In this work, we successfully implemented the unified method and modified direct algebraic method (MDAM) to analyse the dynamical nonlinear wave structures of the predator–prey (PP) model using the conformablederivative that plays a significant role in biology. We secured some novel hyperbolic, trigonometric and rational function solutions. Modulation instability (MI) analysis of the PP model was also observed. Furthermore, the physical behaviours of some of the reported results are shown as three-dimensional, two-dimensional and contour profiles choosing suitable parameters. The computation software Mathematica was used to verify all the reported outcomes by substituting them back into the PP model. The results show that our mentioned methods are simple, efficient and precise to follow and can be used for a variety of further complex problems. The resulting solutionsare novel, intriguing and potentially useful in understanding energy transit and diffusion processes in mathematical models of several disciplines of interest, including biological sciences. The findings are exceptional and unique incomparison to previous findings in the literature. Furthermore, our findings are the first step toward understanding the structure and physical behavior of complicated structures. These solutions define the wave performance of the governing model, actually. We feel that this work is timely and will be of interest to a wide spectrum of experts working on physics, engineering, and biological models.