The predator-prey model with two limit cycles

Abstract

This paper investigates the predator-prey system:

$$\begin{gathered} \dot x = k_1 (x - \alpha x) - k(x)y, \hfill \\ \dot y = ( - k_s + \beta k(x))y \hfill \\ \end{gathered}$$

((1))

with

$$k(x) = \left\{ {\begin{array}{*{20}c} {k_2 x,} & {x \leqslant \tau ,} \\ {k_2 \tau ,} & {x > \tau ,} \\ \end{array} } \right.$$

((2))

whereα, β, τ; k ₁,k ₂,k ₃ are positive constants.

The main results are as follows

• (i) In casek ₃−βk ₂ τ≥0 system (1) has no limit cycle.

• (ii) In casek ₃ −βk ₂ τ<0, k ₁+k₃ −βk ₂ τ>0, and for 0<α≪1, system (1) at least has two limit cycles.