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<p>This study examines a two-dimensional nonlinear degenerate parabolic partial differential equation (PDE) model that characterises biological population dynamics in porous environments, where fluid flow and nutrient diffusion substantially affect population behaviour. We utilise Lie symmetry analysis to explain the model’s intrinsic structure. We establish an optimal system of one-dimensional subalgebras and investigate nonlinear self-adjointness to derive invariant solutions and conservation laws, respectively. We provide new exact solutions that clarify the relationship between population density and the model parameters, including the growth rate (<i>h</i>) and porous constant <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\((\theta )\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. These solutions demonstrate the effect of modifying <i>h</i> and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\theta \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>θ</mi> </math></EquationSource> </InlineEquation> on population distribution patterns. Furthermore, conservation laws are derived from the nonlinear self-adjointness property, explaining the conserved quantities of the system. These findings enhance the understanding of population dynamics in porous media and can be applied to predict long-term population behaviour in conditions such as soil and sediments. The study provides a framework for understanding population behaviour under various environmental conditions.</p>

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Conservation laws and symmetry-driven analysis of a biological population model in porous media

  • 期刊论文

  • 发表日期：

    2026年4月2日

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  • Urvashi Joshi,
  • Aniruddha Kumar Sharma,
  • Rajan Arora

摘要

This study examines a two-dimensional nonlinear degenerate parabolic partial differential equation (PDE) model that characterises biological population dynamics in porous environments, where fluid flow and nutrient diffusion substantially affect population behaviour. We utilise Lie symmetry analysis to explain the model’s intrinsic structure. We establish an optimal system of one-dimensional subalgebras and investigate nonlinear self-adjointness to derive invariant solutions and conservation laws, respectively. We provide new exact solutions that clarify the relationship between population density and the model parameters, including the growth rate (h) and porous constant \((\theta )\) ( θ ) . These solutions demonstrate the effect of modifying h and \(\theta \) θ on population distribution patterns. Furthermore, conservation laws are derived from the nonlinear self-adjointness property, explaining the conserved quantities of the system. These findings enhance the understanding of population dynamics in porous media and can be applied to predict long-term population behaviour in conditions such as soil and sediments. The study provides a framework for understanding population behaviour under various environmental conditions.