American Institute of Mathematical Sciences

Advanced
May  2005, 5(2): 289-298. doi: 10.3934/dcdsb.2005.5.289

Periodic tridiagonal systems modeling competitive-cooperative ecological interactions

Mats Gyllenberg 1, and  Yi Wang 2,

1. 

Department of Mathematics and Statistics, University of Helsinki, FIN-00014 Helsinki, Finland
2. 

Department of Mathematics, University of Turku, FIN-20014 Turku, Finland

Received  November 2003 Revised  July 2004 Published  February 2005

The dynamics of the Poincaré map, associated with a periodic tridiagonal system modeling cooperative-competitive ecological interactions, is investigated. It is shown that the limit-set is either a fixed point or is contained in the boundary of the positive cone and itself contains a cycle of fixed points. Furthermore, the dynamics is trivial if the number of interactive species is not greater than 4.
Keywords: Cooperative-Competitive., Periodic Kolmogorov tridiagonal systems.
Mathematics Subject Classification: 34C15, 34C25, 92D25, 92D4.
Citation: Mats Gyllenberg, Yi Wang. Periodic tridiagonal systems modeling competitive-cooperative ecological interactions. Discrete & Continuous Dynamical Systems - B, 2005, 5 (2) : 289-298. doi: 10.3934/dcdsb.2005.5.289
[1]

Yi Wang, Dun Zhou. Transversality for time-periodic competitive-cooperative tridiagonal systems. Discrete & Continuous Dynamical Systems - B, 2015, 20 (6) : 1821-1830. doi: 10.3934/dcdsb.2015.20.1821
[2]

Isabel Coelho, Carlota Rebelo, Elisa Sovrano. Extinction or coexistence in periodic Kolmogorov systems of competitive type. Discrete & Continuous Dynamical Systems, 2021, 41 (12) : 5743-5764. doi: 10.3934/dcds.2021094
[3]

Zhanyuan Hou, Stephen Baigent. Heteroclinic limit cycles in competitive Kolmogorov systems. Discrete & Continuous Dynamical Systems, 2013, 33 (9) : 4071-4093. doi: 10.3934/dcds.2013.33.4071
[4]

Dan Li, Hui Wan. Coexistence and exclusion of competitive Kolmogorov systems with semi-Markovian switching. Discrete & Continuous Dynamical Systems, 2021, 41 (9) : 4145-4183. doi: 10.3934/dcds.2021032
[5]

Lin Niu, Yi Wang. Non-oscillation principle for eventually competitive and cooperative systems. Discrete & Continuous Dynamical Systems - B, 2019, 24 (12) : 6481-6494. doi: 10.3934/dcdsb.2019148
[6]

G. Donald Allen. A dynamic model for competitive-cooperative species. Conference Publications, 1998, 1998 (Special) : 29-50. doi: 10.3934/proc.1998.1998.29
[7]

B. Cantó, C. Coll, E. Sánchez. The problem of global identifiability for systems with tridiagonal matrices. Conference Publications, 2011, 2011 (Special) : 250-257. doi: 10.3934/proc.2011.2011.250
[8]

Pablo Álvarez-Caudevilla, Julián López-Gómez. The dynamics of a class of cooperative systems. Discrete & Continuous Dynamical Systems, 2010, 26 (2) : 397-415. doi: 10.3934/dcds.2010.26.397
[9]

Xiongxiong Bao, Wenxian Shen, Zhongwei Shen. Spreading speeds and traveling waves for space-time periodic nonlocal dispersal cooperative systems. Communications on Pure & Applied Analysis, 2019, 18 (1) : 361-396. doi: 10.3934/cpaa.2019019
[10]

Tian Hou, Yi Wang, Xizhuang Xie. Instability and bifurcation of a cooperative system with periodic coefficients. Electronic Research Archive, , () : -. doi: 10.3934/era.2021026
[11]

Zhanyuan Hou, Stephen Baigent. Global stability and repulsion in autonomous Kolmogorov systems. Communications on Pure & Applied Analysis, 2015, 14 (3) : 1205-1238. doi: 10.3934/cpaa.2015.14.1205
[12]

Pablo Álvarez-Caudevilla, Julián López-Gómez. Characterizing the existence of coexistence states in a class of cooperative systems. Conference Publications, 2009, 2009 (Special) : 24-33. doi: 10.3934/proc.2009.2009.24
[13]

Anh Tuan Duong, Quoc Hung Phan. A Liouville-type theorem for cooperative parabolic systems. Discrete & Continuous Dynamical Systems, 2018, 38 (2) : 823-833. doi: 10.3934/dcds.2018035
[14]

J. Földes, Peter Poláčik. On cooperative parabolic systems: Harnack inequalities and asymptotic symmetry. Discrete & Continuous Dynamical Systems, 2009, 25 (1) : 133-157. doi: 10.3934/dcds.2009.25.133
[15]

Yubin Liu, Peixuan Weng. Asymptotic spreading of a three dimensional Lotka-Volterra cooperative-competitive system. Discrete & Continuous Dynamical Systems - B, 2015, 20 (2) : 505-518. doi: 10.3934/dcdsb.2015.20.505
[16]

Benedetta Lisena. Dynamic behaviour of a periodic competitive system with pulses. Discrete & Continuous Dynamical Systems - S, 2013, 6 (3) : 723-729. doi: 10.3934/dcdss.2013.6.723
[17]

M. R. S. Kulenović, Orlando Merino. Competitive-exclusion versus competitive-coexistence for systems in the plane. Discrete & Continuous Dynamical Systems - B, 2006, 6 (5) : 1141-1156. doi: 10.3934/dcdsb.2006.6.1141
[18]

Kokum R. De Silva, Tuoc V. Phan, Suzanne Lenhart. Advection control in parabolic PDE systems for competitive populations. Discrete & Continuous Dynamical Systems - B, 2017, 22 (3) : 1049-1072. doi: 10.3934/dcdsb.2017052
[19]

Ryusuke Kon. Dynamics of competitive systems with a single common limiting factor. Mathematical Biosciences & Engineering, 2015, 12 (1) : 71-81. doi: 10.3934/mbe.2015.12.71
[20]

M. R. S. Kulenović, Orlando Merino. Global bifurcation for discrete competitive systems in the plane. Discrete & Continuous Dynamical Systems - B, 2009, 12 (1) : 133-149. doi: 10.3934/dcdsb.2009.12.133

2020 Impact Factor: 1.327

Tools

Metrics

  • PDF downloads (32)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences