Aggregative movement and front propagation for bi-stable population models

Front propagation for the aggregation-diffusion-reaction equation v_τ = [D(v)v_x]_x + f(v) is investigated, where f is a bi-stable reaction-term and D(v) is a diffusion coefficient with changing sign, modeling aggregating-diffusing processes. We provide necessary and sufficient conditions for the existence of traveling wave solutions and classify them according to how or if they attain their equilibria at finite times. We also show that the dynamics can exhibit the phenomena of finite speed of propagation and/or finite speed of saturation.