Simultaneous finite time blow-up in a two-species model for chemotaxis

A system of two classical chemotaxis equations, coupled with an elliptic equation for an attractive chemical, is analyzed. Depending on the parameter values for the three respective diffusion coefficients and the two chemotactic sensitivities in the radial symmetric setting, conditions are given for global existence of solutions and finite time blow-up. A question of interest is, whether there exist parameter regimes, where the two chemotactic species differ in their long time behavior. This questions arises in the context of differential chemotactic behavior in early aggregates of Dictyostelium discoideum.