This paper mainly focuses on the Gauze-type predator-prey model with Crowley-Martin functional response. The local stability of the equilibria is investigated by analyzing the characteristic equation and using the Routh-Hurwitz criterion. Besides, dynamic behavior has been studied by using the center manifold theorem and normal form theory. Finally, several numerical simulations not only verify the theoretical results of Hopf bifurcation but also display more interesting dynamical properties of the model.

In this study, it is aimed to demonstrate the consistency of the previously given formulations of the parameters of 5 or less independent variables of multiple linear regression with data examples. All the values in the parameter formulas of 5 or less independent variables with respect to one dependent variable were found to be the same as all the parameter values obtained from the related equation system. In other words, the values found with the parameter formulas and the parameter values found with the computer program were found to be the same.

This paper is about a class of discrete-time dynamical systems with competitive effects. The local stability of the positive equilibrium point of the system and the conditions for the existence of flip bifurcation and Neimark-Sacker bifurcation are discussed by using the center manifold theorem and bifurcation theory. In addition, the direction of the flip bifurcation and Neimark-Sacker bifurcation is given. Furthermore, a feedback control strategy is employed to control bifurcation and chaos in the system. Finally, flip bifurcation, Neimark-Sacker bifurcation and chaos control strategy are verified with the help of numerical simulations.

This paper deals with the stability of phase locking for the identical Kuramoto model, where each oscillator is influenced sinusoidally by two neighboring oscillators. By studying the model with bidirectionally non-symmetric coupling in a ring configuration, all phase-locked solutions are comprehensively delineated, and the basin of attraction for the stable phase-locked state is estimated. The stability of these phase-locked solutions is clearly established, highlighting that only the synchronized state and splay-state are stable equilibria. The crucial tools in this work are the standard linearization technique and the nonlinear analysis arguments.

The nitrogen uptake model for plant roots is an advection-diffusion equation subject to double Robin boundary conditions in Cartesian coordinates and its analytical method is expected to accurately estimate the quantity of nutrient uptake and fertilization. Firstly, the Michaelis-Menten (MM) kinetics function in the left boundary condition is changed into a function of time by numerical fitting and the nonlinear left Robin boundary condition then becomes a linear one in order to use traditional analytical methods. Based on the eigenfunction expansion method originally built by Golz and Dorroh, the nitrogen uptake model is homogenized and its eigenvalues are obtained from the Sturm-Liouville problem. Because the convergence of this eigenfunction expansion method is slow around the left boundary, i.e., root surface, we additionally consider the Laplace transform to solve the nitrogen uptake model. However, the solution after Laplace transform involves composite functions and numerical inverse Laplace transforms are introduced to obtain the final solutions. The analytical and numerical solutions show that the nitrogen concentration profiles along the distance from the root surface are convex upward and almost horizontal in the middle part with large gradients at both ends. The numerical simulation demonstrates that the eigenfunction expansion method can reach a satisfactory accuracy and the Laplace transform method with Stehfest inversion has higher calculation efficiency.

In this study, we develop a mathematical model through a system of first-order nonlinear ordinary differential equations. This model covers the dynamics between vulnerable cyber accounts and those implicated in cyber vices such as bullying, scams, spreading of misinformation, and the creation of harmful digital footprints. It further explores the mechanisms of recovery and relapse among these accounts. Through some mathematical analysis, we apply relevant theorems to affirm the model’s fundamental properties, which includes its existence, uniqueness, positivity, and boundedness. We also determine the model’s cyber vice-free and endemic equilibrium states, analyzing their local and global asymptotic stability based on when the basic reproduction number $R_{cb}$ is greater or less than one. Simulation exercises are conducted to substantiate our theoretical findings and demonstrate the model’s behavior in relation to $R_{cb}.$ The simulation outcomes reveal an escalating trend in cyber vices, showing the necessity for targeted interventions that promote a more secure online environment for users and the broader cyber space.

Tick-borne diseases pose a potential risk to public health, which is influenced by the stage structure and seasonal reproduction of tick populations. In this paper, a model that explains the transmission dynamics of pathogens among ticks and hosts is formulated and analyzed, considering birth pulse and pesticide pulse on tick population at different moments. Using the stroboscopic mapping for the disease-free system, we prove a globally asymptotically stable positive periodic solution exists when the pulsed pesticide spraying intensity is less than a critical threshold. Applying the comparison theorem for the impulsive differential system, the conditions for global attraction of the disease-free periodic solution to the investigated system are given. Moreover, we demonstrate the persistence of the studied system and give numerical simulations to verify it. Ultimately, we discuss the case with multiple pesticide sprays and conclude that fewer sprays are more favorable for disease extinction.

This article mainly studies some new discrete Hermite-Hadamard inequalities for integer order and fractional order. For this purpose, the definitions of $h$-convexity and preinvexity for a real-valued function $f$ defined on a set of integers $\mathbb{Z}$ are introduced. Under these two new definitions, some new discrete Hermite-Hadamard inequalities for integer order related to the endpoints and the midpoint $\frac{a+b}{2}$ based on the substitution rules are proposed, and they are generalized to fractional order forms. In addition, for the $h$-convex function on the time scale $\mathbb{Z},$ two new discrete Hermite-Hadamard inequalities for integer order by dividing the time scale differently are obtained.

In this manuscript, an approximate periodic solution of a nonlinear Jerk equation concerning the third order time derivative (ODE) is demonstrated. This method succeeds highly in constructing a periodic solution for Jerk nonlinear differential equations. The results investigated from this method are compared with those determined from other analytical and numerical methods. Our results demonstrate a significant consistency with the numerical solution in addition to the analytical methods.

