Dynamic analysis of a modified fast-slow Leslie–Gower aquatic ecosystem with weak Allee effect and piecewise-smooth functional response

In this paper, a modified fast-slow Leslie–Gower model with weak Allee effect and piecewise-smooth functional response is established based on the predator–prey relationship between phycophyta and algivore. The parameter conditions for the existence of a unique internal equilibrium of the system are obtained and the boundary equilibrium bifurcation is discussed in four cases. For the fast-slow system, the dynamic properties near fold point \(Q_2\) $Q_2$ are discussed in two cases, and it is concluded that the system produces a unique relaxation oscillation cycle and an inverse canard explosion phenomenon. Furthermore, the fast-slow phenomenon is presented in four cases when the equilibrium is located on the critical manifold of the left system. The system can generate one, two or three stable or unstable relaxation oscillation cycles. Finally, the relaxation oscillation with double jump points and the influence of the time scale parameter \(\varepsilon \) $\epsilon$ on the fast-slow system are discussed.