Muhammad Sajid Iqbal

This article analyses the biomathematical model that describes the reaction–diffusion predator–prey system with prey-taxis. The analysis includes addressing the question of whether the solution exists and is unique. The next goal is to obtain the exact solutions. The ɸ6 method has been utilized for this purpose. The simulations of the obtained solutions are also incorporated, providing a broader understanding of the solutions.

This work presents the numerical investigations for the deterministic prey–predator model. It is a reaction-diffusion mathematical model for the two-prey interacting with a single predator. The predator–prey model is an important part of the ecosystem, especially for recognizing how the population collaborates in their natural surroundings. The computational numerical scheme, namely, implicit finite difference scheme is developed. The numerical solution of the underlying model is successfully gained — the positivity of the reaction-diffusion model is established. To show the feasibility of the scheme, consistency, positivity, and stability are established by applying mathematical techniques. The numerical scheme yields stable and consistent approximations that remain positive throughout the domain. Moreover, the non-negative initial conditions and homogeneous Neumann conditions are used. For various parameter values, 3D, 2D, and parametric graphs are produced. In addition, the mathematical results are verified through numerical calculations.

This study deals with the time fractional 1D stochastic Poisson–Nernst–Planck (TFSPNP) system under the effect of multiplicative time noise. The M-truncated derivative (MTD) takes into consideration the fractional order time derivative. This is a steady-state Poisson–Nernst–Planck (PNP) equations that have applications in bioelectric dressings and bandages. To obtain the soliton solutions of TFSPNP, we use the generalized exponential rational functional method. These findings are presented in the form of trigonometric, exponential, and hyperbolic functions. Moreover, to show the effect of multiplicative time noise and MTD, we construct the plot of some solutions in the form of three-dimensional, two-dimensional, and their corresponding contours. These plots clearly show the effect of randomness in the wave structures for the exact solitary wave solutions that are attained. In general, the solutions become more stable when a noise term disrupts their symmetry.

This study proposed a modified [ ]-expansion method to seek the new exact traveling wave solutions of the Newell–White–Head–Segel (NWHS) Model. This is an amplitude equation utilized for distributing temperature within a rod that is infinitely thin and long, or determining the flow velocity of a fluid through a pipe that is infinitely long but has a small diameter. The modified [ ]-expansion method is used to extract the new exact solutions. The solutions of this model are categorized in hyperbolic, trigonometric, and rational forms. Moreover, we compare our results with the new auxiliary equation method. The 3D, line and corresponding contour representation of these solutions are depicted by choosing the different values of parameters.

The Biswas–Mollivic equation is a special type of nonlinear Schrödinger equation, which explains the spatio-temporal behaviour of excitable media. In this paper, we investigate the optical soliton solutions of the Biswas–Mollivic equation with cubic–quintic–septic–nonic nonlinearities using the generalized Riccati equation mapping method. This method is efficient and provides new perspectives. It also provides novel insights into the dynamics of excitable media. Our findings add to a better understanding of the complex spatio-temporal patterns that develop in excitable media and have potential applications in the design of new technologies for controlling and manipulating pattern formation. To depict optical soliton solutions graphically, we use the MATLAB software.

This study deals with a stochastic reaction–diffusion biofilm model under quorum sensing. Quorum sensing is a process of communication between cells that permits bacterial communication about cell density and alterations in gene expression. This model produces two results: the bacterial concentration, which over time demonstrates the development and decomposition of the biofilm, and the biofilm bacteria collaboration, which demonstrates the potency of resistance and defense against environmental stimuli. In this study, we investigate numerical solutions and exact solitary wave solutions with the presence of randomness. The finite difference scheme is proposed for the sake of numerical solutions while the generalized Riccati equation mapping method is applied to construct exact solitary wave solutions. The numerical scheme is analyzed by checking consistency and stability. The consistency of the scheme is gained under the mean square sense while the stability condition is gained by the help of the Von Neumann criteria. Exact stochastic solitary wave solutions are constructed in the form of hyperbolic, trigonometric, and rational forms. Some solutions are plots in 3D and 2D form to show dark, bright and solitary wave solutions and the effects of noise as well. Mainly, the numerical results are compared with the exact solitary wave solutions with the help of unique physical problems. The comparison plots are dispatched in three dimensions and line representations as well as by selecting different values of parameters.

In this paper, the exact solutions of classical and stochastic Maccari system is constructed. The exact comparative solutions are examined and plotted. Interesting results in the case of multiplicative noise are formulated and graphically elaborated. The applications of the stochastic Maccari system are added for the physical purpose. The existence of results for the real part of underlying system are discussed first time for a priori estimates. The perturbations, which disturbed the formation of Langmuir waves, are geometrically expressed in this article. Due to the presence of multiplicative noise term, our system brings a real flavor to the dynamics of the problem.