Steady states of a diffusive Lotka-Volterra system with fear effects

In this paper, we focus on the dynamical properties of a diffusive Lotka-Volterra system with fear effects subjected to the Neuman boundary condition in a bounded domain. The nonexistence of nonconstant solutions is obtained when diffusion rate is sufficiently large. We also establish the existence and multiplicity of the nonhomogeneous steady-state solutions bifurcating from the constant solution of the system by means of the Lyapunov-Schmidt reduction method. In addition, we consider the influence of the intrinsic growth rate on the population dynamics of the model and show that not only can the population density of both predator and prey change by changing the intrinsic growth rate, but also the coexistence equilibrium can possibly be destabilized. At the same time, we also consider the fear effect on the stability, we find that there is no obvious impact on the stability when the fear effect k is small. However, the results on stability will obviously change when the fear effect is large, which are very different from the ODE system (Wang et al. in J Math Biol 73:1179-1204, 2016) without the fear effect.