![](/images/graphics-bg.png)
Global Dynamics and Bifurcations Analysis of a Two-Dimensional Discrete-Time Lotka-Volterra Model
Joint Authors
Source
Issue
Vol. 2018, Issue 2018 (31 Dec. 2018), pp.1-18, 18 p.
Publisher
Hindawi Publishing Corporation
Publication Date
2018-01-21
Country of Publication
Egypt
No. of Pages
18
Main Subjects
Abstract EN
In this paper, global dynamics and bifurcations of a two-dimensional discrete-time Lotka-Volterra model have been studied in the closed first quadrant R2.
It is proved that the discrete model has three boundary equilibria and one unique positive equilibrium under certain parametric conditions.
We have investigated the local stability of boundary equilibria O(0,0), A(α1-1)/α3,0, B0,(α4-1)/α6 and the unique positive equilibrium C((α1-1)α6-α2(α4-1))/(α3α6-α2α5),(α3(α4-1)+α5(1-α1))/(α3α6-α2α5), by the method of linearization.
It is proved that the discrete model undergoes a period-doubling bifurcation in a small neighborhood of boundary equilibria A(α1-1)/α3,0,B0,(α4-1)/α6 and a Neimark-Sacker bifurcation in a small neighborhood of the unique positive equilibrium C((α1-1)α6-α2(α4-1))/(α3α6-α2α5),(α3(α4-1)+α5(1-α1))/(α3α6-α2α5).
Further it is shown that every positive solution of the discrete model is bounded and the set 0,α1/α3×0,α4/α6 is an invariant rectangle.
It is proved that if α1<1 and α4<1, then equilibrium O(0,0) of the discrete model is a global attractor.
Finally it is proved that the unique positive equilibrium C((α1-1)α6-α2(α4-1))/(α3α6-α2α5),(α3(α4-1)+α5(1-α1))/(α3α6-α2α5) is a global attractor.
Some numerical simulations are presented to illustrate theoretical results.
American Psychological Association (APA)
Khan, A. Q.& Qureshi, M. N.. 2018. Global Dynamics and Bifurcations Analysis of a Two-Dimensional Discrete-Time Lotka-Volterra Model. Complexity،Vol. 2018, no. 2018, pp.1-18.
https://search.emarefa.net/detail/BIM-1135609
Modern Language Association (MLA)
Khan, A. Q.& Qureshi, M. N.. Global Dynamics and Bifurcations Analysis of a Two-Dimensional Discrete-Time Lotka-Volterra Model. Complexity No. 2018 (2018), pp.1-18.
https://search.emarefa.net/detail/BIM-1135609
American Medical Association (AMA)
Khan, A. Q.& Qureshi, M. N.. Global Dynamics and Bifurcations Analysis of a Two-Dimensional Discrete-Time Lotka-Volterra Model. Complexity. 2018. Vol. 2018, no. 2018, pp.1-18.
https://search.emarefa.net/detail/BIM-1135609
Data Type
Journal Articles
Language
English
Notes
Includes bibliographical references
Record ID
BIM-1135609