Dynamics of a three species competition model

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September 2012, 32(9): 3099-3131. doi: 10.3934/dcds.2012.32.3099

Dynamics of a three species competition model

Yuan Lou ^1, and Daniel Munther ^2,

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

Received January 2012 Revised March 2012 Published April 2012

Abstract
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We investigate the dynamics of a three species competition model, in which all species have the same population dynamics but distinct dispersal strategies. Gejji et al. [15] introduced a general dispersal strategy for two species, termed as an ideal free pair in this paper, which can result in the ideal free distributions of two competing species at equilibrium. We show that if one of the three species adopts a dispersal strategy which produces the ideal free distribution, then none of the other two species can persist if they do not form an ideal free pair. We also show that if two species form an ideal free pair, then the third species in general can not invade. When none of the three species is adopting a dispersal strategy which can produce the ideal free distribution, we find some class of resource functions such that three species competing for the same resource can be ecologically permanent by using distinct dispersal strategies.

Keywords: competitive exclusion, Dispersal, permanence, reaction-diffusion-advection..

Mathematics Subject Classification: Primary: 35K57; Secondary: 92D2.

Citation: Yuan Lou, Daniel Munther. Dynamics of a three species competition model. Discrete and Continuous Dynamical Systems, 2012, 32 (9) : 3099-3131. doi: 10.3934/dcds.2012.32.3099

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), 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]	, Chris Cosner, private communication.
[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," 2^nd 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., 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), 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, 2^nd edition, Springer, Berlin, 2001.
[28]	M. H. Protter and H. F. Weinberger, "Maximum Principles in Differential Equations," 2^nd 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., 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), 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]	, Chris Cosner, private communication.
[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," 2^nd 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., 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), 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, 2^nd edition, Springer, Berlin, 2001.
[28]	M. H. Protter and H. F. Weinberger, "Maximum Principles in Differential Equations," 2^nd 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.

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