Lagrangians for biological models

TL;DR

This paper reveals that methods for deriving Lagrangians in biological models are fundamentally based on the Jacobi Last Multiplier, enabling the construction of both linear and nonlinear Lagrangians even in previously challenging cases.

Contribution

It clarifies the theoretical basis of existing methods and provides a systematic approach to obtain Lagrangians for biological models using the Jacobi Last Multiplier.

Findings

Unified framework for Lagrangian derivation in biological models

Ability to derive nonlinear Lagrangians for complex systems

Application to host-parasite model where previous methods failed

Abstract

We show that a method presented in [S.L. Trubatch and A. Franco, Canonical Procedures for Population Dynamics, J. Theor. Biol. 48 (1974), 299-324] and later in [G.H. Paine, The development of Lagrangians for biological models, Bull. Math. Biol. 44 (1982) 749-760] for finding Lagrangians of classic models in biology, is actually based on finding the Jacobi Last Multiplier of such models. Using known properties of Jacobi Last Multiplier we show how to obtain linear Lagrangians of those first-order systems and nonlinear Lagrangian of the corresponding single second-order equations that can be derived from them, even in the case where those authors failed such as the host-parasite model.