Population dynamics in fresh product markets with no posted prices

We introduce and mathematically study a conceptual model for the dynamics of the buyers population in markets of perishable goods where prices are not posted. Buyers behaviours are driven partly by loyalty to previously visited merchants and partly by sensitivity to merchants intrinsic attractiveness. Moreover, attractiveness evolve in time depending on the relative volumes of buyers, assuming profit/competitiveness optimisation when favourable/unfavourable. While this negative feedback mechanism is a source of instability that promotes oscillatory behaviour, our analysis identifies those critical features that are responsible for the asymptotic stability of stationary states, both in their immediate neighbourhood and globally in phase space. In particular, we show that while full loss of clientele occurs (depending on the initial state) in case of a bounded reactivity rate, it cannot happen when this rate is unbounded and merchants resilience always prevails in this case. Altogether, our analysis provides mathematical insights into the consequences of introducing feedback into buyer-seller interactions and their diversified impacts on the long term levels of clientele in the markets.