Implementing a Method for Stochastization of One-Step Processes in a Computer Algebra System

TL;DR

This paper presents a computer algebra system implementation of a method to convert deterministic one-step models into stochastic differential equations, facilitating modeling of complex phenomena like population dynamics.

Contribution

It introduces an algorithm for stochastization of one-step processes and implements it in SymPy, enabling automated derivation of stochastic models from deterministic ones.

Findings

Successfully derived stochastic equations for Verhulst and Lotka-Volterra models.

Demonstrated the software's capability to refine models iteratively.

Provided a practical tool for modeling complex systems with stochastic dynamics.

Abstract

When modeling such phenomena as population dynamics, controllable ows, etc., a problem arises of adapting the existing models to a phenomenon under study. For this purpose, we propose to derive new models from the rst principles by stochastization of one-step processes. Research can be represented as an iterative process that consists in obtaining a model and its further re nement. The number of such iterations can be extremely large. This work is aimed at software implementation (by means of computer algebra) of a method for stochastization of one-step processes. As a basis of the software implementation, we use the SymPy computer algebra system. Based on a developed algorithm, we derive stochastic di erential equations and their interaction schemes. The operation of the program is demonstrated on the Verhulst and Lotka-Volterra models.