The canard explosion phenomenon in a predator-prey model with Michaelis-Menten functional response is analyzed this paper by employing the geometric singular perturbation theory. First, some turning points, such as, fold point, transcritical pitchfork are identified; then Hopf bifurcation, relaxation oscillation, together transition from bifurcation to oscillation discussed.

In this article, we study the behavior of two species evolving in a domain with free boundary. This system mimics spread invasive or new predator species, which boundaries represent expanding fronts and are described by Stefan condition. A priori estimates for required functions established. These used to prove existence uniqueness solution.

In this paper, we deduce a predator–prey model with discrete time in the interior of R+2 using new method to study its local dynamics and Neimark–Sacker bifurcation. Compared continuous models, ones have many unique properties that help understand changing patterns biological populations from completely perspective. The existence stability three equilibria are analyzed, formation conditions bif...