Boundedness, stability and pattern formation for a predator-prey model with Sigmoid functional response and prey-taxis

This article concerns the structure of the nonconstant steady states for a predator-prey model of Leslie-Gower type with Sigmoid functional and prey-taxis subject to the homogeneous Neumann boundary condition. The existence of bounded classical global solutions is discussed in bounded domains with arbitrary spatial dimension and any prey-taxis sensitivity coefficient. The local stability of the homogeneous steady state is analyzed to show that the prey-taxis sensitivity coefficient destabilizes the stability of the homogeneous steady state when prey defends. Then we study the existence and stability of the nonconstant positive steady state of the system over 1D domain by applying the bifurcation theory and present properties of local branches such as pitchfork and turning direction. Moreover, we discuss global bifurcation, homogeneous steady state solutions, nonconstant steady states solutions, spatio-temporal periodic solutions and spatio-temporal irregular solutions which demonstrate the coexistence and spatial distribution of prey and predator species. Finally, we perform numerical simulations to illustrate and support our theoretical analysis.