Prey and predator are the important factor of the ecosystem. Generally, it is considered that prey–predator models depends on time and it is only required nonlinear system of equations for its dynamical study. But, it is observed that such species can move from one to place to another and in such a way there is a need of nonlinear equations which also depends on spatial as well. The stochastic prey–predator system are investigated numerically and analytically. The proposed stochastic NSFD is used for numerical study; it is consistent with given system and its linear stability analysis showed that it is unconditionally stable. There are two equilibria one is predator free and second is coexistence equilibrium. These equilibria are successfully gained in the numerical case. Extended generalized Riccati equation mapping method is applied for analytical study. The obtained solutions are of the form rational, hyperbolic, and trigonometric. For the comparative study, the unique physical problems are developed and their simulations are drawn for various choices of the parameters. The graphical behavior depicts the efficacy of our study.

In this paper, the Jacobi elliptic function expansion technique is used to obtain the exact solutions of the sixth order (3+1)-dimensional Kadomtsev–Petviashvili–Sawada–Kotera–Ramani equation. Modulation instability is also discussed for this equation. The main purpose is to find novel exact solutions to this equation by means of a finite series expansion of degree n in terms of Jacobi elliptic functions. Single and combined Jacobi elliptic function solutions are obtained. The JEFE method is found to be highly effective for exact analytical solutions of nonlinear partial differential equations and its flexibility permits the development of several variations for specific problem types. The studied equation is reduced to nonlinear ordinary differential equation of integer order by using the traveling wave transformation. We observe that the solutions obtained are precise, and include periodic wave solutions, quasi-periodic wave solutions and solitary waves. Oscillatory phenomena in systems such as plasma physics and optics can be described by periodic wave solutions. Quasi periodic solutions occur in complex systems with multiple interacting frequencies, which are important in turbulence and nonlinear resonance. Solitary waves (solitons) are stable, localized waves that are critical to fluid dynamics, nonlinear optics, and plasma physics, and that model stable wave propagation in many applications. In addition, graphical representations of some solutions are presented to show the direct viewing analysis of the solutions. The results confirm that the proposed technique is a powerful tool for solving a large variety of NPDEs in mathematical physics, and may have applications to other nonlinear evolution equations.

Formulating suitable dynamical models to describe the interplay of different chemical substance in biology and chemistry is a very interesting topic. In this article, we set up a new plankton population dynamical system accompanying delay. Taking advantage of fixed point theorem and appropriate function, we derive the criteria on existence and uniqueness, boundedness of the solutions of the established plankton population dynamical system. By applying Hopf bifurcation and stability theorem of delayed dynamical system, we analyze the emergence of Hopf bifurcation and stability trait of the established plankton population dynamical system. Novel delay-independent criteria guaranteeing the emergence of Hopf bifurcation and stability of the system are derived. By virtue of two different controllers, we have effectively controlled the time of occurrence of Hopf bifurcation and stability domain of the established system. The influence of time delay on Hopf bifurcation and stability of the system is shown. The study shows that when the delay crosses a critical value, then Hopf bifurcations of the established plankton population dynamical system and its controlled systems arise. Numerical experiment outcomes are included to support the derived main fruits. The derived results of this article are completely new and have important theoretical value in balancing and controlling the concentrations of distinct chemical substances.

In this manuscript, the well-known stochastic Burgers’ equation in under investigation numerically and analytically. The stochastic Burgers’ equation plays an important role in the fields of applied mathematics such as fluid dynamics, gas dynamics, traffic flow, and nonlinear acoustics. This study is presented the existence, approximate, and exact stochastic solitary wave results. The existence of results is shown by the help of Schauder fixed point theorem. For the approximate results the proposed stochastic finite difference scheme is constructed. The analysis of the proposed scheme is analyzed by presented the consistency and stability of scheme. The consistency is checked under the mean square sense while the stability condition is gained by the help of Von-Neumann criteria. Meanwhile, the stochastic exact solutions are constructed by using the generalized exponential rational function method. These exact stochastic solutions are obtained in the form of hyperbolic, trigonometric and exponential functions. Mainly, the comparison of both numerical and exact solutions are analyzed via simulations. The unique physical problems are constructed from the newly constructed soliton solutions to compare the numerical results with exact solutions under the presence of randomness. The 3D and line plots are dispatched that are shown the similar behavior by choosing the different values of parameters. These results are the main innovation of this study under the noise effects.