
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., in press. 
[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), 379397. 
[4] 
N. P. Bhatia and G. P. Szegö, "Stability Theory of Dynamical Systems," SpringerVerlag, 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), 247272. doi: 10.1017/S0308210500022721. 
[6] 
R. S. Cantrell and C. Cosner, "Spatial Ecology via ReactionDiffusion 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), 497518. 
[8] 
R. S. Cantrell, C. Cosner and Y. Lou, Evolution of dispersal and the ideal free distribution, Math. Bios. Eng., 7 (2010), 1736. 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), 627658. doi: 10.1512/iumj.2008.57.3204. 
[11] 
M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. Funct. Anal., 8 (1971), 321340. doi: 10.1016/00221236(71)900152. 
[12] 
J. Dockery, V. Hutson, K. Mischaikow and M. Pernarowski, The evolution of slow dispersal rates: A reactiondiffusion model, J. Math. Biol., 37 (1998), 6183. 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), 1636. doi: 10.1007/BF01601953. 
[14] 
A. Friedman, "Partial Differential Equations of Parabolic Type," PrenticeHall, 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), 257299. doi: 10.1007/s1153801196624. 
[16] 
D. Gilbarg and N. Trudinger, "Elliptic Partial Differential Equation of Second Order," 2^{nd} edition, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 224, SpringerVerlag, Berlin, 1983. 
[17] 
J. K. Hale, Dynamical systems and stability, J. Math. Anal. Appl., 26 (1969), 3959. doi: 10.1016/0022247X(69)901759. 
[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), 244251. doi: 10.1016/00405809(83)900278. 
[20] 
P. Hess, "PeriodicParabolic 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), 541557. 
[22] 
K. Y. Lam, Concentration phenomena of a semilinear elliptic equation with large advection in an ecological model, J. Differential Equations, 250 (2011), 161181. 
[23] 
K. Y. Lam, Limiting profiles of semilinear elliptic equations with large advection in population dynamics, II,, SIAM J. Math Anal., in press. 
[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), 10511067. 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), 400426. 
[26] 
W.M. Ni, "The Mathematics of Diffusion," CBMSNSF 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, 2^{nd} edition, Springer, Berlin, 2001. 
[28] 
M. H. Protter and H. F. Weinberger, "Maximum Principles in Differential Equations," 2^{nd} edition, SpringerVerlag, Berlin, 1984. 
[29] 
R. Redlinger, Über die $C^2$kompaktheit der bahn der lösungen semilinearer parabolischer systeme, Proc. Roy. Soc. Edinb. A, 93 (1983), 99103. 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," SpringerVerlag, 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., in press. 
[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), 379397. 
[4] 
N. P. Bhatia and G. P. Szegö, "Stability Theory of Dynamical Systems," SpringerVerlag, 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), 247272. doi: 10.1017/S0308210500022721. 
[6] 
R. S. Cantrell and C. Cosner, "Spatial Ecology via ReactionDiffusion 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), 497518. 
[8] 
R. S. Cantrell, C. Cosner and Y. Lou, Evolution of dispersal and the ideal free distribution, Math. Bios. Eng., 7 (2010), 1736. 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), 627658. doi: 10.1512/iumj.2008.57.3204. 
[11] 
M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. Funct. Anal., 8 (1971), 321340. doi: 10.1016/00221236(71)900152. 
[12] 
J. Dockery, V. Hutson, K. Mischaikow and M. Pernarowski, The evolution of slow dispersal rates: A reactiondiffusion model, J. Math. Biol., 37 (1998), 6183. 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), 1636. doi: 10.1007/BF01601953. 
[14] 
A. Friedman, "Partial Differential Equations of Parabolic Type," PrenticeHall, 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), 257299. doi: 10.1007/s1153801196624. 
[16] 
D. Gilbarg and N. Trudinger, "Elliptic Partial Differential Equation of Second Order," 2^{nd} edition, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 224, SpringerVerlag, Berlin, 1983. 
[17] 
J. K. Hale, Dynamical systems and stability, J. Math. Anal. Appl., 26 (1969), 3959. doi: 10.1016/0022247X(69)901759. 
[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), 244251. doi: 10.1016/00405809(83)900278. 
[20] 
P. Hess, "PeriodicParabolic 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), 541557. 
[22] 
K. Y. Lam, Concentration phenomena of a semilinear elliptic equation with large advection in an ecological model, J. Differential Equations, 250 (2011), 161181. 
[23] 
K. Y. Lam, Limiting profiles of semilinear elliptic equations with large advection in population dynamics, II,, SIAM J. Math Anal., in press. 
[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), 10511067. 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), 400426. 
[26] 
W.M. Ni, "The Mathematics of Diffusion," CBMSNSF 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, 2^{nd} edition, Springer, Berlin, 2001. 
[28] 
M. H. Protter and H. F. Weinberger, "Maximum Principles in Differential Equations," 2^{nd} edition, SpringerVerlag, Berlin, 1984. 
