Dynamic Behavior of the Chavy–Waddy–Kolokolnikov (CWK) Model of Bacterial Clustering in Phototaxis

In this study, we investigate the nonlinear dynamics of the continuity-based Chavy–Waddy–Kolokolnikov (CWK) model for bacterial clustering in phototaxis. The model describes microorganism movement and pattern formation under light stimuli and thus serves as a useful prototype for biological transport processes. We construct and analyze traveling wave solutions that describe diverse aggregation patterns in phototactic systems. To obtain wave solutions, the governing fourth-order nonlinear model is reduced to an ordinary differential equation using a traveling wave transformation. Two analytical techniques, namely the modified auxiliary equation scheme and the extended Π′/Π2-expansion method, are employed to derive these solutions. Thus, a total of 31 distinct analytical solutions are obtained, including breather-type, antikink, plane, antibell-shaped, and localized soliton structures. The graphical representations illustrate the physical characteristics of the solutions, revealing biologically meaningful patterns such as stable aggregation and dispersal fronts. Furthermore, we explicitly compute wave speed, amplitude, and spatial decay rates, providing quantitative analysis of the solution profiles. The comparative analysis demonstrates that only a small subset of solutions overlays with previously reported results, confirming the novelty of the findings. The results enhance the understanding of nonlinear wave propagation and pattern formation in biological systems. They also provide useful insights for modeling bacterial dynamics and other related complex processes.