February  2018, 38(2): 615-650. doi: 10.3934/dcds.2018027

On the classification of generalized competitive Atkinson-Allen models via the dynamics on the boundary of the carrying simplex

1. 

Department of Mathematics and Statistics, University of Helsinki, Helsinki FI-00014, Finland

2. 

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

3. 

School of Sciences, Zhejiang A & F University, Hangzhou 311300, China

* Corresponding author: Lei Niu

Received  May 2017 Revised  August 2017 Published  February 2018

Fund Project: This work is supported by the National Natural Science Foundation of China (NSFC) under Grant No. 11371252 and Grant No. 11771295, Shanghai Gaofeng Project for University Academic Program Development, and the Academy of Finland.

We propose the generalized competitive Atkinson-Allen map
$T_i(x)=\frac{(1+r_i)(1-c_i)x_i}{1+\sum_{j=1}^nb_{ij}x_j}+c_ix_i, 0 <c_i <1, b_{ij}, r_i>0, i, j=1, ···, n, $
which is the classical Atkson-Allen map when
$r_i=1$
and
$c_i=c$
for all
$i=1, ..., n$
and a discretized system of the competitive Lotka-Volterra equations. It is proved that every
$n$
-dimensional map
$T$
of this form admits a carrying simplex Σ which is a globally attracting invariant hypersurface of codimension one. We define an equivalence relation relative to local stability of fixed points on the boundary of Σ on the space of all such three-dimensional maps. In the three-dimensional case we list a total of 33 stable equivalence classes and draw the corresponding phase portraits on each Σ. The dynamics of the generalized competitive Atkinson-Allen map differs from the dynamics of the standard one in that Neimark-Sacker bifurcations occur in two classes for which no such bifurcations were possible for the standard competitive Atkinson-Allen map. We also found Chenciner bifurcations by numerical examples which implies that two invariant closed curves can coexist for this model, whereas those have not yet been found for all other three-dimensional competitive mappings via the carrying simplex. In one class every map admits a heteroclinic cycle; we provide a stability criterion for heteroclinic cycles. Besides, the generalized Atkinson-Allen model is not dynamically consistent with the Lotka-Volterra system.
Citation: 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 & Continuous Dynamical Systems - A, 2018, 38 (2) : 615-650. doi: 10.3934/dcds.2018027
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, Providence, 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]

S. Baigent, a private communication.

[4]

S. Baigent, Convexity-preserving flows of totally competitive planar Lotka-Volterra equations and the geometry of the carrying simplex, Proc. Edinb. Math. Soc., 55 (2012), 53-63. doi: 10.1017/S0013091510000684.

[5]

S. Baigent, Convexity of the carrying simplex for discrete-time planar competitive Kolmogorov systems, J. Difference Equ. Appl., 22 (2016), 609-622. doi: 10.1080/10236198.2015.1125895.

[6]

S. Baigent, Geometry of carrying simplices of 3-species competitive Lotka-Volterra systems, Nonlinearity, 26 (2013), 1001-1029. doi: 10.1088/0951-7715/26/4/1001.

[7]

S. Baigent and Z. Hou, Global stability of interior and boundary fixed points for Lotka-Volterra systems, Differ. Equ. Dyn. Syst., 20 (2012), 53-66. doi: 10.1007/s12591-012-0103-0.

[8]

X. ChenJ. Jiang and L. Niu, On Lotka-Volterra equations with identical minimal intrinsic growth rate, SIAM J. Applied Dyn. Sys., 14 (2015), 1558-1599. doi: 10.1137/15M1006878.

[9]

S. N. Chow and J. K. Hale, Methods of Bifurcation Theory Springer-Verlag, New York, 1982.

[10]

J. M. Cushing, On the fundamental bifurcation theorem for semelparous Leslie models, Chapter 11 in Mathematics of Planet Earth: Dynamics, Games and Science (eds. J. P. Bourguignon, R. Jeltsch, A. Pinto, and M. Viana), CIM Mathematical Sciences Series, Springer, Berlin, 1 (2015), 215–251. doi: 10.1007/978-3-319-16118-1_12.

[11]

N. V. DavydovaO. Diekmann and S. A. van Gils, On circulant populations. I. The algebra of semelparity, Linear Algebra Appl., 398 (2005), 185-243. doi: 10.1016/j.laa.2004.12.020.

[12]

O. DiekmannY. Wang and P. Yan, Carrying simplices in discrete competitive systems and age-structured semelparous populations, Discrete Contin. Dyn. Syst., 20 (2008), 37-52.

[13]

A. GaunersdorferC. H. Hommes and F. O. O. Wagener, Bifurcation routes to volatility clustering under evolutionary learning, Journal of Economic Behavior & Organization, 67 (2008), 27-47. doi: 10.1016/j.jebo.2007.07.004.

[14]

S. A. H. Geritz, Resident-invader dynamics and the coexistence of similar strategies, J. Math. Biol., 50 (2005), 67-82. doi: 10.1007/s00285-004-0280-8.

[15]

S. A. H. GeritzM. GyllenbergF. J. A. Jacobs and K. Parvinen, Invasion dynamics and attractor inheritance, J. Math. Biol., 44 (2002), 548-560. doi: 10.1007/s002850100136.

[16]

S. A. H. Geritz and E. Kisdi, On the mechanistic underpinning of discrete-time population models with complex dynamics, J. Theor. Biol., 228 (2004), 261-269. doi: 10.1016/j.jtbi.2004.01.003.

[17]

S. A. H. GeritzE. KisdiG. Meszéna and J. A. J. Metz, Evolutionarily singular strategies and the adaptive growth and branching of the evolutionary tree, Evolutionary Ecology, 12 (1998), 35-57. doi: 10.1023/A:1006554906681.

[18]

S. A. H. GeritzJ. A. J. MetzE. Kisdi and G. Meszéna, Dynamics of adaptation and evolutionary branching, Phys. Rev. Letters, 78 (1997), 2024-2027. doi: 10.1103/PhysRevLett.78.2024.

[19]

W. GovaertsR. K. GhazianiY. A. Kuznetsov and H. G. E. Meijer, Numerical methods for two-parameter local bifurcation analysis of maps, SIAM J. Sci. Comput., 29 (2007), 2644-2667. doi: 10.1137/060653858.

[20]

W. Govaerts, Y. A. Kuznetsov, H. G. E. Meijer and N. Neirynck, A study of resonance tongues near a Chenciner bifurcation using MatcontM, in European Nonlinear Dynamics Conference, 2011, 24–29.

[21]

M. Gyllenberg and I. I. Hanski, Habitat deterioration, habitat destruction, and metapopulation persistence in a heterogenous landscape, Theor. Popul. Biol., 52 (1997), 198-215. doi: 10.1006/tpbi.1997.1333.

