This study explores the dynamics of a discrete‐time predator–prey system, incorporating a Holling type III functional response and prey refuge, through the piecewise constant argument method. This method keeps things consistent and prevents negative population values, which is often a drawback of older techniques. We take a closer look at fixed points, exploring their existence and stability. We identify the conditions that lead to period‐doubling and Neimark–Sacker bifurcations, and we back up our findings with numerical simulations. Our findings emphasize how important the predation coefficient is for keeping ecological balance and show that, in this model setup, the refuge for prey has a minimal effect on the stability of the system. These insights help us better understand the relationships between predators and their prey, providing valuable guidance for conserving biodiversity and managing ecosystems.

In this work, we apply the Riccati-Bernoulli (RB) sub-ODE approach to provide some vital solitary wave solutions for the nonlinear dispersive modified Benjamin-Bona-Mahony (DMBBM) equation and the Klein-Gordan (KG) equation. The solutions that are provided here are helpful in describing several physical phenomena in inharmonic crystals, cold plasma, compressible fluids and quantum mechanics. The proposed approach is effective and easy, resulting in new generalised solitonic wave profiles. For suitable free parameter values, two-dimensional (2D) and three-dimensional (3D) graphs are depicted to show the shape of the obtained solutions. We also show the effect of the physical parameters on the behaviour of the solutions. Finally, the suggested approach may be extended to different equations appearing in mathematical physics.

The objective of this paper is to investigate the solitonic waves structures to the (2 + 1)-dimensional cubic Klein–Gordon (2D-KG) equation. We employ the Riccati–Bernoulli (RB) sub-ODE approach to extract several types of solitary wave solutions for the 2D-KG model. The RB approach provides solutions in exponential, hyperbolic, trigonometric, and rational forms. This approach avoids laborious calculations and is simple, robust, and efficient. The solutions presented here are beneficial for elucidating various physical phenomena in nature, including plasma electromagnetic interactions, quantum mechanics, and relativistic hydrogen spectrum. Some 2D and 3D graphs are presented to demonstrate the shape of the obtained solutions for suitable values of the free parameters. We additionally demonstrate how the free parameters influence the behavior of the solutions. The results illustrate the simplicity and effectiveness of this method in generating complex nonlinear wave patterns and providing detailed, accurate solutions for nonlinear models in different fields of applied science.

In this work, we consider three nonlinear physical models in mathematical physics through the tanh approach. This method is a strong tool to search for traveling waves resulting from one-dimensional nonlinear wave and nonlinear partial differential equations. This work contributes to a better understanding of kink and anti-kink soliton behavior in nonlinear systems, including dislocation dispersion in crystals, quantum field theory, shallow water, dust-acoustic waves, and nonlinear lattices. Two- and three-dimensional graphs are shown to illustrate the profile of the found solutions for appropriate free parameter values. In addition, we demonstrate how the physical characteristics affect how the solutions behave. Finally, the proposed technique may be applied to many equations emerging in mathematical physics.