This paper is devoted to model (1) for escherichia coli, introduced in [1]. Based on the experimental observations of Budrene and Berg [2, 3], Tyson and coworkers derived (1) with n cell density, c chemotrattactant concentration and s stimulant concentration. Our aim is to study the stability of constant meaning full solution and ultimately boundedness of the solutions. Precisely:
(i) linear and nonlinear stability is proved by using a peculiar Lyapunov function,
(ii) the ultimately boundedness of the solutions in the L2-norm is obtained,
(iii) conditions guaranteeing the global stability are also obtained.