Limit cycles for the competitive three dimensional Lotka-Volterra system
Here, 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.
competitive three-dimensional Lotka-Volterra system
scientific article; zbMATH DE number 1496032