July  2008, 10(1): 109-128. doi: 10.3934/dcdsb.2008.10.109

An investigation of the global properties of a two-dimensional competing species model

1. 

Department of Mathematics and Statistics, University of Limerick, Limerick, Ireland, Ireland

Received  July 2007 Revised  December 2007 Published  April 2008

In this paper we demonstrate how the global dynamics of a biological model can be analysed. In particular, as an example, we consider a competing species population model based on the discretisation of the original Lotka-Volterra equations. We analyse the local and global dynamic properties of the resulting two-dimensional noninvertible dynamical system in the cases when the interspecific competition is considered to be “weak”, “strong” and “mixed”. The main results of this paper are derived from the study of some global bifurcations that change the structure of the attractors and their basins. These bifurcations are investigated by the use of critical curves, a powerful tool for the analysis of the global properties of noninvertible two-dimensional maps.
Citation: Patrick J. Johnson, Mark E. Burke. An investigation of the global properties of a two-dimensional competing species model. Discrete & Continuous Dynamical Systems - B, 2008, 10 (1) : 109-128. doi: 10.3934/dcdsb.2008.10.109
[1]

Claudio Bonanno, Marco Lenci. Pomeau-Manneville maps are global-local mixing. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1051-1069. doi: 10.3934/dcds.2020309

[2]

Yunfeng Geng, Xiaoying Wang, Frithjof Lutscher. Coexistence of competing consumers on a single resource in a hybrid model. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 269-297. doi: 10.3934/dcdsb.2020140

[3]

Knut Hüper, Irina Markina, Fátima Silva Leite. A Lagrangian approach to extremal curves on Stiefel manifolds. Journal of Geometric Mechanics, 2020  doi: 10.3934/jgm.2020031

[4]

Mark F. Demers. Uniqueness and exponential mixing for the measure of maximal entropy for piecewise hyperbolic maps. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 217-256. doi: 10.3934/dcds.2020217

[5]

Zhouchao Wei, Wei Zhang, Irene Moroz, Nikolay V. Kuznetsov. Codimension one and two bifurcations in Cattaneo-Christov heat flux model. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020344

[6]

Meilan Cai, Maoan Han. Limit cycle bifurcations in a class of piecewise smooth cubic systems with multiple parameters. Communications on Pure & Applied Analysis, 2021, 20 (1) : 55-75. doi: 10.3934/cpaa.2020257

[7]

Petr Pauš, Shigetoshi Yazaki. Segmentation of color images using mean curvature flow and parametric curves. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1123-1132. doi: 10.3934/dcdss.2020389

[8]

Qingfang Wang, Hua Yang. Solutions of nonlocal problem with critical exponent. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5591-5608. doi: 10.3934/cpaa.2020253

[9]

Fathalla A. Rihan, Hebatallah J. Alsakaji. Stochastic delay differential equations of three-species prey-predator system with cooperation among prey species. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020468

[10]

Kuo-Chih Hung, Shin-Hwa Wang. Classification and evolution of bifurcation curves for a porous-medium combustion problem with large activation energy. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020281

[11]

Huanhuan Tian, Maoan Han. Limit cycle bifurcations of piecewise smooth near-Hamiltonian systems with a switching curve. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020368

[12]

Christian Aarset, Christian Pötzsche. Bifurcations in periodic integrodifference equations in $ C(\Omega) $ I: Analytical results and applications. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 1-60. doi: 10.3934/dcdsb.2020231

[13]

Karol Mikula, Jozef Urbán, Michal Kollár, Martin Ambroz, Ivan Jarolímek, Jozef Šibík, Mária Šibíková. Semi-automatic segmentation of NATURA 2000 habitats in Sentinel-2 satellite images by evolving open curves. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1033-1046. doi: 10.3934/dcdss.2020231

[14]

Linhao Xu, Marya Claire Zdechlik, Melissa C. Smith, Min B. Rayamajhi, Don L. DeAngelis, Bo Zhang. Simulation of post-hurricane impact on invasive species with biological control management. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 4059-4071. doi: 10.3934/dcds.2020038

[15]

Qian Liu, Shuang Liu, King-Yeung Lam. Asymptotic spreading of interacting species with multiple fronts Ⅰ: A geometric optics approach. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3683-3714. doi: 10.3934/dcds.2020050

[16]

Linfeng Mei, Feng-Bin Wang. Dynamics of phytoplankton species competition for light and nutrient with recycling in a water column. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020359

[17]

Attila Dénes, Gergely Röst. Single species population dynamics in seasonal environment with short reproduction period. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020288

[18]

Michel Chipot, Mingmin Zhang. On some model problem for the propagation of interacting species in a special environment. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020401

[19]

Gi-Chan Bae, Christian Klingenberg, Marlies Pirner, Seok-Bae Yun. BGK model of the multi-species Uehling-Uhlenbeck equation. Kinetic & Related Models, 2021, 14 (1) : 25-44. doi: 10.3934/krm.2020047

[20]

Guoyuan Chen, Yong Liu, Juncheng Wei. Nondegeneracy of harmonic maps from $ {{\mathbb{R}}^{2}} $ to $ {{\mathbb{S}}^{2}} $. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3215-3233. doi: 10.3934/dcds.2019228

2019 Impact Factor: 1.27

Metrics

  • PDF downloads (36)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]