The Yoccoz–Birkeland livestock population model coupled with random price dynamics

Abstract

We study a random version of the population-market model proposed by Arlot, Marmi, and Papini in Arlot et al. (2019). The latter model is based on the Yoccoz–Birkeland integral equation and describes a time evolution of livestock commodities prices which exhibits endogenous deterministic, stochastic behavior. We introduce a stochastic component inspired by the Black–Scholes market model into the price equation, and we prove the existence of a random attractor and a random invariant measure. We compute numerically the fractal dimension and the entropy of the random attractor, and show its convergence to the deterministic one as the volatility in the market equation tends to zero. We also investigate in detail the dependence of the attractor on the choice of the time-discretization parameter. We implement several statistical distances to quantify the similarity between the discretized systems’ attractors and the original ones. In particular, following a work by Cuturi (2013), we use the Sinkhorn distance. This distance is a discrete and penalized version of the Optimal Transport Distance between two measures, given a transport cost matrix.