[22]

M. Gyllenberg and P. Yan, Four limit cycles for a three-dimensional competitive Lotka-Volterra system with a heteroclinic cycle, Comp. Math. Appl., 58 (2009), 649-669. doi: 10.1016/j.camwa.2009.03.111.

[23]

M. Gyllenberg and P. Yan, On the number of limit cycles for three dimensional Lotka-Volterra systems, Discrete Contin. Dyn. Syst. Ser. B, 11 (2009), 347-352. doi: 10.3934/dcdsb.2009.11.347.

[24]

M. GyllenbergP. Yan and Y. Wang, A 3D competitive Lotka-Volterra system with three limit cycles: A falsification of a conjecture by Hofbauer and So, Appl. Math. Lett., 19 (2006), 1-7. doi: 10.1016/j.aml.2005.01.002.

[25]

M. W. Hirsch, On existence and uniqueness of the carrying simplex for competitive dynamical systems, J. Biol. Dyn., 2 (2008), 169-179. doi: 10.1080/17513750801939236.

[26]

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.

[27]

J. Hofbauer and K. Sigmund, Evolutionary Games and Population Dynamics Cambridge University Press, Cambridge, 1998. doi: 10.1017/CBO9781139173179.

[28]

J. Hofbauer and J. W.-H. So, Multiple limit cycles for three dimensional Lotka-Volterra equations, Appl. Math. Lett., 7 (1994), 65-70. doi: 10.1016/0893-9659(94)90095-7.

[29]

Z. Hou and S. Baigent, Fixed point global attractors and repellors in competitive Lotka-Volterra systems, Dyn. Syst., 26 (2011), 367-390. doi: 10.1080/14689367.2011.554384.

[30]

Z. Hou and S. Baigent, Global stability and repulsion in autonomous Kolmogorov systems, Commun. Pure Appl. Anal., 14 (2015), 1205-1238. doi: 10.3934/cpaa.2015.14.1205.

[31]

J. Jiang and L. Niu, On the equivalent classification of three-dimensional competitive Atkinson/Allen models relative to the boundary fixed points, Discrete Contin. Dyn. Syst., 36 (2016), 217-244. doi: 10.3934/dcds.2016.36.217.

[32]

J. Jiang and L. Niu, On the equivalent classification of three-dimensional competitive Leslie/Gower models via the boundary dynamics on the carrying simplex, J. Math. Biol., 74 (2017), 1223-1261. doi: 10.1007/s00285-016-1052-y.

[33]

J. JiangL. Niu and Y. Wang, On heteroclinic cycles of competitive maps via carrying simplices, J. Math. Biol., 72 (2016), 939-972. doi: 10.1007/s00285-015-0920-1.

[34]

J. JiangL. Niu and D. Zhu, On the complete classification of nullcline stable competitive three-dimensional Gompertz models, Nonlinear Anal. R.W.A., 20 (2014), 21-35. doi: 10.1016/j.nonrwa.2014.04.006.

[35]

F. G. W. Jones and J. N. Perry, Modelling populations of cyst-nematodes (nematoda: Heteroderidae), J. Applied Ecology, 15 (1978), 349-371. doi: 10.2307/2402596.

[36]

Y. A. Kuznetsov, Elements of Applied Bifurcation Theory 2$^{nd}$ edition, Springer-Verlag, New York, 1998.

[37]

Y. A. Kuznetsov and R. J. Sacker, Neimark-Sacker bifurcation Scholarpedia 3 (2008), 1845. doi: 10.4249/scholarpedia.1845.

[38]

Z. Lu and Y. Luo, Three limit cycles for a three-dimensional Lotka-Volterra competitive system with a heteroclinic cycle, Comp. Math. Appl., 46 (2003), 231-238. doi: 10.1016/S0898-1221(03)90027-7.

[39]

J. Mierczyński, The $C^1$-property of carrying simplices for a class of competitive systems of ODEs, J. Differential Equations, 111 (1994), 385-409. doi: 10.1006/jdeq.1994.1087.

[40]

A. G. Pakes and R. A. Maller, Mathematical Ecology of Plant Species Competition: A Class of Deterministic Models for Binary Mixtures of Plant Genotypes Cambridge Univ. Press, Cambridge, 1990.

[41]

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.

[42]

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.

[43]

H. L. Smith, Periodic competitive differential equations and the discrete dynamics of competitive maps, J. Differential Equations, 64 (1986), 165-194. doi: 10.1016/0022-0396(86)90086-0.

[44]

H. L. Smith, Planar competitive and cooperative difference equations, J. Diff. Equ. Appl., 3 (1998), 335-357. doi: 10.1080/10236199708808108.

[45]

P. van den Driessche and M. L. Zeeman, Three-dimensional competitive Lotka-Volterra systems with no periodic orbits, SIAM J. Appl. Math., 58 (1998), 227-234. doi: 10.1137/S0036139995294767.

[46]

Y. Wang and J. Jiang, Uniqueness and attractivity of the carrying simplex for discrete-time competitive dynamical systems, J. Differential Equations, 186 (2002), 611-632. doi: 10.1016/S0022-0396(02)00025-6.

[47]

D. Xiao and W. Li, Limit cycles for the competitive three dimensional Lotka-Volterra system, J. Differential Equations, 164 (2000), 1-15. doi: 10.1006/jdeq.1999.3729.

[48]

P. YuM. Han and D. Xiao, Four small limit cycles around a Hopf singular point in 3-dimensional competitive Lotka-Volterra systems, J. Math. Anal. Appl., 436 (2016), 521-555. doi: 10.1016/j.jmaa.2015.12.002.

[49]

E. C. Zeeman and M. L. Zeeman, An n-dimensional competitive Lotka-Volterra system is generically determined by the edges of its carrying simplex, Nonlinearity, 15 (2002), 2019-2032. doi: 10.1088/0951-7715/15/6/312.

[50]

E. C. Zeeman and M. L. Zeeman, From local to global behavior in competitive Lotka-Volterra systems, Trans. Amer. Math. Soc., 355 (2002), 713-734. doi: 10.1090/S0002-9947-02-03103-3.

[51]

E. C. Zeeman and M. L. Zeeman, On the convexity of carrying simplices in competitive Lotka-Volterra systems, in Differential Equations, Dynamical Systems, and Control Science, Lecture Notes in Pure and Appl. Math., 152, Dekker, New York, (1994), 353–364.

[52]

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, Providence, 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]

S. Baigent, a private communication.

[4]

S. Baigent, Convexity-preserving flows of totally competitive planar Lotka-Volterra equations and the geometry of the carrying simplex, Proc. Edinb. Math. Soc., 55 (2012), 53-63. doi: 10.1017/S0013091510000684.

[5]

S. Baigent, Convexity of the carrying simplex for discrete-time planar competitive Kolmogorov systems, J. Difference Equ. Appl., 22 (2016), 609-622. doi: 10.1080/10236198.2015.1125895.

