# American Institute of Mathematical Sciences

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 and Continuous Dynamical Systems, 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, G. Wolkowicz and J. Wu), American Mathematical Society, RI, 21 (1999), 15-30. [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), 157-170. doi: 10.1006/jdeq.1997.3265. [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), 211-226. 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), 1139-1151. doi: 10.1080/10236190410001652739. [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), 37-52. [7] J. Hale and H. Koçak, Dynamics and Bifurcations, Springer-Verlag, New York, 1991. doi: 10.1007/978-1-4612-4426-4. [8] M. W. Hirsch, Systems of differential equations which are competitive or cooperative: III. Competing species, Nonlinearity, 1 (1988), 51-71. doi: 10.1088/0951-7715/1/1/003. [9] J. Hofbauer, V. Hutson and W. Jansen, Coexistence for systems governed by difference equations of Lotka-Volterra type, J. Math. Biol., 25 (1987), 553-570. 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), 573-596. doi: 10.1007/BF00275495. [12] P. Liu and S. N. Elaydi, Discrete competitive and cooperative models of Lotka-Volterra type, J. Comp. Anal. Appl., 3 (2001), 53-73. doi: 10.1023/A:1011539901001. [13] J. E. Marsden and M. McCracken, The Hopf Bifurcation and Its Applications, Springer-Verlag, New York, 1976. [14] J. D. Murray, Mathematical Biology, Springer-Verlag, 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: 10.1016/0020-7519(90)90100-2. [16] L.-I. W. Roeger and L. J. S. Allen, Discrete May-Leonard competition models I, J. Diff. Equ. Appl., 10 (2004), 77-98. doi: 10.1080/10236190310001603662. [17] A. Ruiz-Herrera, Exclusion and dominance in discrete population models via the carrying simplex, J. Diff. Equ. Appl., 19 (2013), 96-113. 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), 165-194. doi: 10.1016/0022-0396(86)90086-0. [19] H. L. Smith, Planar competitive and cooperative difference equations, J. Diff. Equ. Appl., 3 (1998), 335-357. doi: 10.1080/10236199708808108. [20] Y. Wang and J. Jiang, Uniqueness and attractivity of the carrying simplex for discrete-time competitive dynamical systems, J. Diff. Eqns., 186 (2002), 611-632. doi: 10.1016/S0022-0396(02)00025-6. [21] M. L. Zeeman, Hopf bifurcations in competitive three-dimensional Lotka-Volterra systems, Dynam. Stability Systems, 8 (1993), 189-217. 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), 15-30. [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), 157-170. doi: 10.1006/jdeq.1997.3265. [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), 211-226. 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), 1139-1151. doi: 10.1080/10236190410001652739. [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), 37-52. [7] J. Hale and H. Koçak, Dynamics and Bifurcations, Springer-Verlag, New York, 1991. doi: 10.1007/978-1-4612-4426-4. [8] M. W. Hirsch, Systems of differential equations which are competitive or cooperative: III. Competing species, Nonlinearity, 1 (1988), 51-71. doi: 10.1088/0951-7715/1/1/003. [9] J. Hofbauer, V. Hutson and W. Jansen, Coexistence for systems governed by difference equations of Lotka-Volterra type, J. Math. Biol., 25 (1987), 553-570. 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), 573-596. doi: 10.1007/BF00275495. [12] P. Liu and S. N. Elaydi, Discrete competitive and cooperative models of Lotka-Volterra type, J. Comp. Anal. Appl., 3 (2001), 53-73. doi: 10.1023/A:1011539901001. [13] J. E. Marsden and M. McCracken, The Hopf Bifurcation and Its Applications, Springer-Verlag, New York, 1976. [14] J. D. Murray, Mathematical Biology, Springer-Verlag, 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: 10.1016/0020-7519(90)90100-2. [16] L.-I. W. Roeger and L. J. S. Allen, Discrete May-Leonard competition models I, J. Diff. Equ. Appl., 10 (2004), 77-98. doi: 10.1080/10236190310001603662. [17] A. Ruiz-Herrera, Exclusion and dominance in discrete population models via the carrying simplex, J. Diff. Equ. Appl., 19 (2013), 96-113. 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), 165-194. doi: 10.1016/0022-0396(86)90086-0. [19] H. L. Smith, Planar competitive and cooperative difference equations, J. Diff. Equ. Appl., 3 (1998), 335-357. doi: 10.1080/10236199708808108. [20] Y. Wang and J. Jiang, Uniqueness and attractivity of the carrying simplex for discrete-time competitive dynamical systems, J. Diff. Eqns., 186 (2002), 611-632. doi: 10.1016/S0022-0396(02)00025-6. [21] M. L. Zeeman, Hopf bifurcations in competitive three-dimensional Lotka-Volterra systems, Dynam. Stability Systems, 8 (1993), 189-217. doi: 10.1080/02681119308806158.
