Limit Cycles for the Competitive Three Dimensional Lotka–Volterra System

By Dongmei Xiao, Wenxia Li. 2000

96 citations recorded by the source corpus.

Abstract In the first part of this paper, it is proved that the number of limit cycles of the competitive three-dimensional Lotka–Volterra system in R 3 + is finite if this system has not any heteroclinic polycycles in R 3 + . In the second part of this paper, a 3D competitive Lotka–Volterra system with two small parameters is discussed. This system always has a heteroclinic polycycle with three saddles. It is proved that there exists one parameter range in which the system is persistence and has at least two limit cycles, and there exists other parameter ranges in which the system is not persistence and has at least one limit cycle.

Subfields: Geometry and Topology, Public Health, Environmental and Occupational Health

View Paper: https://www.semanticscholar.org/paper/ce1aa088011a187541fa5d6e61d9868efa4ecc4b

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Geometry and Topology

1. George Yin
2. Yong Luo
3. Chao Zhu
4. Zhengyi Lu

In Public Health, Environmental and Occupational Health

1. Jianhai Bao
2. Chenggui Yuan
3. George Yin
4. Chao Zhu
5. Xuerong Mao
6. Meng Liu
7. Yong Luo
8. Zhengyi Lu

Citations in the corpus

• Monotone Dynamical Systems (2013, unknown): “See for example [81, 222, 236, 238, 239, 240, 242] and references therein.”
• On competitive Lotka-Volterra model in random environments (2009, methodology): “of a globally stable equilibrium point in various submodels of the n-dimensional Lotka–Volterra system was obtained in [27] later; limit cycles of some deterministic competitive three-dimensional Lotka–Volterra systems were treated in [53].”
• Two limit cycles in three-dimensional Lotka-Volterra systems☆ (2002, methodology): “Hofbauer and So [7] and Xiao and Li [8] show tha t for Case 27, the local s table posi t ive equi l ibr ium is surrounded by two limit cycles; one is from the Hopf bifurcat ion theorem and the other is guaranteed by the Poincar~-Bendixson theorem.”
• Competitive Lotka–Volterra population dynamics with jumps (2011, background): “n)T,B = (bij)n×n. There is an extensive literature concerned with the dynamics of Eq. (1.2) and we here only mention Gopalsamy [4], Kuang [7], Li et al. [9], Takeuchi and Adachi [22, 23], Xiao and Li [24]. In particular, the books by Gopalsamy [4], and Kuang [7] are good references in this area. On ”
• From local to global behavior in competitive Lotka-Volterra systems (2002, background): “Hofbauer and So [5], Xiao and Li [17] and E.”
• Three limit cycles for a three-dimensional Lotka-Volterra competitive system with a heteroclinic cycle☆ (2003, background): “Hofbauer and So [l] and Xiao and Li [5] construct systems with two limit cycles in class 27.”
• 4. Monotone Dynamical Systems (2005, unknown): “See for example [81, 222, 236, 238, 239, 240, 242] and references therein.”