[6]

S. Baigent, Geometry of carrying simplices of 3-species competitive Lotka-Volterra systems, Nonlinearity, 26 (2013), 1001-1029. doi: 10.1088/0951-7715/26/4/1001.

[7]

S. Baigent and Z. Hou, Global stability of interior and boundary fixed points for Lotka-Volterra systems, Differ. Equ. Dyn. Syst., 20 (2012), 53-66. doi: 10.1007/s12591-012-0103-0.

[8]

X. ChenJ. Jiang and L. Niu, On Lotka-Volterra equations with identical minimal intrinsic growth rate, SIAM J. Applied Dyn. Sys., 14 (2015), 1558-1599. doi: 10.1137/15M1006878.

[9]

S. N. Chow and J. K. Hale, Methods of Bifurcation Theory Springer-Verlag, New York, 1982.

[10]

J. M. Cushing, On the fundamental bifurcation theorem for semelparous Leslie models, Chapter 11 in Mathematics of Planet Earth: Dynamics, Games and Science (eds. J. P. Bourguignon, R. Jeltsch, A. Pinto, and M. Viana), CIM Mathematical Sciences Series, Springer, Berlin, 1 (2015), 215–251. doi: 10.1007/978-3-319-16118-1_12.

[11]

N. V. DavydovaO. Diekmann and S. A. van Gils, On circulant populations. I. The algebra of semelparity, Linear Algebra Appl., 398 (2005), 185-243. doi: 10.1016/j.laa.2004.12.020.

[12]

O. DiekmannY. Wang and P. Yan, Carrying simplices in discrete competitive systems and age-structured semelparous populations, Discrete Contin. Dyn. Syst., 20 (2008), 37-52.

[13]

A. GaunersdorferC. H. Hommes and F. O. O. Wagener, Bifurcation routes to volatility clustering under evolutionary learning, Journal of Economic Behavior & Organization, 67 (2008), 27-47. doi: 10.1016/j.jebo.2007.07.004.

[14]

S. A. H. Geritz, Resident-invader dynamics and the coexistence of similar strategies, J. Math. Biol., 50 (2005), 67-82. doi: 10.1007/s00285-004-0280-8.

[15]

S. A. H. GeritzM. GyllenbergF. J. A. Jacobs and K. Parvinen, Invasion dynamics and attractor inheritance, J. Math. Biol., 44 (2002), 548-560. doi: 10.1007/s002850100136.

[16]

S. A. H. Geritz and E. Kisdi, On the mechanistic underpinning of discrete-time population models with complex dynamics, J. Theor. Biol., 228 (2004), 261-269. doi: 10.1016/j.jtbi.2004.01.003.

[17]

S. A. H. GeritzE. KisdiG. Meszéna and J. A. J. Metz, Evolutionarily singular strategies and the adaptive growth and branching of the evolutionary tree, Evolutionary Ecology, 12 (1998), 35-57. doi: 10.1023/A:1006554906681.

[18]

S. A. H. GeritzJ. A. J. MetzE. Kisdi and G. Meszéna, Dynamics of adaptation and evolutionary branching, Phys. Rev. Letters, 78 (1997), 2024-2027. doi: 10.1103/PhysRevLett.78.2024.

[19]

W. GovaertsR. K. GhazianiY. A. Kuznetsov and H. G. E. Meijer, Numerical methods for two-parameter local bifurcation analysis of maps, SIAM J. Sci. Comput., 29 (2007), 2644-2667. doi: 10.1137/060653858.

[20]

W. Govaerts, Y. A. Kuznetsov, H. G. E. Meijer and N. Neirynck, A study of resonance tongues near a Chenciner bifurcation using MatcontM, in European Nonlinear Dynamics Conference, 2011, 24–29.

[21]

M. Gyllenberg and I. I. Hanski, Habitat deterioration, habitat destruction, and metapopulation persistence in a heterogenous landscape, Theor. Popul. Biol., 52 (1997), 198-215. doi: 10.1006/tpbi.1997.1333.

[22]

M. Gyllenberg and P. Yan, Four limit cycles for a three-dimensional competitive Lotka-Volterra system with a heteroclinic cycle, Comp. Math. Appl., 58 (2009), 649-669. doi: 10.1016/j.camwa.2009.03.111.

[23]

M. Gyllenberg and P. Yan, On the number of limit cycles for three dimensional Lotka-Volterra systems, Discrete Contin. Dyn. Syst. Ser. B, 11 (2009), 347-352. doi: 10.3934/dcdsb.2009.11.347.

[24]

M. GyllenbergP. Yan and Y. Wang, A 3D competitive Lotka-Volterra system with three limit cycles: A falsification of a conjecture by Hofbauer and So, Appl. Math. Lett., 19 (2006), 1-7. doi: 10.1016/j.aml.2005.01.002.

[25]

M. W. Hirsch, On existence and uniqueness of the carrying simplex for competitive dynamical systems, J. Biol. Dyn., 2 (2008), 169-179. doi: 10.1080/17513750801939236.

[26]

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.

[27]

J. Hofbauer and K. Sigmund, Evolutionary Games and Population Dynamics Cambridge University Press, Cambridge, 1998. doi: 10.1017/CBO9781139173179.

[28]

J. Hofbauer and J. W.-H. So, Multiple limit cycles for three dimensional Lotka-Volterra equations, Appl. Math. Lett., 7 (1994), 65-70. doi: 10.1016/0893-9659(94)90095-7.

[29]

Z. Hou and S. Baigent, Fixed point global attractors and repellors in competitive Lotka-Volterra systems, Dyn. Syst., 26 (2011), 367-390. doi: 10.1080/14689367.2011.554384.

[30]

Z. Hou and S. Baigent, Global stability and repulsion in autonomous Kolmogorov systems, Commun. Pure Appl. Anal., 14 (2015), 1205-1238. doi: 10.3934/cpaa.2015.14.1205.

[31]

J. Jiang and L. Niu, On the equivalent classification of three-dimensional competitive Atkinson/Allen models relative to the boundary fixed points, Discrete Contin. Dyn. Syst., 36 (2016), 217-244. doi: 10.3934/dcds.2016.36.217.

[32]

J. Jiang and L. Niu, On the equivalent classification of three-dimensional competitive Leslie/Gower models via the boundary dynamics on the carrying simplex, J. Math. Biol., 74 (2017), 1223-1261. doi: 10.1007/s00285-016-1052-y.

[33]

J. JiangL. Niu and Y. Wang, On heteroclinic cycles of competitive maps via carrying simplices, J. Math. Biol., 72 (2016), 939-972. doi: 10.1007/s00285-015-0920-1.

[34]

J. JiangL. Niu and D. Zhu, On the complete classification of nullcline stable competitive three-dimensional Gompertz models, Nonlinear Anal. R.W.A., 20 (2014), 21-35. doi: 10.1016/j.nonrwa.2014.04.006.

