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On the equivalent classification of threedimensional competitive Atkinson/Allen models relative to the boundary fixed points
1.  Mathematics and Science College, Shanghai Normal University, Shanghai 200234, China, China 
References:
[1] 
L. J. S. Allen, E. J. Allen and D. N. Atkinson, Integrodifference equations applied to plant dispersal, competition, and control, in Differential Equations with Applications to Biology Fields Institute Communications (eds. S. Ruan, G. Wolkowicz and J. Wu), American Mathematical Society, RI, 21 (1999), 1530. 
[2] 
D. N. Atkinson, Mathematical Models for Plant Competition and Dispersal, Master's Thesis, Texas Tech University, Lubbock, TX 79409, 1997. 
[3] 
J. Campos, R. Ortega and A. Tineo, Homeomorphisms of the disk with trivial dynamics and extinction of competitive systems, J. Diff. Eqns., 138 (1997), 157170. doi: 10.1006/jdeq.1997.3265. 
[4] 
C. W. Chi, S. B. Hsu and L. I. Wu, On the asymmetric MayLeonard model of three competing species, SIAM J. Appl. Math., 58 (1998), 211226. doi: 10.1137/S0036139994272060. 
[5] 
J. M. Cushing, S. Levarge, N. Chitnis and S. M. Henson, Some discrete competition models and the competitive exclusion principle, J. Diff. Equ. Appl., 10 (2004), 11391151. doi: 10.1080/10236190410001652739. 
[6] 
O. Diekmann, Y. Wang and P. Yan, Carrying simplices in discrete competitive systems and agestructured semelparous populations, Discrete Contin. Dyn. Syst., 20 (2008), 3752. 
[7] 
J. Hale and H. Koçak, Dynamics and Bifurcations, SpringerVerlag, New York, 1991. doi: 10.1007/9781461244264. 
[8] 
M. W. Hirsch, Systems of differential equations which are competitive or cooperative: III. Competing species, Nonlinearity, 1 (1988), 5171. doi: 10.1088/09517715/1/1/003. 
[9] 
J. Hofbauer, V. Hutson and W. Jansen, Coexistence for systems governed by difference equations of LotkaVolterra type, J. Math. Biol., 25 (1987), 553570. doi: 10.1007/BF00276199. 
[10] 
M. C. Irwin, Smooth Dynamical Systems, Academic Press, New York, 1980. 
[11] 
H. Jiang and T. D. Rogers, The discrete dynamics of symmetric competition in the plane, J. Math. Biol., 25 (1987), 573596. doi: 10.1007/BF00275495. 
[12] 
P. Liu and S. N. Elaydi, Discrete competitive and cooperative models of LotkaVolterra type, J. Comp. Anal. Appl., 3 (2001), 5373. doi: 10.1023/A:1011539901001. 
[13] 
J. E. Marsden and M. McCracken, The Hopf Bifurcation and Its Applications, SpringerVerlag, New York, 1976. 
[14] 
J. D. Murray, Mathematical Biology, SpringerVerlag, New York, 1993. doi: 10.1007/b98869. 
[15] 
A. Pakes and R. Maller, Mathematical Ecology of Plant Species Competition: A Class of Deterministic Models for Binary Mixtures of Plant Genotypes, Cambridge Univ. Press, Cambridge, 1990. doi: 10.1016/00207519(90)901002. 
[16] 
L.I. W. Roeger and L. J. S. Allen, Discrete MayLeonard competition models I, J. Diff. Equ. Appl., 10 (2004), 7798. doi: 10.1080/10236190310001603662. 
[17] 
A. RuizHerrera, Exclusion and dominance in discrete population models via the carrying simplex, J. Diff. Equ. Appl., 19 (2013), 96113. doi: 10.1080/10236198.2011.628663. 
[18] 
H. L. Smith, Periodic competitive differential equations and the discrete dynamics of competitive maps, J. Diff. Eqns., 64 (1986), 165194. doi: 10.1016/00220396(86)900860. 
[19] 
H. L. Smith, Planar competitive and cooperative difference equations, J. Diff. Equ. Appl., 3 (1998), 335357. doi: 10.1080/10236199708808108. 
[20] 
Y. Wang and J. Jiang, Uniqueness and attractivity of the carrying simplex for discretetime competitive dynamical systems, J. Diff. Eqns., 186 (2002), 611632. doi: 10.1016/S00220396(02)000256. 
