January  2016, 36(1): 217-244. doi: 10.3934/dcds.2016.36.217

On the equivalent classification of three-dimensional competitive Atkinson/Allen models relative to the boundary fixed points

1. 

Mathematics and Science College, Shanghai Normal University, Shanghai 200234, China, China

Received  July 2014 Revised  April 2015 Published  June 2015

We study the fixed point index on the carrying simplex of the competitive map. The sum of the indices of the fixed points on the carrying simplex for the three-dimensional competitive map is unit. Based on that, we analyze the asymptotic behavior of the three-dimensional competitive Atkinson/Allen model. We present all the equivalence classes relative to the boundary of the carrying simplex of the low-dimensional (two or three) map, depending upon relationship among the model coefficients. For the two-dimensional case, there are only three dynamic scenarios, and every orbit converges to some fixed point. For the three-dimensional case, there are total $33$ stable equivalence classes, and in $18$ of them all the compact limit sets are fixed points. Further, we focus on the analysis of the dynamics of the other $15$ cases. Hopf bifurcation is studied and a necessary condition for it occurring is given, which implies that the classes $19$-$25$, $28$, $30$ and $32$ do not have any Hopf bifurcation. However, the class $26$ and class $27$ do admit Hopf bifurcations, which means that these two classes may have isolated invariant closed curves in their carrying simplex, and such an invariant closed curve corresponds to either a subharmonic or a quasiperiodic solution in continuous time systems. Each system in class $27$ has a heteroclinic cycle and the numerical simulation also reveals that there exist systems having May-Leonard phenomenon: the existence of nonquasiperiodic oscillation.
Citation: Jifa Jiang, Lei Niu. On the equivalent classification of three-dimensional competitive Atkinson/Allen models relative to the boundary fixed points. Discrete & Continuous Dynamical Systems - A, 2016, 36 (1) : 217-244. doi: 10.3934/dcds.2016.36.217
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, 21 (1999), 15.   Google Scholar

[2]

D. N. Atkinson, Mathematical Models for Plant Competition and Dispersal,, Master's Thesis, (7940).   Google Scholar

[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), 157.  doi: 10.1006/jdeq.1997.3265.  Google Scholar

[4]

C. W. Chi, S. B. Hsu and L. I. Wu, On the asymmetric May-Leonard model of three competing species,, SIAM J. Appl. Math., 58 (1998), 211.  doi: 10.1137/S0036139994272060.  Google Scholar

[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), 1139.  doi: 10.1080/10236190410001652739.  Google Scholar

[6]

O. Diekmann, Y. Wang and P. Yan, Carrying simplices in discrete competitive systems and age-structured semelparous populations,, Discrete Contin. Dyn. Syst., 20 (2008), 37.   Google Scholar

[7]

J. Hale and H. Koçak, Dynamics and Bifurcations,, Springer-Verlag, (1991).  doi: 10.1007/978-1-4612-4426-4.  Google Scholar

[8]

M. W. Hirsch, Systems of differential equations which are competitive or cooperative: III. Competing species,, Nonlinearity, 1 (1988), 51.  doi: 10.1088/0951-7715/1/1/003.  Google Scholar

[9]

J. Hofbauer, V. Hutson and W. Jansen, Coexistence for systems governed by difference equations of Lotka-Volterra type,, J. Math. Biol., 25 (1987), 553.  doi: 10.1007/BF00276199.  Google Scholar

[10]

M. C. Irwin, Smooth Dynamical Systems,, Academic Press, (1980).   Google Scholar

[11]

H. Jiang and T. D. Rogers, The discrete dynamics of symmetric competition in the plane,, J. Math. Biol., 25 (1987), 573.  doi: 10.1007/BF00275495.  Google Scholar

[12]

P. Liu and S. N. Elaydi, Discrete competitive and cooperative models of Lotka-Volterra type,, J. Comp. Anal. Appl., 3 (2001), 53.  doi: 10.1023/A:1011539901001.  Google Scholar

[13]

J. E. Marsden and M. McCracken, The Hopf Bifurcation and Its Applications,, Springer-Verlag, (1976).   Google Scholar

[14]

J. D. Murray, Mathematical Biology,, Springer-Verlag, (1993).  doi: 10.1007/b98869.  Google Scholar

[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, (1990).  doi: 10.1016/0020-7519(90)90100-2.  Google Scholar

[16]

L.-I. W. Roeger and L. J. S. Allen, Discrete May-Leonard competition models I,, J. Diff. Equ. Appl., 10 (2004), 77.  doi: 10.1080/10236190310001603662.  Google Scholar

[17]

A. Ruiz-Herrera, Exclusion and dominance in discrete population models via the carrying simplex,, J. Diff. Equ. Appl., 19 (2013), 96.  doi: 10.1080/10236198.2011.628663.  Google Scholar