[35]

F. G. W. Jones and J. N. Perry, Modelling populations of cyst-nematodes (nematoda: Heteroderidae), J. Applied Ecology, 15 (1978), 349-371. doi: 10.2307/2402596.

[36]

Y. A. Kuznetsov, Elements of Applied Bifurcation Theory 2$^{nd}$ edition, Springer-Verlag, New York, 1998.

[37]

Y. A. Kuznetsov and R. J. Sacker, Neimark-Sacker bifurcation Scholarpedia 3 (2008), 1845. doi: 10.4249/scholarpedia.1845.

[38]

Z. Lu and Y. Luo, Three limit cycles for a three-dimensional Lotka-Volterra competitive system with a heteroclinic cycle, Comp. Math. Appl., 46 (2003), 231-238. doi: 10.1016/S0898-1221(03)90027-7.

[39]

J. Mierczyński, The $C^1$-property of carrying simplices for a class of competitive systems of ODEs, J. Differential Equations, 111 (1994), 385-409. doi: 10.1006/jdeq.1994.1087.

[40]

A. G. Pakes and R. A. Maller, Mathematical Ecology of Plant Species Competition: A Class of Deterministic Models for Binary Mixtures of Plant Genotypes Cambridge Univ. Press, Cambridge, 1990.

[41]

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.

[42]

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.

[43]

H. L. Smith, Periodic competitive differential equations and the discrete dynamics of competitive maps, J. Differential Equations, 64 (1986), 165-194. doi: 10.1016/0022-0396(86)90086-0.

[44]

H. L. Smith, Planar competitive and cooperative difference equations, J. Diff. Equ. Appl., 3 (1998), 335-357. doi: 10.1080/10236199708808108.

[45]

P. van den Driessche and M. L. Zeeman, Three-dimensional competitive Lotka-Volterra systems with no periodic orbits, SIAM J. Appl. Math., 58 (1998), 227-234. doi: 10.1137/S0036139995294767.

[46]

Y. Wang and J. Jiang, Uniqueness and attractivity of the carrying simplex for discrete-time competitive dynamical systems, J. Differential Equations, 186 (2002), 611-632. doi: 10.1016/S0022-0396(02)00025-6.

[47]

D. Xiao and W. Li, Limit cycles for the competitive three dimensional Lotka-Volterra system, J. Differential Equations, 164 (2000), 1-15. doi: 10.1006/jdeq.1999.3729.

[48]

P. YuM. Han and D. Xiao, Four small limit cycles around a Hopf singular point in 3-dimensional competitive Lotka-Volterra systems, J. Math. Anal. Appl., 436 (2016), 521-555. doi: 10.1016/j.jmaa.2015.12.002.

[49]

E. C. Zeeman and M. L. Zeeman, An n-dimensional competitive Lotka-Volterra system is generically determined by the edges of its carrying simplex, Nonlinearity, 15 (2002), 2019-2032. doi: 10.1088/0951-7715/15/6/312.

[50]

E. C. Zeeman and M. L. Zeeman, From local to global behavior in competitive Lotka-Volterra systems, Trans. Amer. Math. Soc., 355 (2002), 713-734. doi: 10.1090/S0002-9947-02-03103-3.

[51]

E. C. Zeeman and M. L. Zeeman, On the convexity of carrying simplices in competitive Lotka-Volterra systems, in Differential Equations, Dynamical Systems, and Control Science, Lecture Notes in Pure and Appl. Math., 152, Dekker, New York, (1994), 353–364.

[52]

M. L. Zeeman, Hopf bifurcations in competitive three-dimensional Lotka-Volterra systems, Dynam. Stability Systems, 8 (1993), 189-217. doi: 10.1080/02681119308806158.

