The survival analysis for single-species system in a polluted environment. (English) Zbl 1111.34038
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.
Reviewer: Antonio Cañada Villar (Granada)
MSC:
34C60 | Qualitative investigation and simulation of ordinary differential equation models |
92D25 | Population dynamics (general) |
34D05 | Asymptotic properties of solutions to ordinary differential equations |