Orbits in a stochastic Goodwin-Lotka-Volterra model

Bernardo Costa-Lima (1) , Adrien Nguyen Huu (2, 3)

1 Department of Mathematics and Statistics
2 FiME Lab - Laboratoire de Finance des Marchés d'Energie
3 Department of Mathematics and Statistics

This paper examines the cycling behavior of a deterministic and a stochastic version of the economical interpretation of the Lotka-Volterra model, the Goodwin model. We provide a characterization of orbits in the deterministic, highly non-linear version of the model. We then study the cycling behavior for a stochastic version, where a Brownian noise is introduced via an heterogeneous productivity factor. Fundamental properties of the system are provided. We start with a perturbation analysis for small range of volatility and prove that the system locally behaves in average as the deterministic counterpart. We then prove with original stochastic Lyapunov techniques that the system produces cycles around an equilibrium point in finite time for general volatility levels. We also conjecture a limit cycle phenomenon via numerical simulations.

Mots clés: Lotka-Volterra model; Goodwin model; stochastic differential equation; recurrent domains

Domaines: Systèmes dynamiques [math.DS]

Bernardo Costa-Lima, Adrien Nguyen Huu. Orbits in a stochastic Goodwin-Lotka-Volterra model. 2013. ⟨hal-00851407v1⟩