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
behaviour. We introduce a stochastic component inspired from the Black-Scholes
market model into the price equation and we prove the existence of a random
attractor and of a random invariant measure. We compute numerically the fractal
dimension and the entropy of the random attractor and we 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 attractors of the discretized
systems and the original one. In particular, following a work by Cuturi (2013),
we use the Sinkhorn distance. This is a discrete and penalized version of the
Optimal Transport Distance between two measures, given a transport cost matrix