[29] 
R. Redlinger, Über die $C^2$kompaktheit der bahn der lösungen semilinearer parabolischer systeme, Proc. Roy. Soc. Edinb. A, 93 (1983), 99103. 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," SpringerVerlag, New York, 1985. 
[1] 
Chris Cosner. Reactiondiffusionadvection models for the effects and evolution of dispersal. Discrete and Continuous Dynamical Systems, 2014, 34 (5) : 17011745. 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) : 11671187. doi: 10.3934/dcdsb.2017057 
[3] 
Azmy S. Ackleh, Keng Deng, Yixiang Wu. Competitive exclusion and coexistence in a twostrain pathogen model with diffusion. Mathematical Biosciences & Engineering, 2016, 13 (1) : 118. doi: 10.3934/mbe.2016.13.1 
[4] 
Xinfu Chen, KingYeung Lam, Yuan Lou. Corrigendum: Dynamics of a reactiondiffusionadvection model for two competing species. Discrete and Continuous Dynamical Systems, 2014, 34 (11) : 49894995. doi: 10.3934/dcds.2014.34.4989 
[5] 
Xinfu Chen, KingYeung Lam, Yuan Lou. Dynamics of a reactiondiffusionadvection model for two competing species. Discrete and Continuous Dynamical Systems, 2012, 32 (11) : 38413859. doi: 10.3934/dcds.2012.32.3841 
[6] 
Kwangjoong Kim, Wonhyung Choi, Inkyung Ahn. Reactionadvectiondiffusion competition models under lethal boundary conditions. Discrete and Continuous Dynamical Systems  B, 2022, 27 (9) : 47494767. 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) : 481493. 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) : 37553764. doi: 10.3934/dcdsb.2018314 
[9] 
Robert Stephen Cantrell, KingYeung Lam. Competitive exclusion in phytoplankton communities in a eutrophic water column. Discrete and Continuous Dynamical Systems  B, 2021, 26 (4) : 17831795. doi: 10.3934/dcdsb.2020361 
[10] 
M. R. S. Kulenović, Orlando Merino. Competitiveexclusion versus competitivecoexistence for systems in the plane. Discrete and Continuous Dynamical Systems  B, 2006, 6 (5) : 11411156. doi: 10.3934/dcdsb.2006.6.1141 
[11] 
H. L. Smith, X. Q. Zhao. Competitive exclusion in a discretetime, sizestructured chemostat model. Discrete and Continuous Dynamical Systems  B, 2001, 1 (2) : 183191. 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 refugemediated selection model. Discrete and Continuous Dynamical Systems  B, 2007, 7 (4) : 683698. doi: 10.3934/dcdsb.2007.7.683 
[13] 
Dan Li, Hui Wan. Coexistence and exclusion of competitive Kolmogorov systems with semiMarkovian switching. Discrete and Continuous Dynamical Systems, 2021, 41 (9) : 41454183. 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) : 10491072. doi: 10.3934/dcdsb.2017052 
[15] 
ZhenHui Bu, ZhiCheng Wang. Curved fronts of monostable reactionadvectiondiffusion equations in spacetime periodic media. Communications on Pure and Applied Analysis, 2016, 15 (1) : 139160. doi: 10.3934/cpaa.2016.15.139 
[16] 
Renhao Cui. Asymptotic profiles of the endemic equilibrium of a reactiondiffusionadvection SIS epidemic model with saturated incidence rate. Discrete and Continuous Dynamical Systems  B, 2021, 26 (6) : 29973022. doi: 10.3934/dcdsb.2020217 
[17] 
Bo Duan, Zhengce Zhang. A twospecies weak competition system of reactiondiffusionadvection with double free boundaries. Discrete and Continuous Dynamical Systems  B, 2019, 24 (2) : 801829. doi: 10.3934/dcdsb.2018208 
[18] 
Mostafa Bendahmane, Kenneth H. Karlsen. Renormalized solutions of an anisotropic reactiondiffusionadvection system with $L^1$ data. Communications on Pure and Applied Analysis, 2006, 5 (4) : 733762. doi: 10.3934/cpaa.2006.5.733 
[19] 
Danhua Jiang, ZhiCheng Wang, Liang Zhang. A reactiondiffusionadvection SIS epidemic model in a spatiallytemporally heterogeneous environment. Discrete and Continuous Dynamical Systems  B, 2018, 23 (10) : 45574578. doi: 10.3934/dcdsb.2018176 
[20] 
ShiLiang Wu, WanTong Li, SanYang Liu. Exponential stability of traveling fronts in monostable reactionadvectiondiffusion equations with nonlocal delay. Discrete and Continuous Dynamical Systems  B, 2012, 17 (1) : 347366. doi: 10.3934/dcdsb.2012.17.347 
2021 Impact Factor: 1.588
Tools
Metrics
Other articles
by authors
[Back to Top]