The predator-prey model with two limit cycles
The paper investigates the predator-prey system: \[ \dot x=k_ 1(x-dx)- k(x)y,\quad \dot y=(-k_ 3+\beta k(x))y \] with \(k(x)=k_ 2(x)\), \(x\leq \tau\); \(k(x)=k_ 2\tau\), \(x>\tau\), where \(\alpha,\beta,\tau;\) \(k_ 1,k_ 2,k_ 3\) are positive constants. The main results are 1. The system has no limit cycle if \(k_ 3-\beta k_ 2\tau \geq 0\) 2. If \(k_ 3-\beta k_ 2\tau <0\), \(k_ 1+k_ 3-\beta k_ 2\tau >0\) and \(0<\alpha \ll 1\), the system has at least two limit cycles in the first quadrant, one of which is around the singular point (x,y), where \(x=k_ 3/\beta k_ 2\), \(y=(k_ 1-\alpha x)/k_ 2.\)
scientific article; zbMATH DE number 4134481