  The spreading phenomena in modified Leslie-Gower reaction-diffusion
predator-prey systems are the topic of this paper. We mainly study the
existence of two different types of traveling waves. Be specific, with the aid
of the upper and lower solutions method, we establish the existence of
traveling wave connecting the prey-present state and the coexistence state or
the prey-present state and the prey-free state by constructing different and
appropriate Lyapunov functions. Moreover, for traveling wave connecting the
prey-present state and the prey-free state, we gain more monotonicity
information on wave profile based on the asymptotic behavior at negative
infinite. Finally, our results are applied to modified Leslie-Gower system with
Holling II type or Lotka-Volterra type, and then a novel Lyapunov function is
constructed for the latter, which further enhances our results. Meanwhile, some
numerical simulations are carried to support our results.
