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In this paper, our aim is mathematical analysis and numerical simulation of a prey-predator model to describe the effect of predation between prey and predator with nonlinear functional response. First, we develop results concerning the boundedness, the existence and uniqueness of the solution. Furthermore, the Lyapunov principle and the Routh–Hurwitz criterion are applied to study respectively the local and global stability results. We also establish the Hopf-bifurcation to show the existence of a branch of nontrivial periodic solutions. Finally, numerical simulations have been accomplished to validate our analytical findings. [ABSTRACT FROM AUTHOR]

This paper is concerned with the dissipativity of Runge-Kutta methods for a class of nonlinear functional-integro-differential equations (FIDEs). The dissipativity results of Runge-Kutta methods for the FIDEs are given. It is shown under a suitable condition that an algebraically stable Runge-Kutta method is dissipative when applied to the FIDEs. Numerical examples are given to illustrate the correctness of our theoretical results. [ABSTRACT FROM AUTHOR]

By using the classical fixed point theorem for operators on a cone, in this paper, some results of one and two positive solutions to a class of nonlinear first-order periodic boundary value problems of impulsive dynamic equations on time scales are obtained. [ABSTRACT FROM AUTHOR]

The objective of this paper is to study oscillation of a forced second-order neutral differential equation. By using the generalized Riccati substitution and integral technique, a new sufficient condition is obtained which insures that all solutions to the studied equation are oscillatory. An illustrative example is included. [ABSTRACT FROM AUTHOR]

In this paper, we use topological degree theory to establish new results on the existence of periodic solutions for a p-Laplacian neutral Rayleigh equation. [ABSTRACT FROM AUTHOR]

In this paper, we consider a time-dependent diffusion problem with two-sided Riemann-Liouville fractional derivatives. By introducing a fractional-order flux as auxiliary variable, we establish the saddle-point variational formulation, based on which we employ a locally conservative mixed finite element method to approximate the unknown function, its derivative and the fractional flux in space and use the backward Euler scheme to discrete the time derivative, and thus propose a fully discrete expanded mixed finite element procedure. We prove the well-posedness and the optimal order error estimates of the proposed procedure for a sufficiently smooth solution. Numerical experiments are presented to confirm our theoretical findings. [ABSTRACT FROM AUTHOR]

This paper investigates the impacts of state-dependent impulses on the stability of switching Cohen-Grossberg neural networks (CGNN) by means of B-equivalence method. Under certain conditions, the state-dependent impulsive switching systems can be reduced to the fixed-time ones. Furthermore, a stability criterion for the considered CGNN using the proposed comparison system is established. Finally, two numerical examples are provided to illustrate the efficiency of the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the generalized Hyers-Ulam stability of a reciprocal type functional equation in several variables in matrix non-Archimedean random normed spaces by direct and fixed point methods. [ABSTRACT FROM AUTHOR]

In this paper, applying the properties of matrix Schur complement and matrix inverse, via some matrix equalities and inequalities, we present new lower and upper solution bounds of the discrete algebraic Riccati equation. Then, by the compressed image principle and a matrix norm inequality, we offer an existence uniqueness condition and a fixed point iteration algorithm for the solution of the discrete algebraic Riccati equation. Finally, a corresponding numerical example demonstrates the effectiveness of the developed results. [ABSTRACT FROM AUTHOR]

In this paper, we use the analysis of relatively dense sets to point out some deficiencies and inaccuracies in the definition of uniformly almost periodic functions which has been proposed in recent works, and we correct it. Some new generalizations of invariance under translation time scales and almost periodic functions are established. Our study will ensure that now we can study almost periodic problems precisely on time scales. [ABSTRACT FROM AUTHOR]

In this paper, a new Monod type chemostat model with delay and impulsive input on two substrates is considered. By using the global attractivity of a k times periodically pulsed input chemostat model, we obtain the bound of the system. By the means of a fixed point in a Poincaré map for the discrete dynamical system, we obtain a semi-trivial periodic solution; further, we establish the sufficient conditions for the global attractivity of the semi-trivial periodic solution. Using the theory on delay functional and impulsive differential equations, we obtain a sufficient condition with time delay for the permanence of the system. [ABSTRACT FROM AUTHOR]

