Novel predator-prey model admitting exact analytical solution

TL;DR

This paper introduces a new predator-prey model within a Hamiltonian framework that admits an exact analytical solution, overcoming the limitations of the classical Lotka-Volterra model which requires numerical methods.

Contribution

It develops a Hamiltonian formalism for predator-prey models and identifies a unique model with an explicit analytical solution, extending to power-law and multi-component systems.

Findings

Identified a predator-prey model with an explicit solution

Derived the solution using elementary functions

Extended the model to power-law and N-component systems

Abstract

The Lotka-Volterra predator-prey model still represents the paradigm for the description of the competition in population dynamics. Despite its extreme simplicity, it does not admit an analytical solution, and for this reason, numerical integration methods are usually adopted to apply it to various fields of science. The aim of the present work is to investigate the existence of new predator-prey models sharing the broad features of the standard Lotka-Volterra model and, at the same time, offer the advantage of possessing exact analytical solutions. To this purpose, a general Hamiltonian formalism, which is suitable for treating a large class of predator-prey models in population dynamics within the same framework, has been developed as a first step. The only existing model having the property of admitting a simple exact analytical solution, is identified within the above class of models. The solution of this special predator-prey model is obtained explicitly, in terms of known elementary functions, and its main properties are studied. Finally, the generalization of this model, based on the concept of power-law competition, as well as its extension to the case of $N$-component competition systems, are considered.