[21] 
M. L. Zeeman, Hopf bifurcations in competitive threedimensional LotkaVolterra systems, Dynam. Stability Systems, 8 (1993), 189217. doi: 10.1080/02681119308806158. 
show all references
References:
[1] 
L. J. S. Allen, E. J. Allen and D. N. Atkinson, Integrodifference equations applied to plant dispersal, competition, and control, in Differential Equations with Applications to Biology Fields Institute Communications (eds. S. Ruan, G. Wolkowicz and J. Wu), American Mathematical Society, RI, 21 (1999), 1530. 
[2] 
D. N. Atkinson, Mathematical Models for Plant Competition and Dispersal, Master's Thesis, Texas Tech University, Lubbock, TX 79409, 1997. 
[3] 
J. Campos, R. Ortega and A. Tineo, Homeomorphisms of the disk with trivial dynamics and extinction of competitive systems, J. Diff. Eqns., 138 (1997), 157170. doi: 10.1006/jdeq.1997.3265. 
[4] 
C. W. Chi, S. B. Hsu and L. I. Wu, On the asymmetric MayLeonard model of three competing species, SIAM J. Appl. Math., 58 (1998), 211226. doi: 10.1137/S0036139994272060. 
[5] 
J. M. Cushing, S. Levarge, N. Chitnis and S. M. Henson, Some discrete competition models and the competitive exclusion principle, J. Diff. Equ. Appl., 10 (2004), 11391151. doi: 10.1080/10236190410001652739. 
[6] 
O. Diekmann, Y. Wang and P. Yan, Carrying simplices in discrete competitive systems and agestructured semelparous populations, Discrete Contin. Dyn. Syst., 20 (2008), 3752. 
[7] 
J. Hale and H. Koçak, Dynamics and Bifurcations, SpringerVerlag, New York, 1991. doi: 10.1007/9781461244264. 
[8] 
M. W. Hirsch, Systems of differential equations which are competitive or cooperative: III. Competing species, Nonlinearity, 1 (1988), 5171. doi: 10.1088/09517715/1/1/003. 
[9] 
J. Hofbauer, V. Hutson and W. Jansen, Coexistence for systems governed by difference equations of LotkaVolterra type, J. Math. Biol., 25 (1987), 553570. doi: 10.1007/BF00276199. 
[10] 
M. C. Irwin, Smooth Dynamical Systems, Academic Press, New York, 1980. 
[11] 
H. Jiang and T. D. Rogers, The discrete dynamics of symmetric competition in the plane, J. Math. Biol., 25 (1987), 573596. doi: 10.1007/BF00275495. 
[12] 
P. Liu and S. N. Elaydi, Discrete competitive and cooperative models of LotkaVolterra type, J. Comp. Anal. Appl., 3 (2001), 5373. doi: 10.1023/A:1011539901001. 
[13] 
J. E. Marsden and M. McCracken, The Hopf Bifurcation and Its Applications, SpringerVerlag, New York, 1976. 
[14] 
J. D. Murray, Mathematical Biology, SpringerVerlag, New York, 1993. doi: 10.1007/b98869. 
[15] 
A. Pakes and R. Maller, Mathematical Ecology of Plant Species Competition: A Class of Deterministic Models for Binary Mixtures of Plant Genotypes, Cambridge Univ. Press, Cambridge, 1990. doi: 10.1016/00207519(90)901002. 
[16] 
L.I. W. Roeger and L. J. S. Allen, Discrete MayLeonard competition models I, J. Diff. Equ. Appl., 10 (2004), 7798. doi: 10.1080/10236190310001603662. 
[17] 
A. RuizHerrera, Exclusion and dominance in discrete population models via the carrying simplex, J. Diff. Equ. Appl., 19 (2013), 96113. doi: 10.1080/10236198.2011.628663. 
[18] 
H. L. Smith, Periodic competitive differential equations and the discrete dynamics of competitive maps, J. Diff. Eqns., 64 (1986), 165194. doi: 10.1016/00220396(86)900860. 
[19] 
H. L. Smith, Planar competitive and cooperative difference equations, J. Diff. Equ. Appl., 3 (1998), 335357. doi: 10.1080/10236199708808108. 
[20] 
Y. Wang and J. Jiang, Uniqueness and attractivity of the carrying simplex for discretetime competitive dynamical systems, J. Diff. Eqns., 186 (2002), 611632. doi: 10.1016/S00220396(02)000256. 
[21] 
M. L. Zeeman, Hopf bifurcations in competitive threedimensional LotkaVolterra systems, Dynam. Stability Systems, 8 (1993), 189217. doi: 10.1080/02681119308806158. 
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