Conventional  preytaxis systems assume that preytactic velocity is proportional to the prey  density gradient. However many experiments exploring the predator-prey  interactions show it is the predator's acceleration instead of velocity that is  proportional to the prey density gradient in the preytactic movement, which we  call non-conventional preytaxis. Mathematical models of non-conventional  preytaxis were proposed in the literature to interpret the spatial heterogeneity  of predators and prey observed in experiments. We shall explore the qualitative  behavior of such non-conventional preytaxis systems and establish the global  existence of classical solutions with uniform-in-time bound in any spatial  dimensions. Moreover we prove the global stability of spatially homogeneous  prey-only and coexistence steady states with decay rates under certain  conditions on system parameters. For the parameters outside the stability  regime, we perform linear stability analysis to find the possible patterning  regimes and use numerical simulations to demonstrate that spatially  inhomogeneous time-periodic patterns will typically arise from the  non-conventional preytaxis system. Noticing that conventional preytaxis systems  are unable to produce spatial patterns, our results imply that the  non-conventional preytaxis is indeed more appropriate than conventional  preytaxis to interpret the spatial heterogeneity resulting from the  predator-prey interactions.