The paper deals with the study of a model for the effect of toxicant levels on a single-species ecosystem in the case where there is a constant emission of a toxicant. This model is a particular case of a PDE system which was proposed by B. Buonomo, A. Di Liddo and I. Sgura [A diffusive-convective model for the dynamics of population-toxicant intentions: some analytical and numerical results, Math. Bios. 157, 37–64 (1999)]. In the present paper, the model is given by the ODE system \[ \begin{aligned} \frac{dx}{dt} &= x(b_0-d_0-\alpha c_0-fx), \\ \frac{dc_0}{dt} &= kc_e-(r+m+b_0-fx)c_0, \\ \frac{dc_e}{dt} &= -kc_ex+(r+d_0+\alpha c_0)c_0x-hc_e+u(t),\end{aligned} \] where all constants are positive. Sufficient conditions for weak persistence and extinction are found. Moreover, the threshold between weak persistence in the mean and extinction is established in some cases.