We consider a reaction-diffusion system modeling chemotaxis, which describes the situation of two species of bacteria competing for the same nutrient. We use Moser-Alikakos iteration to prove the global existence of the solution. We also study the existence of nontrivial steady state solutions and their stability.

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In this paper we consider a nonlocal differential equation, which is a limiting equation of one dimensional Gierer-Meinhardt model. We study the existence of spike steady states and their stability. We also construct a single-spike quasi-equilibrium solution and investigate the dynamics of spikelike solutions.