Stability in the process with lag of interaction between mining and processing industries

TL;DR

This paper develops a mathematical model using delay differential equations to analyze the stability of interactions between mining and processing industries, identifying conditions for steady cooperation.

Contribution

It introduces a stability criterion for the dynamic interaction modeled by delay differential equations, specifically for the case of two industries with lagged interactions.

Findings

Derived a stability criterion for the equilibrium point.

Identified time intervals for raw material deliveries to maintain stability.

Reduced the problem to analyzing a second-order quasi-polynomial.

Abstract

A mathematical model of dynamic interaction between mining and processing industries is formalized and studied in the paper. The process of interaction is described by a system of two delay differential equations. The criterion for asymptotic stability of nontrivial equilibrium point is obtained when both industries co-work steadily. The problem is reduced to finding stability criterion for quasi-polynomial of second order. Time intervals between deliveries of raw materials which make it possible to preserve stable interaction between the two industries are found.