The super- and subsoltuion theory is developed when the nonlinear reaction function is discontinuous at stable steady states. The solution is defined in a weak sense using a notion of setvalued integral. The existence and the uniqueness of the weak solution are obtained together with a comparison principle. The lack of Lipschitz continuity of the problem forces the solution to reach such stable steady states in a finite time. This discontinuity driven dynamics produces physically interesting phenomena such as finite time extinction, free boundaries, and compactly supported solutions. The developed theory is applied to the Allee effect and a few criteria for the initial population distribution are found, which decide the extinction, survival, expansion, and blowup of the population.