Search

Novel explicit wave solutions are constructed for the Kudryashov–Sinelshchikov (KS) equation through liquid–gas bubbles mix under the thermodynamic conditions. A new fractional definition (Atangana–Baleanu derivative operator) is employed through the modified Khater method to get new wave solutions in distinct types of this model that is used to describe the phenomena of pressure waves through liquid–gas bubbles mix under the thermodynamic conditions. The stability property of the obtained solutions is tested to show the ability of our obtained solutions through the physical experiments. The novelty and advantage of the proposed method are illustrated by applying to this model. Some sketches are plotted to show more about the dynamical behavior of this model. [ABSTRACT FROM AUTHOR]

The delayed logistic equation (also known as Hutchinson's equation or Wright's equation) was originally introduced to explain oscillatory phenomena in ecological dynamics. While it motivated the development of a large number of mathematical tools in the study of nonlinear delay differential equations, it also received criticism from modellers because of the lack of a mechanistic biological derivation and interpretation. Here, we propose a new delayed logistic equation, which has clear biological underpinning coming from cell population modelling. This nonlinear differential equation includes terms with discrete and distributed delays. The global dynamics is completely described, and it is proven that all feasible non-trivial solutions converge to the positive equilibrium. The main tools of the proof rely on persistence theory, comparison principles and an L 2 -perturbation technique. Using local invariant manifolds, a unique heteroclinic orbit is constructed that connects the unstable zero and the stable positive equilibrium, and we show that these three complete orbits constitute the global attractor of the system. Despite global attractivity, the dynamics is not trivial as we can observe long-lasting transient oscillatory patterns of various shapes. We also discuss the biological implications of these findings and their relations to other logistic-type models of growth with delays. [ABSTRACT FROM AUTHOR]

This article is about Ulam's type stability of nth order nonlinear differential equations with fractional integrable impulses. It is a best procession to the stability of higher order fractional integrable impulsive differential equations in quasi–normed Banach space. Different Ulam's type stability results are obtained by using the definitions of Riemann–Liouville fractional integral, Hölder's inequality and the beta integral inequality. [ABSTRACT FROM AUTHOR]

This paper analyzes the stability of fractional differential equations with Prabhakar derivative, which is a generalization of the fractional differential equation with Caputo and Riemann-Liouville derivative. As a result, the sufficient condition for asymptotic stability has been obtained by studying the eigenvalues of system related matrix and the position of these eigenvalues in the complex plane. To show the application of the results a two-dimensional predator–prey model has been studied with Prabhakar derivative and the dynamic behavior has been extracted. Then, a numerical method along with its error has been presented for solving differential equations with Prabhakar derivative. The predator–prey model simulation has been conducted by using this numerical method. [ABSTRACT FROM AUTHOR]

This paper is concerned with establishing global asymptotic stability results for a class of non-linear PDE, in which the nonlinearity enters as a scalar multiple which is non-local. The method of proof does not involve a Lyapounov function. It is shown that stability for the PDE is equivalent to stability for a differential delay equation. Stability for the delay equation is proven by exploiting certain monotonicity properties. These are established by using the methods of optimal control theory. [ABSTRACT FROM AUTHOR]

In this paper, we consider nonlinear evolution equations of second order in Banach spaces involving unbounded delay, which can model an elastic system with structural damping involving infinite delays. By using fixed point for condensing maps, we prove the existence and exponential decay of mild solutions. The obtained results can be applied to the nonlinear vibration equation of elastic beams with structural damping and infinite delay. [ABSTRACT FROM AUTHOR]

Based on a notion of Stieltjes derivative of a function with respect to another function, we provide Ulam–Hyers type stability results for nonlinear differential equations driven by measures on compact or on unbounded intervals, in the lack of Lipschitz continuity assumptions. In particular, one can deduce stability results for generalized differential equations, dynamic equations on time scales or impulsive differential equations (including the case of an infinite number of impulses that accumulate in the considered interval, thus allowing the study of Zeno hybrid systems). [ABSTRACT FROM AUTHOR]

The paper deals with the problems of the global existence, boundedness and integrability, and solutions of an integro-differential equation (IDE) of second order with n-constant retardations. Three new theorems, Theorems 2.1–2.3 and a corollary, Corollary 2.1, have been proved, which consist of new sufficient criteria on global existence, stability, asymptotic stability, boundedness of solutions and square integrability of the first derivatives of solutions of the considered nonlinear retarded IDE of second order (RIDE). The techniques used in the proofs are based on the construction of a suitable Lyapunov–Krasovskii functional (LK functional). The results of this work include and get better results that are available in the literature. The given criteria have very simple forms, and they can be easily verified with applications. A numerical example is given and the orbits of the exact solutions are drawn with assigned initial conditions to demonstrate the effectiveness of the obtained results. [ABSTRACT FROM AUTHOR]

