The global existence and the instability of constant steady states are obtained together for a Keller-Segel type chemotactic aggregation model. Organisms are assumed to change their motility depending only on the chemical density but not on its gradient. However, the resulting model is closely related to the logarithmic model, u(t) = Delta(gamma(v)u) = del. (gamma(v))). u(t)=epsilon Delta v-v+u, where is the motility function. The global existence is shown for all chemosensitivity constant with a smallness assumption on . On the other hand constant steady states are shown to be unstable only if and is small. Furthermore, the threshold diffusivity is found that, if , any constant steady state is unstable and an aggregation pattern appears. Numerical simulations are given for radial cases.