Lagrangian Formalism in Biology: II. Non-Standard and Null Lagrangians and their Role in Population Dynamics

TL;DR

This paper develops a new method for deriving non-standard Lagrangians in nonlinear dynamical systems, applies it to population models, and discusses their implications in biological population dynamics.

Contribution

A novel method for deriving non-standard Lagrangians for nonlinear systems with damping is introduced and applied to biological population models.

Findings

Derived non-standard Lagrangians for population models

Identified limitations related to linearizability of systems

Discussed the role of null Lagrangians in population dynamics

Abstract

Non-standard Lagrangians do not display any discernible energy-like terms, yet they give the same equations of motion as standard Lagrangians, which have easily identifiable energy-like terms. A new method to derive non-standard Lagrangians for second-order nonlinear differential equations with damping is developed and the limitations of this method are explored. It is shown that the limitations do not exist only for those nonlinear dynamical systems that can be converted into linear ones. The obtained results are applied to selected population dynamics models for which non-standard Lagrangians and their corresponding null Lagrangians and gauge functions are derived, and their roles in the population dynamics are discussed.