In this paper, we consider a single population system with impulsively unilateral diffusion and impulsive input toxins in a polluted environment. All solutions of the investigated system are proved to be uniformly bounded. By mathematical analysis methods and the theory of impulsive differential equations, the condition of the globally asymptotically stable population-extinction solution of the investigated system is obtained. The permanent condition of the investigated system is also obtained. Finally, numerical analysis is carried out to illustrate our results. Our results provide a reliable theory basis for exploring biological resource management in a polluted environment. [ABSTRACT FROM AUTHOR]

In this paper, we study a relation between exponential dichotomy on time scales and admissibility of the pair $(\mathrm{C}_{\mathrm{rd}}^{b}(\mathbb{T}^{+}, X),L^{p}(\mathbb{T}^{+},X))$ for an evolution family on time scales. We establish a sufficient criterion for the existence of exponential dichotomy on time scales in terms of the admissibility of the pair $(\mathrm{C}_{\mathrm{rd}}^{b}(\mathbb{T}^{+}, X),L^{p}(\mathbb{T}^{+},X))$ for the evolution family. Conversely, with the help of exponential dichotomy on time scales, we give the admissibility of the pair $(\mathrm{C}_{\mathrm{rd}}^{b}(\mathbb{T}^{+}, X),L^{p}(\mathbb{T}^{+},X))$ for an input-output equation on time scales. [ABSTRACT FROM AUTHOR]

In this work, we address the periodic coverage phenomenon on arbitrary unbounded time scales and initiate a new idea, namely, we introduce the concept of changing-periodic time scales. We discuss some properties of this new concept and illustrate several examples. Specially, we establish a basic decomposition theorem of time scales which provides bridges between periodic time scales and an arbitrary time scale with a bounded graininess function μ. Based on this result, we introduce local-almost periodic and local-almost automorphic functions on changing-periodic time scales and study some related properties. The concept of changing-periodic time scales introduced in this paper will help in understanding and removing the serious deficiency which arises in the study of classical functions on time scales. [ABSTRACT FROM AUTHOR]

The paper focuses on the degenerate Euler numbers, the degenerate Euler polynomials and the degenerate Bernoulli polynomials. By adopting the method of recurrences, two explicit expressions have been established for sums of products of the degenerate Euler polynomials and the degenerate Bernoulli polynomials. As a special case of the degenerate Euler polynomials, an expression can be obtained for... [ABSTRACT FROM AUTHOR]

This paper is concerned with a nonlocal reaction-diffusion equation with nonlocal source and interior absorption ut = ∫ RN J(x-y)(u(y,t) - u(x,t)) dy + λ ∫ Ω uq dx - up, x ∈Ω, t > 0, u(x,t) = 0,x ∉ Ω, t ≥ 0, u(x,0) = u0(x), x ∈ Ω. We investigate the critical extinction exponents for the problem based on some adequate supersolutions and subsolutions. [ABSTRACT FROM AUTHOR]

In this paper, we study a delayed predator-prey model with impulse and, in particular, the existence of the predator-free periodic solution. We employ the approach and techniques coming from epidemiology and calculate the basic reproduction number for the predator. Using the basic reproduction number, we consider the global attraction of the predator-free periodic solution and uniform persistence of the predator. Our results improve the results by Li and Liu (Adv. Differ. Equ. 2016:42, 2016), where they left the open problem of finding a threshold value that determines the eradication and uniform persistence of the predator. Furthermore, we give some numerical simulations to illustrate our results. [ABSTRACT FROM AUTHOR]

This paper is concerned with the existence of entire solutions for a reaction-diffusion equation with doubly degenerate nonlinearity. Here the entire solutions are the classical solutions that exist for all (x,t)∈R2. With the aid of the comparison theorem and the sup-sub solutions method, we construct some entire solutions that behave as two opposite traveling front solutions with critical speeds moving towards each other from both sides of x-axis and then annihilating. In addition, we apply the existence theorem to a specially doubly degenerate case. [ABSTRACT FROM AUTHOR]

In this paper, we present the asymptotic behavior of the solutions for a general class of difference equations. We introduce general theorems in order to study the stability and periodicity of the solutions. Moreover, we use a new technique to study the existence of periodic solutions of this general equation. By using our general results, we can study many special cases that have not been studied previously and some problems that were raised previously. Some numerical examples are provided to illustrate the new results. [ABSTRACT FROM AUTHOR]

