Limit cycles for the competitive three dimensional Lotka-Volterra system.

*(English)*Zbl 0960.34022Here, the following competitive three-dimensional Lotka-Volterra system is investigated
\[
\dot x= X(b- Ax),
\]
where \(x= \text{col}(x_1, x_2, x_3)\) is a three-dimensional state vector \(X= \text{diag}(x_1, x_2, x_3)\) is a \(3\times 3\) diagonal matrix, \(b= \text{col}(b_1, b_2, b_3)\) is a positive real vector and \(A= (a_{ij})_{3\times 3}\) is a positive matrix. First, it is proved that the number of limit cycles of the system in \(\mathbb{R}^3_+\) is finite if the system has not any heteroclinic polycycles in \(\mathbb{R}^3_+\). Second, a particular 3-dimensional competitive Lotka-Volterra system with two small parameters is discussed. It is proved that there exists one parameter range in which the system is persistent and has at least two limit cycles, and there exists other parameter ranges in which the system is not persistent and has at least one limit cycle. Hence, some open questions are answered partly in this paper.

Reviewer: Chen Lan Sun (Beijing)

##### MSC:

34C05 | Topological structure of integral curves, singular points, limit cycles of ordinary differential equations |

92D25 | Population dynamics (general) |

34C60 | Qualitative investigation and simulation of ordinary differential equation models |

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\textit{D. Xiao} and \textit{W. Li}, J. Differ. Equations 164, No. 1, 1--15 (2000; Zbl 0960.34022)

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