The stability of monotone traveling waves to a stream-population model is established in a particular weighted function space via the method of upper and lower solutions and a squeezing technique. By analyzing the behaviors of the traveling wave for a large time period under a small perturbation, we obtain the results of the local stability. The comparison principle and the squeeze theorem also allows us to prove the global stability of the positive steady-state solutions in the special weighted function space by constructing suitable upper and lower solutions.