  Diffusion-driven instability and bifurcation analysis are studied in a
predator-prey model with herd behavior and quadratic mortality by incorporating
multiple Allee effect into prey species. The existence and stability of the
equilibria of the system are studied. And bifurcation behaviors of the system
without diffusion are shown. The sufficient and necessary conditions for Turing
instability occurring are obtained. And the stability and the direction of Hopf
and steady state bifurcations are explored by using the normal form method.
Furthermore, some numerical simulations are presented to support our
theoretical analysis. We found that too large diffusion rate of prey prevents
Turing instability from emerging. Finally, we summarize our findings in the
conclusion.
