Slow–fast analysis of zooplankton and toxin-producing phytoplankton dynamics: Exploring singular Hopf bifurcations, Canard explosions and oscillatory regime shifts

In most ecological systems, species grow on different time scale, for which the ecological model reveals complex oscillatory dynamics. In this paper, we have considered a phytoplankton (prey)-zooplankton (predator) model with a modified Holling type II functional response, where the predator species are influenced by toxins produced by the prey species. The model is initially analyzed within a deterministic framework, revealing the occurrence of Transcritical, Saddle-node, Hopf, Cusp, Bogdanov–Takens, generalized Hopf and Homoclinic bifurcations. The model is transformed into a slow–fast predator–prey system when the prey and predator species are considered to grow on fast and slow time scales, respectively. We have analyzed the system using Geometric singular perturbation theory (GSPT), asymptotic expansion technique, Blow up and Blow down techniques and observed a rich and complex dynamics such as Canard cycle near singular Hopf bifurcation threshold, Canard explosion, maximal Canard and relaxation oscillations. Additionally, we have investigated the transient dynamics to analyze the system’s short-term behavior before it converges to the ultimate attractor in the asymptotic limit. It is also analyzed how toxicity affects on the transient dynamics. Furthermore, Numerical simulations are conducted to support the theoretical results.