Coupling the Yoccoz-Birkeland population model with price dynamics: Chaotic livestock commodities market cycles

We propose a new model for the time evolution of livestock commodities prices which exhibits endogenous deterministic stochastic behaviour. The model is based on the Yoccoz–Birkeland integral equation, a model first developed for studying the time-evolution of single species with high average fertility, a relatively short mating season and density-dependent reproduction rates. This equation is then coupled with a differential equation describing the price of a livestock commodity driven by the unbalance between its demand and supply. At its birth the cattle population is split into two parts: reproducing females and cattle for butchery. The relative amount of the two is determined by the spot price of the meat. We prove the existence of an attractor (theorem A) and of a non-trivial periodic solution (theorem B) and we investigate numerically the properties of the attractor: the strange attractor existing for the original Yoccoz–Birkeland model is persistent but its chaotic behaviour depends also on the time evolution of the price in an essential way.