 [1] Mats Gyllenberg, Jifa Jiang, Lei Niu, Ping Yan. On the classification of generalized competitive Atkinson-Allen models via the dynamics on the boundary of the carrying simplex. Discrete and Continuous Dynamical Systems, 2018, 38 (2) : 615-650. doi: 10.3934/dcds.2018027 [2] Mats Gyllenberg, Jifa Jiang, Lei Niu, Ping Yan. Permanence and universal classification of discrete-time competitive systems via the carrying simplex. Discrete and Continuous Dynamical Systems, 2020, 40 (3) : 1621-1663. doi: 10.3934/dcds.2020088 [3] Stephen Baigent. Convex geometry of the carrying simplex for the May-Leonard map. Discrete and Continuous Dynamical Systems - B, 2019, 24 (4) : 1697-1723. doi: 10.3934/dcdsb.2018288 [4] Lin Niu, Yi Wang, Xizhuang Xie. Carrying simplex in the Lotka-Volterra competition model with seasonal succession with applications. Discrete and Continuous Dynamical Systems - B, 2021, 26 (4) : 2161-2172. doi: 10.3934/dcdsb.2021014 [5] Chunqing Wu, Patricia J.Y. Wong. Global asymptotical stability of the coexistence fixed point of a Ricker-type competitive model. Discrete and Continuous Dynamical Systems - B, 2015, 20 (9) : 3255-3266. doi: 10.3934/dcdsb.2015.20.3255 [6] Marie-Claude Arnaud. A nondifferentiable essential irrational invariant curve for a $C^1$ symplectic twist map. Journal of Modern Dynamics, 2011, 5 (3) : 583-591. doi: 10.3934/jmd.2011.5.583 [7] Robert Skiba, Nils Waterstraat. The index bundle and multiparameter bifurcation for discrete dynamical systems. Discrete and Continuous Dynamical Systems, 2017, 37 (11) : 5603-5629. doi: 10.3934/dcds.2017243 [8] Dmitriy Yu. Volkov. The Hopf -- Hopf bifurcation with 2:1 resonance: Periodic solutions and invariant tori. Conference Publications, 2015, 2015 (special) : 1098-1104. doi: 10.3934/proc.2015.1098 [9] Hans Koch. A renormalization group fixed point associated with the breakup of golden invariant tori. Discrete and Continuous Dynamical Systems, 2004, 11 (4) : 881-909. doi: 10.3934/dcds.2004.11.881 [10] Yu-Hsien Chang, Guo-Chin Jau. The behavior of the solution for a mathematical model for analysis of the cell cycle. Communications on Pure and Applied Analysis, 2006, 5 (4) : 779-792. doi: 10.3934/cpaa.2006.5.779 [11] Bing Zeng, Pei Yu. A hierarchical parametric analysis on Hopf bifurcation of an epidemic model. Discrete and Continuous Dynamical Systems - S, 2022  doi: 10.3934/dcdss.2022069 [12] Hongyong Zhao, Daiyong Wu. Point to point traveling wave and periodic traveling wave induced by Hopf bifurcation for a diffusive predator-prey system. Discrete and Continuous Dynamical Systems - S, 2020, 13 (11) : 3271-3284. doi: 10.3934/dcdss.2020129 [13] Runxia Wang, Haihong Liu, Fang Yan, Xiaohui Wang. Hopf-pitchfork bifurcation analysis in a coupled FHN neurons model with delay. Discrete and Continuous Dynamical Systems - S, 2017, 10 (3) : 523-542. doi: 10.3934/dcdss.2017026 [14] Fabien Crauste. Global Asymptotic Stability and Hopf Bifurcation for a Blood Cell Production Model. Mathematical Biosciences & Engineering, 2006, 3 (2) : 325-346. doi: 10.3934/mbe.2006.3.325 [15] Zhihua Liu, Hui Tang, Pierre Magal. Hopf bifurcation for a spatially and age structured population dynamics model. Discrete and Continuous Dynamical Systems - B, 2015, 20 (6) : 1735-1757. doi: 10.3934/dcdsb.2015.20.1735 [16] Hui Miao, Zhidong Teng, Chengjun Kang. Stability and Hopf bifurcation of an HIV infection model with saturation incidence and two delays. Discrete and Continuous Dynamical Systems - B, 2017, 22 (6) : 2365-2387. doi: 10.3934/dcdsb.2017121 [17] Pengmiao Hao, Xuechen Wang, Junjie Wei. Global Hopf bifurcation of a population model with stage structure and strong Allee effect. Discrete and Continuous Dynamical Systems - S, 2017, 10 (5) : 973-993. doi: 10.3934/dcdss.2017051 [18] Stephen Pankavich, Nathan Neri, Deborah Shutt. Bistable dynamics and Hopf bifurcation in a refined model of early stage HIV infection. Discrete and Continuous Dynamical Systems - B, 2020, 25 (8) : 2867-2893. doi: 10.3934/dcdsb.2020044 [19] Jixun Chu, Pierre Magal. Hopf bifurcation for a size-structured model with resting phase. Discrete and Continuous Dynamical Systems, 2013, 33 (11&12) : 4891-4921. doi: 10.3934/dcds.2013.33.4891 [20] Jisun Lim, Seongwon Lee, Yangjin Kim. Hopf bifurcation in a model of TGF-$\beta$ in regulation of the Th 17 phenotype. Discrete and Continuous Dynamical Systems - B, 2016, 21 (10) : 3575-3602. doi: 10.3934/dcdsb.2016111

2021 Impact Factor: 1.588