We investigate the behavior of positive solutions of \begin{equation*} \left\{ \ \begin {aligned} &\partial_t u = \partial_x^2 (\log u) + u(1-u), \qquad \quad x\in \mathbb {R}, \\ & \lim_{x\to -\infty } \partial_x (\log u) = \alpha, \quad \lim_{x\to \infty } \partial_x (\log u) = - \beta, \end {aligned} \right. \nonumber \end{equation*} where are given nonnegative constants. This equation has a traveling pulse solution if and only if , and any solution vanishes in a finite time when . For the case of , we successfully classified the behavior of solutions into three categories: spreading, boundedness of total mass, and vanishing. Then, we describe in detail the asymptotic behavior of solutions in each category. First, the transition layer of the spreading solution propagates with the speed of a traveling front. Second, the mass-bounded solution converges to a traveling pulse solution. Third, a rescaled profile of vanishing solution converges to some traveling pulse in an asymptotically self-similar manner. In addition, the heteroclinic orbits that connect the traveling pulse to are constructed.