] |
Renhao Cui. Asymptotic profiles of the endemic equilibrium of a reaction-diffusion-advection SIS epidemic model with saturated incidence rate. Discrete and Continuous Dynamical Systems - B, 2021, 26 (6) : 2997-3022. doi: 10.3934/dcdsb.2020217 |
[17] |
Bo Duan, Zhengce Zhang. A two-species weak competition system of reaction-diffusion-advection with double free boundaries. Discrete and Continuous Dynamical Systems - B, 2019, 24 (2) : 801-829. doi: 10.3934/dcdsb.2018208 |
[18] |
Mostafa Bendahmane, Kenneth H. Karlsen. Renormalized solutions of an anisotropic reaction-diffusion-advection system with $L^1$ data. Communications on Pure and Applied Analysis, 2006, 5 (4) : 733-762. doi: 10.3934/cpaa.2006.5.733 |
[19] |
Danhua Jiang, Zhi-Cheng Wang, Liang Zhang. A reaction-diffusion-advection SIS epidemic model in a spatially-temporally heterogeneous environment. Discrete and Continuous Dynamical Systems - B, 2018, 23 (10) : 4557-4578. doi: 10.3934/dcdsb.2018176 |
[20] |
Shi-Liang Wu, Wan-Tong Li, San-Yang Liu. Exponential stability of traveling fronts in monostable reaction-advection-diffusion equations with non-local delay. Discrete and Continuous Dynamical Systems - B, 2012, 17 (1) : 347-366. doi: 10.3934/dcdsb.2012.17.347 |
2020 Impact Factor: 1.392
Tools
Metrics
Other articles
by authors
[Back to Top]