Motile biological cells in tissue often display the phenomenon of durotaxis, i.e. they tend to move towards stiffer parts of substrate tissue. The mechanism for this behavior is not understood. We consider simplified models for durotaxis based on the classic persistent random walker scheme. Even a one- dimensional model of this type sheds interesting light on the classes of behavior cells might exhibit. Our results strongly indicate that cells must be able to sense the gradient of stiffness in order to show the effects observed in experiment. This is in contrast to the claims in recent publications that it is sufficient for cells to be more persistent in their motion on stiff substrates to show durotaxis: i.e., if would be enough to sense the value of the stiffness.