In the present article, we propose and analyze a new mathematical model for a predator–prey system including the following terms: a Monod–Haldane functional response (a generalized Holling type IV), a term describing the anti-predator behavior of prey populations and one for an impulsive control strategy. In particular, we establish the existence condition under which the system has a locally asymptotically stable prey-eradication periodic solution. Violating such a condition, the system turns out to be permanent. Employing bifurcation theory, some conditions, under which the existence and stability of a positive periodic solution of the system occur but its prey-eradication periodic solution becomes unstable, are provided. Furthermore, numerical simulations for the proposed model are given to confirm the obtained theoretical results.

Heat transfer engineering is significant in many applications, especially in buoyancy natural convection in concentric and eccentric cavities. The biggest practical challenges, in this context, are capturing the self-natural flow, estimating the mixing performance, and determining what parameters affect the temperature distribution in the cavity. In this paper, we focus on the improvement of a mathematical model, in order to enhance the accuracy of the solution, by investigating a new source term in the SST $k-\omega$ turbulence model based on the finite volume technique. The commercial numerical simulation software ANSYS Fluent 2021R1 is implemented to validate the accuracy. A concentric cavity was chosen for validation, the obtained temperature profiles at $\theta =0^{°}$, $\theta ={30}^{°}$, $\theta ={60}^{°}$, $\theta ={90}^{°}$, $\theta ={120}^{°}$, $\theta ={150}^{°}$, and $\theta ={180}^{°}$ were compared with previous experimental data. We applied this model to four eccentric rotating scenarios, including inner counterclockwise rotation, outer counterclockwise rotation, inner–outer clockwise rotation, and inner clockwise–outer counterclockwise rotation. The numerical simulation results reveal that the new source term in the momentum equation can produce superior results in the concentric test-case. The proposed mathematical model can describe the heat transfer under the eccentric co-rotation scenario well. Furthermore, the results for eccentric cases confirm that the rotational direction affects the mixing temperature by generating a large vortex in the cavity, which increases the temperature mixing performance.

Integral equations are essential tools in various areas of applied mathematics. A computational approach to solving an integral equation is important in scientific research. The Haar wavelet collocation method (HWCM) with operational matrices of integration is one famous method which has been applied to solve systems of linear integral equations. In this paper, an approximated analytical method based on the Haar wavelet collocation method is applied to the system of diffusion convection partial differential equations with initial and boundary conditions. This system determines the enzymatic glucose fuel cell with the chemical reaction rate of the Morrison equation. The enzymatic glucose fuel cell model describes the concentration of glucose and hydrogen ion that can be converted into energy. During the process, the model reduces to the linear integral equation system including computational Haar matrices. The computational Haar matrices can be computed by HWCM coding in the Maple program. Illustrated examples are provided to demonstrate the preciseness and effectiveness of the proposed method. The results are shown as numerical solutions of glucose and hydrogen ion.