[18]

H. L. Smith, Periodic competitive differential equations and the discrete dynamics of competitive maps,, J. Diff. Eqns., 64 (1986), 165.  doi: 10.1016/0022-0396(86)90086-0.  Google Scholar

[19]

H. L. Smith, Planar competitive and cooperative difference equations,, J. Diff. Equ. Appl., 3 (1998), 335.  doi: 10.1080/10236199708808108.  Google Scholar

[20]

Y. Wang and J. Jiang, Uniqueness and attractivity of the carrying simplex for discrete-time competitive dynamical systems,, J. Diff. Eqns., 186 (2002), 611.  doi: 10.1016/S0022-0396(02)00025-6.  Google Scholar

[21]

M. L. Zeeman, Hopf bifurcations in competitive three-dimensional Lotka-Volterra systems,, Dynam. Stability Systems, 8 (1993), 189.  doi: 10.1080/02681119308806158.  Google Scholar

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, 21 (1999), 15.   Google Scholar

[2]

D. N. Atkinson, Mathematical Models for Plant Competition and Dispersal,, Master's Thesis, (7940).   Google Scholar

[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), 157.  doi: 10.1006/jdeq.1997.3265.  Google Scholar

[4]

C. W. Chi, S. B. Hsu and L. I. Wu, On the asymmetric May-Leonard model of three competing species,, SIAM J. Appl. Math., 58 (1998), 211.  doi: 10.1137/S0036139994272060.  Google Scholar

[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), 1139.  doi: 10.1080/10236190410001652739.  Google Scholar

[6]

O. Diekmann, Y. Wang and P. Yan, Carrying simplices in discrete competitive systems and age-structured semelparous populations,, Discrete Contin. Dyn. Syst., 20 (2008), 37.   Google Scholar

[7]

J. Hale and H. Koçak, Dynamics and Bifurcations,, Springer-Verlag, (1991).  doi: 10.1007/978-1-4612-4426-4.  Google Scholar

[8]

M. W. Hirsch, Systems of differential equations which are competitive or cooperative: III. Competing species,, Nonlinearity, 1 (1988), 51.  doi: 10.1088/0951-7715/1/1/003.  Google Scholar

[9]

J. Hofbauer, V. Hutson and W. Jansen, Coexistence for systems governed by difference equations of Lotka-Volterra type,, J. Math. Biol., 25 (1987), 553.  doi: 10.1007/BF00276199.  Google Scholar

[10]

M. C. Irwin, Smooth Dynamical Systems,, Academic Press, (1980).   Google Scholar

[11]

H. Jiang and T. D. Rogers, The discrete dynamics of symmetric competition in the plane,, J. Math. Biol., 25 (1987), 573.  doi: 10.1007/BF00275495.  Google Scholar

[12]

P. Liu and S. N. Elaydi, Discrete competitive and cooperative models of Lotka-Volterra type,, J. Comp. Anal. Appl., 3 (2001), 53.  doi: 10.1023/A:1011539901001.  Google Scholar

[13]

J. E. Marsden and M. McCracken, The Hopf Bifurcation and Its Applications,, Springer-Verlag, (1976).   Google Scholar

[14]

J. D. Murray, Mathematical Biology,, Springer-Verlag, (1993).  doi: 10.1007/b98869.  Google Scholar

[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, (1990).  doi: 10.1016/0020-7519(90)90100-2.  Google Scholar

[16]

L.-I. W. Roeger and L. J. S. Allen, Discrete May-Leonard competition models I,, J. Diff. Equ. Appl., 10 (2004), 77.  doi: 10.1080/10236190310001603662.  Google Scholar

[17]

A. Ruiz-Herrera, Exclusion and dominance in discrete population models via the carrying simplex,, J. Diff. Equ. Appl., 19 (2013), 96.  doi: 10.1080/10236198.2011.628663.  Google Scholar

[18]

H. L. Smith, Periodic competitive differential equations and the discrete dynamics of competitive maps,, J. Diff. Eqns., 64 (1986), 165.  doi: 10.1016/0022-0396(86)90086-0.  Google Scholar

[19]

H. L. Smith, Planar competitive and cooperative difference equations,, J. Diff. Equ. Appl., 3 (1998), 335.  doi: 10.1080/10236199708808108.  Google Scholar

[20]

Y. Wang and J. Jiang, Uniqueness and attractivity of the carrying simplex for discrete-time competitive dynamical systems,, J. Diff. Eqns., 186 (2002), 611.  doi: 10.1016/S0022-0396(02)00025-6.  Google Scholar

[21]

M. L. Zeeman, Hopf bifurcations in competitive three-dimensional Lotka-Volterra systems,, Dynam. Stability Systems, 8 (1993), 189.  doi: 10.1080/02681119308806158.  Google Scholar

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