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In this paper, we mainly investigate the dynamics of traveling wavefronts for a model describing host tissue degradation by bacteria. We first establish the existence of spreading speed, and show that the spreading speed coincides with the minimal wave speed of traveling wavefronts. Moreover, a lower bound estimate of the spreading speed is given. Then, we prove that the traveling wavefronts with large speeds are globally exponentially stable, when the initial perturbation around the traveling wavefronts decays exponentially as x → − ∞ , but the initial perturbation can be arbitrarily large in other locations. The adopted methods are the weighted energy and the squeezing technique. [ABSTRACT FROM AUTHOR]