Figure 8.  The orbits emanating from x0 = (1.004, 0.9927, 1.48) and x0 = (1.001, 1.002, 1.001) for the map T ∈ CGAA(3) with the parameters B[29], ri[29] and c1 = 0.999655, c2 = 0.339655, c3 = 0.2 are asymptotic to the bigger quasiperiodic curve and the smaller one respectively, where B[29] and ri [29] are given in Example 4.1.
Figure 1.  The dynamics in Σ replaced by ∆1. A closed dot • stands for a fixed point attracting on Σ, and an open dot ° stands for the one repelling on Σ. Each Σ denotes an equivalence class.
Figure 2.  The orbit emanating from x0 = (0.7667, 0.7667, 1) for the map T ∈ CGAA(3) with the parameters B[26], ri [26] and ci, i = 1, 2, 3 tends to an attracting quasiperiodic curve (the blue circle), where B[26] and ri [26] are given in Example 4.1 and c1 = 0.81, c2 = 0.5, c3 = 0.5.
Figure 3.  The orbit emanating from x0 = (0.7667, 1, 0.7667) for the map T ∈ CGAA(3) with the parameters B[27], ri [27] and ci, i = 1, 2, 3 tends to an attracting quasiperiodic curve (the blue boundary), where B[27] and ri [27] are given in Example 4.1 and c1 = 0.2, c2 = 0.8, c3 = 0.8.
Figure 4.  The orbit emanating from x0 = (0.9333, 1, 0.9333) for the map T ∈ CGAA(3) with the parameters B[29], ri [29] and ci, i = 1, 2, 3 tends to an attracting quasiperiodic curve (the blue circle), where B[29], ri [29], i = 1, 2, 3 are given in Example 4.1 and c1 = 0.89, c2 = 0.9995, c3 = 0.8.
Figure 5.  The orbit emanating from x0 = (0.3333, 1, 0.3333) for the map T ∈ CGAA(3) with the parameters B[31], ri [31] and ci, i = 1, 2, 3 tends to an attracting quasiperiodic curve (the blue circle), where B[31], ri [31] are given in Example 4.1 and c1 = 0.9, c2 = 0.9962, c3 = 0.75.
Figure 6.  Bifurcation diagram of the Chenciner bifurcation in the (µ1, µ2)-plane for the case L2(0) < 0. The origin is the Chenciner bifurcation point. The vertical dashed line µ1 = 0 is the NeimarkSacker bifurcation curve. In the region Ⅰ below the curve Tc, there is only one fixed point which is stable; in the region Ⅱ (µ1 > 0), there is a unique invariant closed curve which is stable; in the region Ⅲ between the curve Tc and the positive µ2-axis, a stable invariant closed curve (outer) and an unstable invariant closed curve (inner) coexist; on the solid curve Tc, these two circles coincide.
Figure 7.  A possible phase portrait on the carrying simplex for the map $T\in{\rm{CGAA}}(3)$ in class $26$. A stable invariant closed curve, the smaller red circle $\Gamma_1$, and an unstable invariant closed curve, the bigger red circle $\Gamma_2$ coexist. All the orbits in $\dot{\Sigma}\setminus (\Gamma_2\cup R_p(\Gamma_2))$ except those on the stable manifold restricted to $\Sigma$ of $v_{\{1\}}$ converge to the axial fixed point $q_{\{2\}}$, where $R_p(\Gamma_2)$ denotes the component of $\Sigma\setminus \Gamma_2$ containing $p$.
Figure 9.  The orbits emanating from x0 = (1, 0.0333, 0.0333), x0 = (1, 0.1, 0.1) and x0 = (1, 0.2, 0.2) for the map T ∈ CGAA(3) in Example 4.3 lead away from Σ and tend to the positive fixed point p.
Figure 10.  The orbits emanating from $x_0=(0.8333,0.8333,1)$, $x_0=(0.9,0.9,1)$ and $x_0=(0.9333,0.9333,1)$ for the map $\hat{T}\in {\rm{CGAA}}(3)$ in Example 4.3 approach to $\partial\Sigma$.
Figure 11.  The orbit emanating from $x_0=(0.7667,0.7667,1)$ converges to the axial steady state $q_{\{2\}}=(0,8,0)$ for system (21).
Figure 12.  The orbit emanating from x0 = (0.7667, 1, 0.7667) approaches the heteroclinic cycle for system (21).
Figure 13.  The orbit emanating from x0 = (0.3333, 1, 0.3333) tends to p = (1, 1, 1) for LV system (21).
Figure 14.  The orbits emanating from x0 = (1, 0.0333, 0.0333), x0 = (1, 0.1, 0.1) and x0 = (1, 0.2, 0.2) for LV system (21) with the parameters B, ci and ri given in Example 4.3 tend to p = $\left( {\frac{{258}}{{385}},\frac{{346}}{{385}},\frac{{236}}{{385}}} \right)$
Figure 15.  The orbits emanating from $x_0=(0.8333,0.8333,1)$, $x_0=(0.9,0.9,1)$ and $x_0=(0.9333,0.9333,1)$ for LV system (21) with parameters $\hat{B}$, $\hat{c}_i$ and $\hat{r}_i$ given in Example 4.3 approach to the heteroclinic cycle.
