Canard phenomena for a slow-fast predator-prey system with group defense of the prey

In this study, we consider that the predator reproduces much slower than the prey and propose a slow-fast predator-prey system with the group defense of the prey, which is described by the Ivlev-type functional response. We analyze the canard phenomena (relaxation oscillation and canard cycles) of our proposed model by applying the singular perturbation theory. Firstly, we apply the blow-up techniques and visualize the behavior near the special fold point, which allows us to prove the existence of the unique attractive limit cycle and imply the appearance of the canard phenomenon when crossing the fold point, and the numerical simulations verify the theoretical results. Then we further use the singular perturbation theory to investigate the existence and cyclicity of the canard cycle without head and the existence of the canard cycle with head. Our main results of the complex canard phenomenon reveal the fact that some certain species in the ecosystem suddenly burst back many years after being about extinct. & COPY; 2023 Elsevier Inc. All rights reserved.