Stability for a Non-Smooth Filippov Ratio-Dependent Predator-Prey System through a Smooth Lyapunov Function

For nonsmooth Filippov systems, the stability of system is assumed to be proved by Lyapunov functions, such as piecewise smooth functions. This extension was based on solution and Clarke generalized gradient. However, it difficult estimate gradient a non-smooth function. In some cases, can divided into continuous discontinuous components. If Lebesgue measure components zero, function utilized prove owing inner product being zero. this paper, we apply ratio-dependent predator-prey system. contrast existing literature, in although components, part not zero but negative. proof stability, negative value condition stricter than zero-value condition. method only needs construct function, which simpler or other methods.