Table 1.  The 33 equivalence classes in CGAA(3), where $\gamma_{ij}:=r_j-b_{ji}\frac{r_i}{b_{ii}}$, $\beta_{ij}=\frac{r_ib_{jj}-r_jb_{ij}}{b_{ii}b_{jj}-b_{ij}b_{ji}}$ for $i, j=1, 2, 3$ and $i\neq j$, and each $\Sigma$ is given by a representative model of that class. A fixed point is represented by a closed dot $\bullet$ if it attracts on $\Sigma$, by an open dot $\circ$ if it repels on $\Sigma$, and by the intersection of its hyperbolic manifolds if it is a saddle on $\Sigma$.
Class The Corresponding Parameters Phase Portrait in $\Sigma$
1 $\gamma_{12} <0, \gamma_{13}<0, \gamma_{21}>0, \gamma_{23}>0, \gamma_{31}>0, \gamma_{32}<0$
2 (ⅰ) $\gamma_{12}<0, \gamma_{13}<0, \gamma_{21}<0, \gamma_{23}>0, \gamma_{31}>0, \gamma_{32}<0$
(ⅱ) $b_{31}\beta_{12}+b_{32}\beta_{21}-r_3<0$
3 (ⅰ) $\gamma_{12}<0, \gamma_{13}<0, \gamma_{21}>0, \gamma_{23}<0, \gamma_{31}>0, \gamma_{32}<0$
(ⅱ) $b_{12}\beta_{23}+b_{13}\beta_{32}-r_1<0$
4 (ⅰ) $\gamma_{12}>0, \gamma_{13}<0, \gamma_{21}>0, \gamma_{23}<0, \gamma_{31}>0, \gamma_{32}<0$
(ⅱ) $b_{12}\beta_{23}+b_{13}\beta_{32}-r_1<0$
(ⅲ) $b_{31}\beta_{12}+b_{32}\beta_{21}-r_3>0$
5 (ⅰ) $\gamma_{12}>0, \gamma_{13}>0, \gamma_{21}>0, \gamma_{23}<0, \gamma_{31}<0, \gamma_{32}>0$
(ⅱ) $b_{31}\beta_{12}+b_{32}\beta_{21}-r_3>0$
6 (ⅰ) $\gamma_{12}>0, \gamma_{13}>0, \gamma_{21}<0, \gamma_{23}>0, \gamma_{31}<0, \gamma_{32}>0$
(ⅱ) $b_{12}\beta_{23}+b_{13}\beta_{32}-r_1>0$
7 (ⅰ) $\gamma_{12}>0, \gamma_{13}>0, \gamma_{21}>0, \gamma_{23}>0, \gamma_{31}<0, \gamma_{32}<0$
(ⅱ) $b_{31}\beta_{12}+b_{32}\beta_{21}-r_3<0$
8 (ⅰ) $\gamma_{12}>0, \gamma_{13}>0, \gamma_{21}>0, \gamma_{23}<0, \gamma_{31}<0, \gamma_{32}<0$
(ⅱ) $b_{12}\beta_{23}+b_{13}\beta_{32}-r_1<0$
(ⅲ) $b_{31}\beta_{12}+b_{32}\beta_{21}-r_3<0$
9 (ⅰ) $\gamma_{12}>0, \gamma_{13}>0, \gamma_{21}>0, \gamma_{23}>0, \gamma_{31}<0, \gamma_{32}>0$
(ⅱ) $b_{12}\beta_{23}+b_{13}\beta_{32}-r_1>0$
(ⅲ) $b_{31}\beta_{12}+b_{32}\beta_{21}-r_3<0$
10 (ⅰ) $\gamma_{12}>0, \gamma_{13}>0, \gamma_{21}>0, \gamma_{23}>0, \gamma_{31}<0, \gamma_{32}>0$
(ⅱ) $b_{12}\beta_{23}+b_{13}\beta_{32}-r_1<0$
(ⅲ) $b_{31}\beta_{12}+b_{32}\beta_{21}-r_3>0$
11 (ⅰ) $\gamma_{12}>0, \gamma_{13}>0, \gamma_{21}>0, \gamma_{23}<0, \gamma_{31}>0, \gamma_{32}<0$
(ⅱ) $b_{12}\beta_{23}+b_{13}\beta_{32}-r_1<0$
(ⅲ) $b_{21}\beta_{13}+b_{23}\beta_{31}-r_2<0$
(ⅳ) $b_{31}\beta_{12}+b_{32}\beta_{21}-r_3>0$
12 (ⅰ) $\gamma_{12}>0, \gamma_{13}>0, \gamma_{21}>0, \gamma_{23}>0, \gamma_{31}>0, \gamma_{32}>0$
(ⅱ) $b_{12}\beta_{23}+b_{13}\beta_{32}-r_1<0$
(ⅲ) $b_{21}\beta_{13}+b_{23}\beta_{31}-r_2<0$
(ⅳ) $b_{31}\beta_{12}+b_{32}\beta_{21}-r_3>0$
13 (ⅰ) $\gamma_{12}<0, \gamma_{13}<0, \gamma_{21}<0, \gamma_{23}<0, \gamma_{31}>0, \gamma_{32}>0$
(ⅱ) $b_{31}\beta_{12}+b_{32}\beta_{21}-r_3>0$
14 (ⅰ) $\gamma_{12}<0, \gamma_{13}<0, \gamma_{21}<0, \gamma_{23}>0, \gamma_{31}>0, \gamma_{32}>0$
(ⅱ) $b_{12}\beta_{23}+b_{13}\beta_{32}-r_1>0$
(ⅲ) $b_{31}\beta_{12}+b_{32}\beta_{21}-r_3>0$
15 (ⅰ) $\gamma_{12}<0, \gamma_{13}<0, \gamma_{21}<0, \gamma_{23}<0, \gamma_{31}>0, \gamma_{32}<0$
(ⅱ) $b_{12}\beta_{23}+b_{13}\beta_{32}-r_1<0$
(ⅲ) $b_{31}\beta_{12}+b_{32}\beta_{21}-r_3>0$
16 (ⅰ) $\gamma_{12}<0, \gamma_{13}<0, \gamma_{21}<0, \gamma_{23}<0, \gamma_{31}>0, \gamma_{32}<0$
(ⅱ) $b_{12}\beta_{23}+b_{13}\beta_{32}-r_1>0$
(ⅲ) $b_{31}\beta_{12}+b_{32}\beta_{21}-r_3<0$
17 (ⅰ) $\gamma_{12}<0, \gamma_{13}<0, \gamma_{21}<0, \gamma_{23}>0, \gamma_{31}<0, \gamma_{32}>0$
(ⅱ) $b_{12}\beta_{23}+b_{13}\beta_{32}-r_1>0$
(ⅲ) $b_{21}\beta_{13}+b_{23}\beta_{31}-r_2>0$
(ⅳ) $b_{31}\beta_{12}+b_{32}\beta_{21}-r_3<0$
18 (ⅰ) $\gamma_{12}<0, \gamma_{13}<0, \gamma_{21}<0, \gamma_{23}<0, \gamma_{31}<0, \gamma_{32}<0$
(ⅱ) $b_{12}\beta_{23}+b_{13}\beta_{32}-r_1>0$
(ⅲ) $b_{21}\beta_{13}+b_{23}\beta_{31}-r_2>0$
(ⅳ) $b_{31}\beta_{12}+b_{32}\beta_{21}-r_3<0$
19 (ⅰ) $\gamma_{12}>0, \gamma_{13}>0, \gamma_{21}<0, \gamma_{23}<0, \gamma_{31}<0, \gamma_{32}<0$
(ⅱ) $b_{12}\beta_{23}+b_{13}\beta_{32}-r_1<0$
20 (ⅰ) $\gamma_{12}<0, \gamma_{13}<0, \gamma_{21}<0, \gamma_{23}<0, \gamma_{31}>0, \gamma_{32}<0$
(ⅱ) $b_{12}\beta_{23}+b_{13}\beta_{32}-r_1<0$
(ⅲ) $b_{31}\beta_{12}+b_{32}\beta_{21}-r_3<0$
21 (ⅰ) $\gamma_{12}<0, \gamma_{13}<0, \gamma_{21}<0, \gamma_{23}>0, \gamma_{31}<0, \gamma_{32}>0$
(ⅱ) $b_{12}\beta_{23}+b_{13}\beta_{32}-r_1>0$
