Global stability of a population dynamics model with inhibition and negative feedback

Reactions or interactions with the rate which is inhibited by the product or a by-product of the reaction are fairly common in biology and chemical kinetics. Biological examples of such interactions include selfpoisoning of bacteria, the non-lytic immune response and the antiviral (and in particular antiretroviral) therapy. As a case study, in this notice, we consider global asymptotic properties for a simple model with negative feedback (the Wodarz model) where the interaction is inhibited by a by-product of the reaction. The objective of this notice is an extending of a technique that was developed during last decade for the global analysis of models with positive feedback to the systems, where the feedback is negative. Using the direct Lyapunov method with Volterra type Lyapunov functions, we establish conditions for the global stability of this model. This result enables us to evaluate the comparative impacts of the lytic and nonlytic components, the efficiency of the antiviral therapy and the possibility of self-poisoning for bacteria. The same approach can be successfully applied to more complex models with negative feedback.