Abstract
In this paper, we are concerned with the existence of traveling pulse solutions of one-dimensional generalized Keller-Segel system with nonlinear chemical gradients and small cell diffusion by using the dynamical systems approach, especially based on geometric singular perturbation theory and Poincaré-Bendixson theorem. To show the existence of traveling pulse solutions, we first analyze the dynamics of the system by geometric singular perturbation theory. And then we seek an invariant region for the associated traveling wave equation. Finally, we apply Poincaré-Bendixson theorem to analyze the flow on this invariant region to obtain the existence of traveling pulse solutions in this bounded invariant region. As applications, we present two examples to illustrate our main results.
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