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The parasitoid is a broad evolutionary association of hymenopteran insects which are well‐known as biological control agents. Parasites are different from predators because parasites only take resources from one host, whereas predators eat many preys. The preeminent target of this study is to present a fractional model of host–parasitoid population dynamical system through the Caputo operator. The host–parasitoid population dynamical model is a system of three dimensional coupled differential equations. This research also investigates the possibility for obtaining new chaotic behaviors with singular fractional operator and shows the chaotic behavior at different values of fractional order. Furthermore, two numerical schemes Adam–Bashforth–Moulton and new Toufik–Atangana mechanism were suggested to solve fractional host–parasitoid population numerically. Again, some numerical simulations are perform to assess the efficiency of the newly proposed method. Some attractive illustration is presented through graphically. We strongly believe that the proposed work will open various new doors of investigations toward modeling on parasitoid.

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This paper presents many new complex combined dark-bright soliton solutions obtained with the help of the accurate sine-Gordon expansion method to the B-type Kadomtsev-Petviashvili-Boussinesq equation with binary power order nonlinearity. With the use of some computational programs, we plot many new surfaces of the results obtained in this paper. In addition, we present the interactions between complex travelling wave patterns and their solitons.

In this research paper, we implement the sine-Gordon expansion method to two governing models which are the (2+1)-dimensional Nizhnik–Novikov–Veselov equation and the Caudrey–Dodd–Gibbon–Sawada–Kotera equation. We use conformable derivative to transform these nonlinear partial differential models to ordinary differential equations. We find some wave solutions having trigonometric function, hyperbolic function. Under the strain conditions of these solutions obtained in this paper, various simulations are plotted.

The main aim of this paper is to investigate the various dimensional nonlinear Fokas and Breaking soliton equations via a powerful analytical method, namely, sine-Gordon expansion method. Many new solutions such as complex combined dark-bright soliton solutions, singular and hyperbolic functions are derived. Choosing the suitable values of these parameters, various novel simulations are also plotted. Such results explain the wave behavior of the governing models, physically.