Integrability of a Verhulst biological population model via Lie point symmetries

Abstract

Purpose This study focuses on understanding species population dynamics through a (3 + 1)-dimensional biological population model governed by Verhulst's logistic growth law. The main objective is to explore how nonlinear growth and spatial diffusion mechanisms influence the evolution of populations over time and space. By modeling the system with a nonlinear degenerate parabolic partial differential equation (PDE), the research seeks to provide a deeper theoretical framework for interpreting complex biological and ecological processes. Design/methodology/approach The investigation is carried out using Lie symmetry analysis as the principal analytical tool. The symmetry properties of the governing nonlinear PDE are systematically determined, followed by the construction of one-dimensional optimal subalgebras. These subalgebras enable the reduction of the original high-dimensional PDE into ordinary differential equations (ODEs), making the problem more tractable. Exact invariant solutions are then derived from the reduced equations and further expanded using symmetry group transformations to generate broader classes of solutions and visual interpretations of the system's dynamics. Findings The study successfully obtains several invariant solutions that describe the behavior of the Verhulst population model under different symmetry reductions. These solutions reveal important dynamical features of population spread, interaction and regulation under nonlinear and degenerate diffusion effects. The visual and analytical results demonstrate how successive symmetry reductions can simplify complex PDE models while preserving essential biological characteristics, thereby offering meaningful insight into the spatial–temporal evolution of populations. Originality/value The originality of this work lies in the application of Lie symmetry methods to a high-dimensional, degenerate biological population model incorporating logistic growth. The systematic use of optimal subalgebras and successive reductions provides a structured pathway from complex PDEs to exact analytical solutions. This approach not only advances the mathematical treatment of nonlinear population models but also strengthens the connection between symmetry analysis and practical ecological modeling, offering new perspectives for population control and environmental management strategies.