The investigator continues research on applications of dynamical systems ideas to problems in ecology and population biology, focussing on the notion of persistence and models of the chemostat. The persistence theorem is studied in the context of reaction-diffusion equations and functional differential equations. The chemostat is a laboratory apparatus that is used in ecology to study exploitative competition. (It is a model of a simple lake.) It is one place where both the mathematics and the biological experiments are tractable; all the parameters needed for the equations of the mathematical model are measurable in the laboratory. The basic model does not allow for the delat between nutrient uptake by the organisms in the chemostat and reproduction (cell division). A standard assumption in the model is that the tank is well stirred; hence there are differences in the amounts of nutrient present in different parts of the tank. To study the effect of a nutrient gradient, another device called the gradostat has been proposed by biologists. The mathematics of that device is now fairly well understood. Another way to induce a gradient is to allow the nutrient to enter the tank at one point and to diffuse along with the organisms. The notion of persistence tries to capture the ecological idea that all components of a model ecosystem survive. Such theorems have proved useful in the context of ordinary differential equation models; e.g., in the analysis of the gradostat and of a chemostat with an inhibitor. The concept has now found its way into the mathematical literature in a general form, in particular for an asymptotically smooth semi-dynamical system. The aim of this project is to use the full potential of the persistence theorem by applying it in an infinite dimensional context, to reaction-diffusion equations and differential equations with delays. If the assumptions about the chemostat are changed to remove the well-stirred hypothesis and to let the nutrient and the cells move by diffusion or convection, then a reaction-diffusion equation or a gradostat results. Delayed chemostat models have been studied, but no physiological mechanism has been introduced to account for the delay. The obvious mechanism is the cell cycle, the time between nutrient uptake and cell division.
