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

TL;DR

This paper introduces a stochastic extension to a livestock population-price model, analyzing the effects of market volatility on the system's long-term behavior and attractor properties through numerical and statistical methods.

Contribution

It incorporates a Black-Scholes inspired stochastic component into the Yoccoz-Birkeland model and studies the resulting random attractor and invariant measure.

Findings

Existence of a random attractor and invariant measure proven.

Numerical computation of fractal dimension and entropy of the attractor.

Attractor convergence to the deterministic case as market volatility decreases.

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 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.