  In this article, we have considered a planar slow-fast modified Leslie-Gower
predator-prey model with a weak Allee effect in the predator, based on the
natural assumption that the prey reproduces far more quickly than the predator.
We present a thorough mathematical analysis demonstrating the existence of
homoclinic orbits, heteroclinic orbits, singular Hopf bifurcation, canard limit
cycles, relaxation oscillations, the birth of canard explosion by combining the
normal form theory of slow-fast systems, Fenichel's theorem and blow-up
technique near non-hyperbolic point. We have obtained very rich dynamical
phenomena of the model, including the saddle-node, Hopf, transcritical
bifurcation, generalized Hopf, cusp point, homoclinic orbit, heteroclinic
orbit, and Bogdanov-Takens bifurcations. Moreover, we have investigated the
global stability of the unique positive equilibrium, as well as bistability,
which shows that the system can display either 'prey extinction', 'stable
coexistence', or 'oscillating coexistence' depending on the initial population
size and values of the system parameters. The theoretical findings are verified
by numerical simulations.
