Persistence and extinction of single population in a polluted environment.

*(English)*Zbl 1084.92032From the introduction: Today, the most threatening problem to the society is the change in environment caused by pollution, affecting the long term survival of species, human life style and biodiversity of the habitat. Therefore the study of the effects of toxicant on the population and the assessment of the risk to populations is becoming more important.

Recently, a first attempt to consider a spatial structure has been carried out [see B. Buonomo and A. di Liddo, Dyn. Syst. Appl. 8, 181–196 (1999; Zbl 0936.35087); Nonlinear Anal., Real World Appl. 5, No. 4, 749–762 (2004; Zbl 1074.92036)] where a reaction-diffusion model is proposed to describe the the dynamics of a living population interacting with a toxicant present in the environment (external toxicant) through the amount of toxicant stored into the bodies of the living organisms (internal toxicant). However, as the authors pointed out, even if the resulting model presents many features which make stimulating its study, such a modelling approach is a rough approximation to the biological phenomena at hand. Buonomo et.al. viewed the internal toxicant as drifted by the living population and then, by balance arguments, they derived a PDE system consisting of two reaction diffusion equations coupled with a first-order convection equation, and the corresponding ODE system was obtained as well [see Math. Biosci. 157, 37–64 (1999)]. This model is the most realistic by now but the analysis of it is so difficult that they only used some analytic and numerical approaches. Obviously, more clear work is deserved to do. We use some new methods to investigate the model made by Buonomo et al. and conditions of survival and extinction are obtained.

Recently, a first attempt to consider a spatial structure has been carried out [see B. Buonomo and A. di Liddo, Dyn. Syst. Appl. 8, 181–196 (1999; Zbl 0936.35087); Nonlinear Anal., Real World Appl. 5, No. 4, 749–762 (2004; Zbl 1074.92036)] where a reaction-diffusion model is proposed to describe the the dynamics of a living population interacting with a toxicant present in the environment (external toxicant) through the amount of toxicant stored into the bodies of the living organisms (internal toxicant). However, as the authors pointed out, even if the resulting model presents many features which make stimulating its study, such a modelling approach is a rough approximation to the biological phenomena at hand. Buonomo et.al. viewed the internal toxicant as drifted by the living population and then, by balance arguments, they derived a PDE system consisting of two reaction diffusion equations coupled with a first-order convection equation, and the corresponding ODE system was obtained as well [see Math. Biosci. 157, 37–64 (1999)]. This model is the most realistic by now but the analysis of it is so difficult that they only used some analytic and numerical approaches. Obviously, more clear work is deserved to do. We use some new methods to investigate the model made by Buonomo et al. and conditions of survival and extinction are obtained.