In this article we consider a coupled system of differential equations to describe the evolution of a biological species. The system consists of two equations, a second order parabolic PDE of nonlinear type coupled to an ODE. The system contains chemotactic terms with constant chemotaxis coefficient describing the evolution of a biological species "$u$" which moves towards a higher concentration of a chemical species "$v$" in a bounded domain of $ \mathbb{R}^n$. The chemical "$v$" is assumed to be a non-diffusive substance or with neglectable diffusion properties, satisfying the equation
$v_t = h(u, v).$
We obtain results concerning the bifurcation of constant steady states under the assumption
$ h_v+χ u h_u>0 $
with growth terms $g$. The Parabolic-ODE problem is also considered for the case $h_v+χ u h_u = 0$ without growth terms, i.e. $g \equiv 0$. Global existence of solutions is obtained for a range of initial data.
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