Paper-Scissors-Stone Model for Interacting Population and its Limit Theorem

TL;DR

This paper introduces a stochastic model of three interacting species resembling a paper-scissors-stone game, deriving differential equations from probabilistic laws and analyzing its stochastic structure.

Contribution

The paper develops a novel stochastic collision model for three species and derives associated differential equations using stochastic calculus, including a weak law and a central limit theorem.

Findings

Weak law of large numbers for the model

Central limit theorem established

Derivation of differential equations from stochastic laws

Abstract

This paper treats a random collision model of three species, which is represented by the random time change of three standard Poisson processes. The prey-predator relation in the random collision model looks like paper-scissors-stone game, and the model is called the paper-scissors model. At first, we investigate the stochastic structure of our model. By using stochastic calculus, the model is decomposed into a semi-martingale, and we prove a weak law of large numbers and a central limit theorem. The main purpose of this paper is to obtain an ordinary differential equation from the weak law and a stochastic differential equation from the central limit theorem.