Nonlinear Dynamical Analysis of a Diffusion-Driven Bacterial Density Model: Integrability and Bifurcation Analysis

This work investigates the dynamical properties of the Kolmogorov–Petrovskii–Piskunov (KPP) equation. We begin by establishing its non-integrability through the Painlevé test. Using a traveling wave transformation, we reduce the equation to a planar dynamical system, which we identify as non-conservative. A subsequent bifurcation analysis, supported by Bendixson’s criterion, rules out the existence of periodic orbits and, thus, periodic solutions—a finding further validated by phase portraits. Furthermore, we classify the types and co-dimensions of the bifurcations present in the system. We demonstrate that under certain conditions, the system can exhibit saddle-node, transcritical, and pitchfork bifurcations, while Hopf and Bogdanov–Takens bifurcations cannot occur. This study concludes by systematically deriving a power series solution for the reduced equation.