Invariance Analysis, Symmetry Reduction and Conservation Laws for Biological Population in Porous Media

TL;DR

This paper applies Lie symmetry analysis to a biological population model in porous media, uncovering invariance properties, conservation laws, and invariant solutions to understand long-term population behavior.

Contribution

It introduces a symmetry-based framework for analyzing population dynamics in porous media, revealing conservation laws and invariant solutions not previously identified.

Findings

Identification of Lie symmetries in the population model

Derivation of conservation laws and invariant solutions

Insights into parameters influencing population distribution

Abstract

This research paper talks about using complex mathematical tools to study and figure out the behavior of biological populations in porous media. Porous media offer a unique environment where various factors, including fluid flow and nutrient diffusion, significantly influence population dynamics. The theory of Lie symmetries is used to find inherent symmetries in the governing equation of the population model, helping to find conservation laws and invariant solutions. The derivation and analysis of the optimal system provide insights into the most influential parameters affecting population growth and distribution. Furthermore, the study explores the construction of invariant solutions, which aid in characterizing long-term population behavior. The article concludes with the non-linear self-adjointness property and conservation laws for the model.