(ⅲ) $b_{21}\beta_{13}+b_{23}\beta_{31}-r_2<0$
(ⅳ) $b_{31}\beta_{12}+b_{32}\beta_{21}-r_3<0$
22 (ⅰ) $\gamma_{12}>0, \gamma_{13}>0, \gamma_{21}<0, \gamma_{23}<0, \gamma_{31}>0, \gamma_{32}<0$
(ⅱ) $b_{12}\beta_{23}+b_{13}\beta_{32}-r_1<0$
(ⅲ) $b_{21}\beta_{13}+b_{23}\beta_{31}-r_2>0$
23 (ⅰ) $\gamma_{12}>0, \gamma_{13}>0, \gamma_{21}>0, \gamma_{23}>0, \gamma_{31}<0, \gamma_{32}<0$
(ⅱ) $b_{31}\beta_{12}+b_{32}\beta_{21}-r_3>0$
24 (ⅰ) $\gamma_{12}>0, \gamma_{13}>0, \gamma_{21}>0, \gamma_{23}>0, \gamma_{31}<0, \gamma_{32}>0$
(ⅱ) $b_{12}\beta_{23}+b_{13}\beta_{32}-r_1>0$
(ⅲ) $b_{31}\beta_{12}+b_{32}\beta_{21}-r_3>0$
25 (ⅰ) $\gamma_{12}>0, \gamma_{13}>0, \gamma_{21}>0, \gamma_{23}<0, \gamma_{31}>0, \gamma_{32}<0$
(ⅱ) $b_{12}\beta_{23}+b_{13}\beta_{32}-r_1<0$
(ⅲ) $b_{21}\beta_{13}+b_{23}\beta_{31}-r_2>0$
(ⅳ) $b_{31}\beta_{12}+b_{32}\beta_{21}-r_3>0$
26 (ⅰ) $\gamma_{12}>0, \gamma_{13}>0, \gamma_{21}<0, \gamma_{23}<0, \gamma_{31}>0, \gamma_{32}<0$
(ⅱ) $b_{12}\beta_{23}+b_{13}\beta_{32}-r_1>0$
(ⅲ) $b_{21}\beta_{13}+b_{23}\beta_{31}-r_2<0$
27 $\gamma_{12}>0, \gamma_{13}<0, \gamma_{21}<0, \gamma_{23}>0, \gamma_{31}>0, \gamma_{32}<0$
28 (ⅰ) $\gamma_{12}<0, \gamma_{13}<0, \gamma_{21}<0, \gamma_{23}>0, \gamma_{31}>0, \gamma_{32}<0$
(ⅱ) $b_{31}\beta_{12}+b_{32}\beta_{21}-r_3>0$
29 (ⅰ) $\gamma_{12}>0, \gamma_{13}>0, \gamma_{21}>0, \gamma_{23}<0, \gamma_{31}<0, \gamma_{32}>0$
(ⅱ) $b_{31}\beta_{12}+b_{32}\beta_{21}-r_3<0$
30 (ⅰ) $\gamma_{12}<0, \gamma_{13}<0, \gamma_{21}<0, \gamma_{23}<0, \gamma_{31}>0, \gamma_{32}<0$
(ⅱ) $b_{12}\beta_{23}+b_{13}\beta_{32}-r_1>0$
(ⅲ) $b_{31}\beta_{12}+b_{32}\beta_{21}-r_3>0$
31 (ⅰ) $\gamma_{12}>0, \gamma_{13}>0, \gamma_{21}>0, \gamma_{23}>0, \gamma_{31}<0, \gamma_{32}>0$
(ⅱ) $b_{12}\beta_{23}+b_{13}\beta_{32}-r_1<0$
(ⅲ) $b_{31}\beta_{12}+b_{32}\beta_{21}-r_3<0$
32 (ⅰ) $\gamma_{12}<0, \gamma_{13}<0, \gamma_{21}<0, \gamma_{23}<0, \gamma_{31}<0, \gamma_{32}<0$
(ⅱ) $b_{12}\beta_{23}+b_{13}\beta_{32}-r_1>0$
(ⅲ) $b_{21}\beta_{13}+b_{23}\beta_{31}-r_2>0$
(ⅳ) $b_{31}\beta_{12}+b_{32}\beta_{21}-r_3>0$
33 (ⅰ) $\gamma_{12}>0, \gamma_{13}>0, \gamma_{21}>0, \gamma_{23}>0, \gamma_{31}>0, \gamma_{32}>0$
(ⅱ) $b_{12}\beta_{23}+b_{13}\beta_{32}-r_1<0$
(ⅲ) $b_{21}\beta_{13}+b_{23}\beta_{31}-r_2<0$
(ⅳ) $b_{31}\beta_{12}+b_{32}\beta_{21}-r_3<0$
Class The Corresponding Parameters Phase Portrait in $\Sigma$
1 $\gamma_{12} <0, \gamma_{13}<0, \gamma_{21}>0, \gamma_{23}>0, \gamma_{31}>0, \gamma_{32}<0$
2 (ⅰ) $\gamma_{12}<0, \gamma_{13}<0, \gamma_{21}<0, \gamma_{23}>0, \gamma_{31}>0, \gamma_{32}<0$
(ⅱ) $b_{31}\beta_{12}+b_{32}\beta_{21}-r_3<0$
3 (ⅰ) $\gamma_{12}<0, \gamma_{13}<0, \gamma_{21}>0, \gamma_{23}<0, \gamma_{31}>0, \gamma_{32}<0$
(ⅱ) $b_{12}\beta_{23}+b_{13}\beta_{32}-r_1<0$
4 (ⅰ) $\gamma_{12}>0, \gamma_{13}<0, \gamma_{21}>0, \gamma_{23}<0, \gamma_{31}>0, \gamma_{32}<0$
(ⅱ) $b_{12}\beta_{23}+b_{13}\beta_{32}-r_1<0$
(ⅲ) $b_{31}\beta_{12}+b_{32}\beta_{21}-r_3>0$
5 (ⅰ) $\gamma_{12}>0, \gamma_{13}>0, \gamma_{21}>0, \gamma_{23}<0, \gamma_{31}<0, \gamma_{32}>0$
(ⅱ) $b_{31}\beta_{12}+b_{32}\beta_{21}-r_3>0$
6 (ⅰ) $\gamma_{12}>0, \gamma_{13}>0, \gamma_{21}<0, \gamma_{23}>0, \gamma_{31}<0, \gamma_{32}>0$
(ⅱ) $b_{12}\beta_{23}+b_{13}\beta_{32}-r_1>0$
7 (ⅰ) $\gamma_{12}>0, \gamma_{13}>0, \gamma_{21}>0, \gamma_{23}>0, \gamma_{31}<0, \gamma_{32}<0$
(ⅱ) $b_{31}\beta_{12}+b_{32}\beta_{21}-r_3<0$
8 (ⅰ) $\gamma_{12}>0, \gamma_{13}>0, \gamma_{21}>0, \gamma_{23}<0, \gamma_{31}<0, \gamma_{32}<0$
(ⅱ) $b_{12}\beta_{23}+b_{13}\beta_{32}-r_1<0$
(ⅲ) $b_{31}\beta_{12}+b_{32}\beta_{21}-r_3<0$
9 (ⅰ) $\gamma_{12}>0, \gamma_{13}>0, \gamma_{21}>0, \gamma_{23}>0, \gamma_{31}<0, \gamma_{32}>0$
(ⅱ) $b_{12}\beta_{23}+b_{13}\beta_{32}-r_1>0$
(ⅲ) $b_{31}\beta_{12}+b_{32}\beta_{21}-r_3<0$
10 (ⅰ) $\gamma_{12}>0, \gamma_{13}>0, \gamma_{21}>0, \gamma_{23}>0, \gamma_{31}<0, \gamma_{32}>0$
(ⅱ) $b_{12}\beta_{23}+b_{13}\beta_{32}-r_1<0$
(ⅲ) $b_{31}\beta_{12}+b_{32}\beta_{21}-r_3>0$
11 (ⅰ) $\gamma_{12}>0, \gamma_{13}>0, \gamma_{21}>0, \gamma_{23}<0, \gamma_{31}>0, \gamma_{32}<0$
(ⅱ) $b_{12}\beta_{23}+b_{13}\beta_{32}-r_1<0$
(ⅲ) $b_{21}\beta_{13}+b_{23}\beta_{31}-r_2<0$
(ⅳ) $b_{31}\beta_{12}+b_{32}\beta_{21}-r_3>0$
12 (ⅰ) $\gamma_{12}>0, \gamma_{13}>0, \gamma_{21}>0, \gamma_{23}>0, \gamma_{31}>0, \gamma_{32}>0$
