Boundedness and global stability in a three-species predator-prey system with prey-taxis

This paper deals with a three-species predator-prey system with prey-taxis {u(t) = d(1)Delta u - chi(1)del . (u del w) + gamma(1)uw - theta(1)u - beta(1)uv, (x, t) is an element of Omega x(0, infinity), u(t) = d(2)Delta v - chi(2)del . (u del w) + gamma(2)vw - theta(2)v - beta(2)uv, (x, t) is an element of Omega x(0, infinity), w(t) = d(3)Delta w - (u + v)w + mu w(1 - omega) (x, t) is an element of Omega x(0, infinity), under homogeneous Neumann boundary conditions in a smooth bounded domain Omega subset of R-2, with the nonnegative initial data (u(0), v(0), w(0)) is an element of (W-1,W-p(Omega))(3) with p > 2. By means of the entropy-like inequality and boundedness criterion, we can establish the uniform boundedness of global classical solutions for the above model. Moreover, we show the asymptotic stabilization of the preyonly, semi-coexistence and coexistence steady states under certain conditions by constructing some appropriate Lyapunov functionals and using LaSalle's invariance principle.