The fractional Riccati differential equation has a wide application in various areas, for instance, economics and the description of solar activity. In this paper, we focus on the numerical approach of the fractional Riccati differential equations. Two different types of fractional operators are considered under the Riemann-Liouville and Caputo senses. From the numerical simulations, we observe that the explicit finite difference method is not stable. Instead, we employ the implicit finite difference methods to discretize the complicated systems such that stability can be guaranteed. We also exhibit the total error estimations for our algorithms to ensure good approximations. Compared with the other polynomial numerical methods, we can properly extend the model into a larger domain with a large terminal time, which can be verified by numerical examples. Further, we discuss some complex numerical examples to demonstrate the performance of our methods and indicate that our approaches are applicable and tractable to other fractional Riccati equations.

The investigation of predator cooperative hunting and prey refuge is crucial for understanding ecological dynamics. In recent years, the role of cooperative hunting or prey refuge in predator-prey systems has received much attention from researchers. However, the study on the combined effects of predator cooperation and predator-dependent prey refuge in the predation system has not yet been investigated. Therefore, in this paper, we propose a prey-predator model with both the factors of predator-dependent prey refuge and cooperative hunting. The positivity and boundedness of the system’s solutions are investigated, followed by the existence and local stability of the equilibrium points. Sufficient conditions for the existence of Hopf bifurcation of the system are obtained. The direction of Hopf bifurcation in the system is investigated by using the center manifold theorem and normal form method. From the analysis of the model, we find that the dependence coefficient $m$ of the prey refuge ratio on the number of predators may be responsible for the stability of the system. The results also indicate that suitable competition coefficients $s$ and cooperative hunting coefficients $\alpha$ between predators may enable the species to coexist in the long run. Furthermore, we observe two limit cycles of the system when the parameters satisfy certain conditions. Finally, the dynamical behavior of the model is performed through intriguing numerical simulations.

The concept of strong $β^∗$−$I$− open sets in ideal topological spaces is investigated and some of their properties are obtained, also we study these in relation to some other types of sets. Furthermore, by using the new notion, we define the strong $β^∗$−$I$− interior and strong $β^∗$−$I$− closure operators.

In this paper, we introduce the concept of square-mean pseudo $S$-asymptotically $(ω,c)$-periodic for stochastic processes and establish some composition and convolution theorems for such stochastic processes. In addition, we investigate the existence and uniqueness of square-mean pseudo $S$-asymptotically $(ω,c)$-periodic mild solutions to some stochastic fractional integrodifferential equations. We illustrate our main results with an application to stochastic Weyl fractional integrodifferential equations.

This article aims to develop a quick and easy Taylor wavelet collocation method with the help of an operational matrix of integration of the Taylor wavelets. Solving epidemiological models ensures the necessary accuracy for relatively small grid points. Finding the appropriate approximations with a new numerical design is challenging. This study examines the fractional Pine wilt disease (PWD) model. Using the Caputo fractional derivative for the fractional order, we developed the novel wavelet scheme known as the Taylor wavelet collocation technique (TWCM) to approximate the PWD model numerically. The results have been compared between the developed method, the Homotopy analysis transform method (HATM), the RK4 method, and the ND solver. The numerical outcomes demonstrate that (TWCM) is incredibly effective and precise for solving the PWD model of fractional order. The approach under consideration is a powerful tool for obtaining numerical solutions to fractional-order nonlinear differential equations. The fractional order differential operator provides a more advanced way to study the dynamic behavior of different complex systems than the integer order differential operator does. The proposed wavelet method suits solutions with sharp edge/jump discontinuities. Fractional differential equations, delay differential equations, and stiff systems can be solved using this method directly without using any control parameters. For highly nonlinear problems, the TWCM technique yields accurate solutions close to exact solutions by avoiding data rounding and just computing a few terms. Mathematical software Mathematica has been used for numerical computations and implementation.

This paper introduces some new variants of abstract critical point theorems that do not rely on any compactness condition of Palais Smale type. The focus is on locally Lipschitz continuous functional $\Phi$ : $E → R,$ where $E$ is a reflexive banach space. The theorems are established through the utilization of the least action principle, the perturbation argument, the reduction method, and the properties of sub-differential and generalized gradients in the sense of F.H. Clarke. These approaches have been instrumental in advancing the theory of critical points, providing a new perspective that eliminates the need for traditional compactness constraints. The implications of these results are far-reaching, with potential applications in optimization, control theory, and partial differential equations.

The Burgers’ equation has widespread applications across various fields. In this paper, we propose an efficient approach for obtaining the numerical solution to the time-fractional Burgers’ equation. We extend the classical Burgers’ equation to its fractional form by introducing Caputo derivatives. Using the Cole-Hopf transform, we reformulate the problem into a fractional diffusion equation. The Laplace transform method is then applied to convert the equation into an ordinary differential equation (ODE), which can be solved analytically. However, due to the lack of an inverse Laplace transform for this specific form, numerical approximation methods are then utilised to approximate the true solution. Numerical simulations are provided to demonstrate the stability and accuracy of the proposed method.

We consider the Cauchy problem for a general Korteweg-de Vries-Burgers equation and the Cauchy problem for the corresponding linear equation. We will couple together a few novel ideas, several existing ideas and existing results and use rigorous mathematical analysis to accomplish several very important and very interesting results for these Cauchy problems.