Modeling Dynamics of Complex System with Solutions of the Generalized Lotka-Volterra Equations

TL;DR

This paper develops a nonlinear differential equation model using generalized Lotka-Volterra equations to describe evolution in complex systems, analyzing solutions under different forcing conditions and comparing them with climate data.

Contribution

It introduces a new modeling framework for complex systems with forcing functions, extending the classical Lotka-Volterra equations and analyzing their solutions in various contexts.

Findings

Existence of unstable periodic solutions when F=0.

Presence of attractor solutions under periodic forcing.

Qualitative match between model solutions and climate data.

Abstract

A system of nonlinear ordinary differential equations with forcing function is developed to model evolution processes in complex systems. In this system R, C, and P are the resource, consumption, and production functions correspondingly. F is the forcing function. Two cases F=0, and F=A+Bsin(omega x t) are considered. It is shown that if F=0, there exist an unstable periodic solution for a certain set of system coefficients. In the case of periodic forcing, system has an attractor solution. The system developed may have a wide range of applications: in biological and social sciences, in economics, in ecology, and, as well, in modeling climate. A correspondence between the theoretical (numerical) solution and Late Pleistocene climate data dynamics is analyzed on a qualitative level.