(ⅱ) $b_{12}\beta_{23}+b_{13}\beta_{32}-r_1<0$
(ⅲ) $b_{21}\beta_{13}+b_{23}\beta_{31}-r_2<0$
(ⅳ) $b_{31}\beta_{12}+b_{32}\beta_{21}-r_3>0$
13 (ⅰ) $\gamma_{12}<0, \gamma_{13}<0, \gamma_{21}<0, \gamma_{23}<0, \gamma_{31}>0, \gamma_{32}>0$
(ⅱ) $b_{31}\beta_{12}+b_{32}\beta_{21}-r_3>0$
14 (ⅰ) $\gamma_{12}<0, \gamma_{13}<0, \gamma_{21}<0, \gamma_{23}>0, \gamma_{31}>0, \gamma_{32}>0$
(ⅱ) $b_{12}\beta_{23}+b_{13}\beta_{32}-r_1>0$
(ⅲ) $b_{31}\beta_{12}+b_{32}\beta_{21}-r_3>0$
15 (ⅰ) $\gamma_{12}<0, \gamma_{13}<0, \gamma_{21}<0, \gamma_{23}<0, \gamma_{31}>0, \gamma_{32}<0$
(ⅱ) $b_{12}\beta_{23}+b_{13}\beta_{32}-r_1<0$
(ⅲ) $b_{31}\beta_{12}+b_{32}\beta_{21}-r_3>0$
16 (ⅰ) $\gamma_{12}<0, \gamma_{13}<0, \gamma_{21}<0, \gamma_{23}<0, \gamma_{31}>0, \gamma_{32}<0$
(ⅱ) $b_{12}\beta_{23}+b_{13}\beta_{32}-r_1>0$
(ⅲ) $b_{31}\beta_{12}+b_{32}\beta_{21}-r_3<0$
17 (ⅰ) $\gamma_{12}<0, \gamma_{13}<0, \gamma_{21}<0, \gamma_{23}>0, \gamma_{31}<0, \gamma_{32}>0$
(ⅱ) $b_{12}\beta_{23}+b_{13}\beta_{32}-r_1>0$
(ⅲ) $b_{21}\beta_{13}+b_{23}\beta_{31}-r_2>0$
(ⅳ) $b_{31}\beta_{12}+b_{32}\beta_{21}-r_3<0$
18 (ⅰ) $\gamma_{12}<0, \gamma_{13}<0, \gamma_{21}<0, \gamma_{23}<0, \gamma_{31}<0, \gamma_{32}<0$
(ⅱ) $b_{12}\beta_{23}+b_{13}\beta_{32}-r_1>0$
(ⅲ) $b_{21}\beta_{13}+b_{23}\beta_{31}-r_2>0$
(ⅳ) $b_{31}\beta_{12}+b_{32}\beta_{21}-r_3<0$
19 (ⅰ) $\gamma_{12}>0, \gamma_{13}>0, \gamma_{21}<0, \gamma_{23}<0, \gamma_{31}<0, \gamma_{32}<0$
(ⅱ) $b_{12}\beta_{23}+b_{13}\beta_{32}-r_1<0$
20 (ⅰ) $\gamma_{12}<0, \gamma_{13}<0, \gamma_{21}<0, \gamma_{23}<0, \gamma_{31}>0, \gamma_{32}<0$
(ⅱ) $b_{12}\beta_{23}+b_{13}\beta_{32}-r_1<0$
(ⅲ) $b_{31}\beta_{12}+b_{32}\beta_{21}-r_3<0$
21 (ⅰ) $\gamma_{12}<0, \gamma_{13}<0, \gamma_{21}<0, \gamma_{23}>0, \gamma_{31}<0, \gamma_{32}>0$
(ⅱ) $b_{12}\beta_{23}+b_{13}\beta_{32}-r_1>0$
(ⅲ) $b_{21}\beta_{13}+b_{23}\beta_{31}-r_2<0$
(ⅳ) $b_{31}\beta_{12}+b_{32}\beta_{21}-r_3<0$
22 (ⅰ) $\gamma_{12}>0, \gamma_{13}>0, \gamma_{21}<0, \gamma_{23}<0, \gamma_{31}>0, \gamma_{32}<0$
(ⅱ) $b_{12}\beta_{23}+b_{13}\beta_{32}-r_1<0$
(ⅲ) $b_{21}\beta_{13}+b_{23}\beta_{31}-r_2>0$
23 (ⅰ) $\gamma_{12}>0, \gamma_{13}>0, \gamma_{21}>0, \gamma_{23}>0, \gamma_{31}<0, \gamma_{32}<0$
(ⅱ) $b_{31}\beta_{12}+b_{32}\beta_{21}-r_3>0$
24 (ⅰ) $\gamma_{12}>0, \gamma_{13}>0, \gamma_{21}>0, \gamma_{23}>0, \gamma_{31}<0, \gamma_{32}>0$
(ⅱ) $b_{12}\beta_{23}+b_{13}\beta_{32}-r_1>0$
(ⅲ) $b_{31}\beta_{12}+b_{32}\beta_{21}-r_3>0$
25 (ⅰ) $\gamma_{12}>0, \gamma_{13}>0, \gamma_{21}>0, \gamma_{23}<0, \gamma_{31}>0, \gamma_{32}<0$
(ⅱ) $b_{12}\beta_{23}+b_{13}\beta_{32}-r_1<0$
(ⅲ) $b_{21}\beta_{13}+b_{23}\beta_{31}-r_2>0$
(ⅳ) $b_{31}\beta_{12}+b_{32}\beta_{21}-r_3>0$
26 (ⅰ) $\gamma_{12}>0, \gamma_{13}>0, \gamma_{21}<0, \gamma_{23}<0, \gamma_{31}>0, \gamma_{32}<0$
(ⅱ) $b_{12}\beta_{23}+b_{13}\beta_{32}-r_1>0$
(ⅲ) $b_{21}\beta_{13}+b_{23}\beta_{31}-r_2<0$
27 $\gamma_{12}>0, \gamma_{13}<0, \gamma_{21}<0, \gamma_{23}>0, \gamma_{31}>0, \gamma_{32}<0$
28 (ⅰ) $\gamma_{12}<0, \gamma_{13}<0, \gamma_{21}<0, \gamma_{23}>0, \gamma_{31}>0, \gamma_{32}<0$
(ⅱ) $b_{31}\beta_{12}+b_{32}\beta_{21}-r_3>0$
29 (ⅰ) $\gamma_{12}>0, \gamma_{13}>0, \gamma_{21}>0, \gamma_{23}<0, \gamma_{31}<0, \gamma_{32}>0$
(ⅱ) $b_{31}\beta_{12}+b_{32}\beta_{21}-r_3<0$
30 (ⅰ) $\gamma_{12}<0, \gamma_{13}<0, \gamma_{21}<0, \gamma_{23}<0, \gamma_{31}>0, \gamma_{32}<0$
(ⅱ) $b_{12}\beta_{23}+b_{13}\beta_{32}-r_1>0$
(ⅲ) $b_{31}\beta_{12}+b_{32}\beta_{21}-r_3>0$
31 (ⅰ) $\gamma_{12}>0, \gamma_{13}>0, \gamma_{21}>0, \gamma_{23}>0, \gamma_{31}<0, \gamma_{32}>0$
(ⅱ) $b_{12}\beta_{23}+b_{13}\beta_{32}-r_1<0$
(ⅲ) $b_{31}\beta_{12}+b_{32}\beta_{21}-r_3<0$
32 (ⅰ) $\gamma_{12}<0, \gamma_{13}<0, \gamma_{21}<0, \gamma_{23}<0, \gamma_{31}<0, \gamma_{32}<0$
(ⅱ) $b_{12}\beta_{23}+b_{13}\beta_{32}-r_1>0$
(ⅲ) $b_{21}\beta_{13}+b_{23}\beta_{31}-r_2>0$
(ⅳ) $b_{31}\beta_{12}+b_{32}\beta_{21}-r_3>0$
33 (ⅰ) $\gamma_{12}>0, \gamma_{13}>0, \gamma_{21}>0, \gamma_{23}>0, \gamma_{31}>0, \gamma_{32}>0$
(ⅱ) $b_{12}\beta_{23}+b_{13}\beta_{32}-r_1<0$
(ⅲ) $b_{21}\beta_{13}+b_{23}\beta_{31}-r_2<0$
(ⅳ) $b_{31}\beta_{12}+b_{32}\beta_{21}-r_3<0$
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