In this paper, we deal with a class of Gilpin-Ayala ecological models with discrete and distributed time delays. By employing a fixed point theorem of strict-set-contraction and inequality techniques, some sufficient conditions for the existence of periodic solutions are established. As an application, one example is given to illustrate the validity of our main results.

By making a special product Banach space and using the famous result of Covitz and Nadler on fixed point of multifunctions we investigate the existence of a solution for a system of fractional finite difference inclusions via some boundary conditions. We provide an example to illustrate our main result.

In this paper, a diffusive Leslie-type predator-prey model is investigated. The existence of a global positive solution, persistence, stability of the equilibria and Hopf bifurcation are studied respectively. By calculating the normal form on the center manifold, the formulas determining the direction and the stability of Hopf bifurcations are explicitly derived. Finally, our...

In this work, we introduce some new results on the Lyapunov inequality, uniqueness and multiplicity results of nontrivial solutions of the nonlinear fractional Sturm-Liouville problems { D 0 + q ( p ( t ) u ′ ( t ) ) + Λ ( t ) f ( u ( t ) ) = 0 , 1 < q ≤ 2 , t ∈ ( 0 , 1 ) , α u ( 0 ) − β p ( 0 ) u ′ ( 0 ) = 0 , γ u ( 1 ) + δ p ( 1 ) u ′ ( 1 ) = 0 , $$\textstyle\begin{cases} D_{0...

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...

In this article, we extend fractional calculus with nonsingular exponential kernels, initiated recently by Caputo and Fabrizio, to higher order. The extension is given to both left and right fractional derivatives and integrals. We prove existence and uniqueness theorems for the Caputo (CFC) and Riemann (CFR) type initial value problems by using Banach contraction theorem. Then...

In this paper, we study the effect of time delay and the scaled Rayleigh number on chaotic convection in porous media with fractional order. The stability analysis for different fractional-order cases is investigated and the effective chaotic range of the fractional order is determined by a general synchronization of nonidentical chaotic systems based on the active control...

We study the existence of compact almost automorphic solutions for a class of integral equations with time-dependent and state-dependent delay. An application to a blowflies model and a transmission lines model is carried out to support the theoretical finding.

The present paper deals with the Cauchy problem for the multi-term time-space fractional diffusion equation in one dimensional space. The time fractional derivatives are defined as Caputo fractional derivatives and the space fractional derivative is defined in the Riesz sense. Firstly the domain of the fractional Laplacian is extended to a Banach space. Then the analytical...

In this paper, using the algebraic structure of the Abelian group, we introduce the concept of a matched space for time scales, and we construct the algebraic structure of matched spaces to solve the closedness of time scales under non-translational shifts. Using a matched space for time scales, a new concept of periodic time scales is introduced. Based on it, new concepts of...

In this paper the synchronization of fractional-order chaotic systems and a new property of fractional derivatives are studied. Then we propose a new fractional-order extension of Lyapunov direct method to control the fractional-order chaotic systems. A new synchronization method and a linear feedback controller are given to achieve the synchronization of fractional-order chaotic...

The main purpose of this paper is by using the definitions and properties of Chebyshev polynomials to study the power sum problems involving Fibonacci polynomials and Lucas polynomials and to obtain some interesting divisible properties.

In this paper, synchronization of a network with time-varying topology and delay is investigated. Firstly, proper controllers are designed for achieving synchronization by adopting impulsive control scheme. Based on the linear matrix inequality technique and the Lyapunov function method, a synchronization criterion is derived and analytically proved. Secondly, controllers are...

In this paper, the variational iteration method (VIM) is applied to solve the time and space-time fractional Burgers’ equation for various initial conditions. VIM solutions are computed for the fractional Burgers’ equation to show the behavior of VIM solutions as the fractional derivative parameter is changed. The results obtained by VIM are compared with exact solutions and also...

We propose a mathematical model that can control the stability of an unstable system. Periodicity is an important feature of the system. We add a continuous control to the first half of each period of the system and then add an impulse control J1 at the 1 / 2 $1/2$ period time. Again, we do not control the rest half of each period of the system. Finally, we add an impulse control...

This paper studies the existence and blowup of solutions for the modified Klein-Gordon-Zakharov equations for plasmas with a quantum correction, which describe the interaction between high frequency Langmuir waves and low frequency ion-acoustic waves in a plasma considering the quantum effects. Firstly the existence and uniqueness of the local smooth solutions are obtained by the...

A stochastic predator-prey system with time-dependent delays is considered. Firstly, we show the existence of a global positive solution and stochastically ultimate boundedness. Secondly, the critical value between weak persistence and extinction of the prey is obtained and we also give the asymptotic pathwise estimation. Finally, we simulate the model to illustrate our results.

We propose a new stochastic competition chemostat system with saturated growth rate and impulsive toxicant input. The main purpose of this paper is to study the stochastic dynamics of a high-dimensional impulsive stochastic chemostat model and find the threshold between persistence and extinction for the impulsive stochastic chemostat system. First, we investigate the stability...

In this paper, a fractional-order model of palm trees, the lesser date moth and the predator is presented. Existence conditions of the local asymptotic stability of the equilibrium points of the fractional system are analyzed. We prove that the positive equilibrium point is globally stable also. The numerical simulations come to illustrate the dynamical behaviors of the model...

We propose and study a discrete competitive system of the following form: x 1 ( n + 1 ) = x 1 ( n ) exp [ r 1 − a 1 x 1 ( n ) − b 1 x 2 ( n ) 1 + c 2 x 2 ( n ) ] , x 2 ( n + 1 ) = x 2 ( n ) exp [ r 2 − a 2 x 2 ( n ) − b 2 x 1 ( n ) 1 + c 1 x 1 ( n ) ] . $$\begin{aligned} &x_{1}(n+1)=x_{1}(n)\exp{\biggl[r_{1}-a_{1}x_{1}(n)- \frac {b_{1}x_{2}(n)}{1+c_{2}x_{2}(n)}\biggr]}, \\ &x_{2...