This paper is concerned with the Murray–Thomas model for interacting chemicals or species, under Robin boundary data. It is shown that the solutions are bounded and asymptotically converging toward an absorbing set of the phase-space. The stability of the positive constant steady states is discussed via a symmetrization procedure.

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.