Analysis on existence of bifurcation solutions for a predator-prey model with herd behavior

Abstract

• A predator-prey model with diffusion and herd behavior is investigated. • The local and global stability are determined. • The steady-state bifurcations from both simple and double eigenvalues are studied. • Numerical simulations illustrate and supplement our theoretical analysis. This paper is concerned with a diffusive predator-prey model with herd behavior. The local and global stability of the unique homogeneous positive steady state U * is obtained. Treating the conversion or consumption rate γ as the bifurcation parameter, the steady-state bifurcations both from simple and double eigenvalues are studied near U *. The techniques include the Lyapunov function, the spectrum analysis of operators, the bifurcation theory, space decompositions and the implicit function theorem.