# American Institute of Mathematical Sciences

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

## 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

Received  January 2012 Revised  March 2012 Published  April 2012

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.
Citation: Yuan Lou, Daniel Munther. Dynamics of a three species competition model. Discrete & 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., ().   Google Scholar [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. Google Scholar [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.  Google Scholar [4] N. P. Bhatia and G. P. Szegö, "Stability Theory of Dynamical Systems," Springer-Verlag, New York, 1970. Google Scholar [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.  Google Scholar [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.  Google Scholar [7] R. S. Cantrell, C. Cosner and Y. Lou, Advection mediated coexistence of competing species, Proc. Roy. Soc. Edinb. A, 137 (2007), 497-518. Google Scholar [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.  Google Scholar [9] , Chris Cosner,, private communication., ().   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [14] A. Friedman, "Partial Differential Equations of Parabolic Type," Prentice-Hall, 1964. Google Scholar [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.  Google Scholar [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.  Google Scholar [17] J. K. Hale, Dynamical systems and stability, J. Math. Anal. Appl., 26 (1969), 39-59. doi: 10.1016/0022-247X(69)90175-9.  Google Scholar [18] G. H. Hardy, J. E. Littlewood and G. Pólya, "Inequalities," University Press, Cambridge, UK, 1952. Google Scholar [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.  Google Scholar [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.  Google Scholar [21] J. Huska, Harnack inequality and exponential separation for oblique derivative problems on Lipschitz domains, Journal of Differential Equations, 226 (2006), 541-557. Google Scholar [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. Google Scholar [23] K. Y. Lam, Limiting profiles of semilinear elliptic equations with large advection in population dynamics, II,,, SIAM J. Math Anal., ().   Google Scholar [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.  Google Scholar [25] Y. Lou, On the effects of migration and spatial heterogeneity on single and multiple species, Journal of Differential Equations, 223 (2006), 400-426. Google Scholar [26] W.-M. Ni, "The Mathematics of Diffusion," CBMS-NSF Regional Conference Series in Applied Mathematics, 82, SIAM, 2011. Google Scholar [27] A. Okubo and S. A. Levin, "Diffusion and Ecological Problems: Modern Perspectives, Interdisciplinary Applied Mathematics," Vol. 14, 2nd edition, Springer, Berlin, 2001. Google Scholar [28] M. H. Protter and H. F. Weinberger, "Maximum Principles in Differential Equations," 2nd edition, Springer-Verlag, Berlin, 1984. Google Scholar [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.  Google Scholar [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. Google Scholar [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.  Google Scholar [32] P. Turchin, "Qualitative Analysis of Movement," Sinauer Press, Sunderland, MA, 1998. Google Scholar [33] E. Zeidler, "Nonlinear Functional Analysis and its Applications. I. Fixed Point Theorems," Springer-Verlag, New York, 1985. Google Scholar

show all references

##### References:
 [1] I. Averill, Y. Lou and D. Munther, On several conjectures from evolution of dispersal,, J. Biol. Dyn., ().   Google Scholar [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. Google Scholar [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.  Google Scholar [4] N. P. Bhatia and G. P. Szegö, "Stability Theory of Dynamical Systems," Springer-Verlag, New York, 1970. Google Scholar [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.  Google Scholar [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.  Google Scholar [7] R. S. Cantrell, C. Cosner and Y. Lou, Advection mediated coexistence of competing species, Proc. Roy. Soc. Edinb. A, 137 (2007), 497-518. Google Scholar [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.  Google Scholar [9] , Chris Cosner,, private communication., ().   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [14] A. Friedman, "Partial Differential Equations of Parabolic Type," Prentice-Hall, 1964. Google Scholar [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.  Google Scholar [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.  Google Scholar [17] J. K. Hale, Dynamical systems and stability, J. Math. Anal. Appl., 26 (1969), 39-59. doi: 10.1016/0022-247X(69)90175-9.  Google Scholar [18] G. H. Hardy, J. E. Littlewood and G. Pólya, "Inequalities," University Press, Cambridge, UK, 1952. Google Scholar [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.  Google Scholar [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.  Google Scholar [21] J. Huska, Harnack inequality and exponential separation for oblique derivative problems on Lipschitz domains, Journal of Differential Equations, 226 (2006), 541-557. Google Scholar [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. Google Scholar [23] K. Y. Lam, Limiting profiles of semilinear elliptic equations with large advection in population dynamics, II,,, SIAM J. Math Anal., ().   Google Scholar [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.  Google Scholar [25] Y. Lou, On the effects of migration and spatial heterogeneity on single and multiple species, Journal of Differential Equations, 223 (2006), 400-426. Google Scholar [26] W.-M. Ni, "The Mathematics of Diffusion," CBMS-NSF Regional Conference Series in Applied Mathematics, 82, SIAM, 2011. Google Scholar [27] A. Okubo and S. A. Levin, "Diffusion and Ecological Problems: Modern Perspectives, Interdisciplinary Applied Mathematics," Vol. 14, 2nd edition, Springer, Berlin, 2001. Google Scholar [28] M. H. Protter and H. F. Weinberger, "Maximum Principles in Differential Equations," 2nd edition, Springer-Verlag, Berlin, 1984. Google Scholar [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.  Google Scholar [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. Google Scholar [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.  Google Scholar [32] P. Turchin, "Qualitative Analysis of Movement," Sinauer Press, Sunderland, MA, 1998. Google Scholar [33] E. Zeidler, "Nonlinear Functional Analysis and its Applications. I. Fixed Point Theorems," Springer-Verlag, New York, 1985. Google Scholar
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