Lagrangians and Newtonian analogues for Biological Systems

TL;DR

This paper explores the application of variational principles and Lagrangian mechanics to biological population systems, proposing a novel framework to model their dynamics and identify conservation laws.

Contribution

It introduces an algorithm to generate Lagrangian functions for population models and establishes a Newtonian analogy for two-dimensional systems.

Findings

Lagrangian functions can effectively model population dynamics

Two-dimensional models are equivalent to one-dimensional Newtonian systems

Conservation laws are identified using Noether's theorems

Abstract

This study investigates the potential for biological systems to be governed by a variational principle, suggesting that such systems may evolve to minimize or optimize specific quantities. To explore this idea, we focus on identifying Lagrange functions that can effectively model the dynamics of selected population systems. These functions provide a deeper understanding of population evolution by framing their behavior in terms of energy-like variables. We present an algorithm for generating Lagrangian functions applicable to a family of population dynamics models and demonstrate the equivalence between two-dimensional population models and a one-dimensional Newtonian mechanical analog. Furthermore, we explore the existence of conservation laws for these models, utilizing Noether's theorems to investigate their implications.