A three-species cooperative system with time delays and Lévy jumps is proposed in this paper. Firstly, by comparison method and inequality techniques, we discuss the stability in mean and extinction of species, and the stochastic permanence of this system. Secondly, by applying asymptotic method, we investigate the stability in distribution of solutions. Thirdly, utilizing ergodic method, we obtain the optimal harvesting policy of this system. Finally, some numerical examples are given to illustrate our main results. [ABSTRACT FROM AUTHOR]

The purpose of this article is to discuss the dynamics of the spread of Ebola virus disease (EVD), a kind of fever commonly known as Ebola hemorrhagic fever. It is rare but severe and is considered to be extremely dangerous. Ebola virus transmits to people through domestic and wild animals, called transmitting agents, and then spreads into the human population through close and direct contact among individuals. To study the dynamics and to illustrate the stability pattern of Ebola virus in human population, we have developed an SEIR type model consisting of coupled nonlinear differential equations. These equations provide a good tool to discuss the mode of impact of Ebola virus on the human population through domestic and wild animals. We first formulate the proposed model and obtain the value of threshold parameter R 0 for the model. We then determine both the disease-free equilibrium (DFE) and endemic equilibrium (EE) and discuss the stability of the model. We show that both the equilibrium states are locally asymptotically stable. Employing Lyapunov functions theory, global stabilities at both the levels are carried out. We use the Runge–Kutta method of order 4 (RK4) and a non-standard finite difference (NSFD) scheme for the susceptible–exposed–infected–recovered (SEIR) model. In contrast to RK4, which fails for large time step size, it is found that the NSFD scheme preserves the dynamics of the proposed model for any step size used. Numerical results along with the comparison, using different values of step size h, are provided. [ABSTRACT FROM AUTHOR]

A fractional-order eco-epidemiological model with disease in the prey population is formulated and analyzed. Mathematical analysis and numerical simulations are performed to clarify the characteristics of the proposed fractional-order model. The existence, uniqueness, non-negativity and boundedness of the solutions are proved. The local and global asymptotic stability of all equilibrium points are investigated. Finally, numerical simulations are conducted to illustrate the analytical results. The occurrence of Hopf bifurcations and transcritical bifurcations for the fractional-order eco-epidemiological model are demonstrated. It is observed that the fractional order has a stabilization effect and it may help to control the coexistence between susceptible prey, infected prey and predator populations. [ABSTRACT FROM AUTHOR]

In this article,we present some new sufficient conditions for the oscillation of all solutions of a second order difference equation with several super-linear neutral terms. The results obtained here extend or complement some of the known results reported in the literature. Examples illustrating the importance of the main results are included. [ABSTRACT FROM AUTHOR]

We apply an iterative reproducing kernel Hilbert space method to get the solutions of fractional Riccati differential equations. The analysis implemented in this work forms a crucial step in the process of development of fractional calculus. The fractional derivative is described in the Caputo sense. Outcomes are demonstrated graphically and in tabulated forms to see the power of the method. Numerical experiments are illustrated to prove the ability of the method. Numerical results are compared with some existing methods. [ABSTRACT FROM AUTHOR]

The boundedness character of positive solutions of the following system of difference equations: $x_{n+1}=A+\frac{y^{p}_{n}}{x_{n-3}^{r}}$, $y_{n+1}=A+\frac {x^{p}_{n}}{y_{n-3}^{r}}$, $n\in{\mathbb{N}}_{0}$, when $\min\{A,r\}>0$ and $p\ge0$, is studied. [ABSTRACT FROM AUTHOR]

We establish an explicit formula for q-analog of Morley's congruence. [ABSTRACT FROM AUTHOR]

We investigate the hyper-order of solutions of two class of complex linear differential equations. We investigate the growth of solutions of higher order and certain second order linear differential equations, and we obtain some results which improve and extend some previous results in complex oscillations. [ABSTRACT FROM AUTHOR]

We will expand the scope of application of a fixed point theorem due to Krasnosel'skiĭ and Zabreiko to the family of second-order dynamic equations described by uΔΔ(t) = f (uσ (t)), t ∈ [0,1] ∩ 핋, with multipoint boundary conditions u(0) = 0, u(σ²(1)) = ∑i=1nαiu(ηi), and ∑i=1nαi ≤ 1 for the purpose of establishing existence results. We will determine sufficient conditions on our function ƒ such that the assumptions of the fixed point theorem are satisfied, which in return gives us the existence of solutions. [ABSTRACT FROM AUTHOR]