Discrete and Continuous Dynamical Systems - Series A (DCDS-A)

On the classification of generalized competitive Atkinson-Allen models via the dynamics on the boundary of the carrying simplex
Pages: 615 - 650, Issue 2, February 2018

doi:10.3934/dcds.2018027      Abstract        References        Full text (3057.0K)           Related Articles

Mats Gyllenberg - Department of Mathematics and Statistics, University of Helsinki, FIN-00014 Helsinki, Finland (email)
Jifa Jiang - Mathematics and Science College, Shanghai Normal University, Shanghai 200234, China (email)
Lei Niu - Department of Mathematics and Statistics, University of Helsinki, Helsinki FI-00014, Finland (email)
Ping Yan - Department of Mathematics and Statistics, University of Helsinki, Helsinki FI-00014, Finland (email)

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.       
5 S. Baigent, Convexity of the carrying simplex for discrete-time planar competitive Kolmogorov systems, J. Difference Equ. Appl., 22 (2016), 609-622.       
6 S. Baigent, Geometry of carrying simplices of 3-species competitive Lotka-Volterra systems, Nonlinearity, 26 (2013), 1001-1029.       
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.       
8 X. Chen, J. Jiang and L. Niu, On Lotka-Volterra equations with identical minimal intrinsic growth rate, SIAM J. Applied Dyn. Sys., 14 (2015), 1558-1599.       
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.
11 N. V. Davydova, O. Diekmann and S. A. van Gils, On circulant populations. I. The algebra of semelparity, Linear Algebra Appl., 398 (2005), 185-243.       
12 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.       
13 A. Gaunersdorfer, C. H. Hommes and F. O. O. Wagener, Bifurcation routes to volatility clustering under evolutionary learning, Journal of Economic Behavior & Organization, 67 (2008), 27-47.
14 S. A. H. Geritz, Resident-invader dynamics and the coexistence of similar strategies, J. Math. Biol., 50 (2005), 67-82.       
15 S. A. H. Geritz, M. Gyllenberg, F. J. A. Jacobs and K. Parvinen, Invasion dynamics and attractor inheritance, J. Math. Biol., 44 (2002), 548-560.       
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.       
17 S. A. H. Geritz, E. Kisdi, G. 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.
18 S. A. H. Geritz, J. A. J. Metz, E. Kisdi and G. Meszéna, Dynamics of adaptation and evolutionary branching, Phys. Rev. Letters, 78 (1997), 2024-2027.
19 W. Govaerts, R. K. Ghaziani, Y. 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.       
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.
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.       
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.       
24 M. Gyllenberg, P. 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.       
25 M. W. Hirsch, On existence and uniqueness of the carrying simplex for competitive dynamical systems, J. Biol. Dyn., 2 (2008), 169-179.       
26 M. W. Hirsch, Systems of differential equations which are competitive or cooperative: III. Competing species, Nonlinearity, 1 (1988), 51-71.       
27 J. Hofbauer and K. Sigmund, Evolutionary Games and Population Dynamics, Cambridge University Press, Cambridge, 1998.       
28 J. Hofbauer and J. W.-H. So, Multiple limit cycles for three dimensional Lotka-Volterra equations, Appl. Math. Lett., 7 (1994), 65-70.       
29 Z. Hou and S. Baigent, Fixed point global attractors and repellors in competitive Lotka-Volterra systems, Dyn. Syst., 26 (2011), 367-390.       
30 Z. Hou and S. Baigent, Global stability and repulsion in autonomous Kolmogorov systems, Commun. Pure Appl. Anal., 14 (2015), 1205-1238.       
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.       
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.       
33 J. Jiang, L. Niu and Y. Wang, On heteroclinic cycles of competitive maps via carrying simplices, J. Math. Biol., 72 (2016), 939-972.       
34 J. Jiang, L. 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.       
35 F. G. W. Jones and J. N. Perry, Modelling populations of cyst-nematodes (nematoda: Heteroderidae), J. Applied Ecology, 15 (1978), 349-371.
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.
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.       
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.       
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.
42 A. Ruiz-Herrera, Exclusion and dominance in discrete population models via the carrying simplex, J. Diff. Equ. Appl., 19 (2013), 96-113.       
43 H. L. Smith, Periodic competitive differential equations and the discrete dynamics of competitive maps, J. Differential Equations, 64 (1986), 165-194.       
44 H. L. Smith, Planar competitive and cooperative difference equations, J. Diff. Equ. Appl., 3 (1998), 335-357.       
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.       
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.       
47 D. Xiao and W. Li, Limit cycles for the competitive three dimensional Lotka-Volterra system, J. Differential Equations, 164 (2000), 1-15.       
48 P. Yu, M. 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.       
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.       
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.       
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.       

Go to top