This paper is concerned with the travelling wavefronts of a nonlocal dispersal cooperation model with harvesting and state-dependent delay, which is assumed to be an increasing function of the population density with lower and upper bound. Especially, state-dependent delay is introduced into a nonlocal reaction-diffusion model. The conditions of Schauder's fixed point theorem are proved by constructing a reasonable set of functions Γ (see Section 2) and a pair of upper-lower solutions, so the existence of traveling wavefronts is established. The present study is continuation of a previous work that highlights the Laplacian diffusion. [ABSTRACT FROM AUTHOR]

This paper is concerned with a competition and cooperation model of two enterprises with multiple delays and feedback controls. With the aid of the difference inequality theory, we have obtained some sufficient conditions which guarantee the permanence of the model. Under a suitable condition, we prove that the system has global stable periodic solution. The paper ends with brief conclusions.

In this paper, an age-structured heroin epidemic model, where the susceptibility of individuals and the relapse of heroin users in treatment are described by two age-dependent variables, is formulated and analyzed. The basic reproduction ratio of the model is derived and proved to be a threshold condition, which completely determines the global behaviors of the model. The asymptotic smoothness of the semiflow generated by the family of solutions, uniform persistence and existence of an interior global attractor have been presented for establishing and defining a Lyapunov functional on this attractor. Some control strategies of heroin and two special cases of the model formulation are addressed. [ABSTRACT FROM AUTHOR]

Linear delayed differential systems x ˙ i (t) = ∑ j = 1 m ∑ k = 1 r ij a ij k (t) x j h ij k (t) , i = 1 , ⋯ , m are considered on a half-infinity interval t ≥ 0 . It is assumed that m and r ij , i , j = 1 , ⋯ , m are natural numbers and the coefficients a ij k : [ 0 , ∞) → R and delays h ij k : [ 0 , ∞) → R are Lebesgue measurable functions. New explicit results on uniform exponential stability, depending on all delays, are derived. The conditions obtained do not require the dominance of diagonal terms over the off-diagonal terms as most of the existing stability tests for non-autonomous delay differential systems do. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with a non-autonomous competing model with time delays and feedback controls. Applying the comparison theorem of differential equations and by constructing a suitable Lyapunov functional, some sufficient conditions which guarantee the existence of a unique globally asymptotically stable nonnegative almost periodic solution of the system are established. An example with its numerical simulations is given to illustrate the feasibility of our results. [ABSTRACT FROM AUTHOR]

We are interested in the approximation of the ruin probability of a classical risk model using the strong stability method. Particularly, we study the sensitivity of the stability bound for ruin probabilities of two risk models to approach (a simpler ideal model and a complex real one, which must be close in some sense) regarding to different large claims (heavy-tailed distributions). In a first case, we study the impact of the tail of some claim distributions on the quality of this approximation using the strong stability of a Markov chain. In a second case, we look at the sensitivity of the stability bound for the ruin probability regarding to different large claims, using two versions of the strong stability method: strong stability of a Markov chain and strong stability of a Lindley process. In both cases, comparative studies based on numerical examples and simulation results, involving different heavy-tailed distributions, are performed. [ABSTRACT FROM AUTHOR]

We prove the existence and uniqueness of the periodic solution as well as its conditional stability for the semilinear partial neutral function differential equation of the form ∂ F u t ∂ t = A (t) F u t + g (t , u t) , t ∈ (0 , ∞) ; u 0 = ϕ. Here the operator-valued function t ↦ A (t) is T-periodic; the nonlinear delay operator g (t , ψ) is T-periodic with respect to t and Lipschitz continuous with respect to ψ . In our strategy, on the one hand, we prove a Massera-type theorem for the linearized neutral equation, then pass to the semilinear equation using fixed-point arguments and Neumann series. On the other hand, our abstract results fit perfectly with the case that the family (A (t)) t ≥ 0 generates an evolution family having exponential dichotomy. In such a case, we can apply our abstract results to show the existence, uniqueness, and conditional stability of periodic solution to the above equation. [ABSTRACT FROM AUTHOR]

In this paper, we prove some results on the existence of solutions and the mean square asymptotic stability for a class of impulsive neutral stochastic differential systems with variable delays by using a contraction mapping principle. Namely, a sufficient condition ensuring the asymptotic stability is proved. The assumptions do not impose any restrictions neither on boundedness nor on the differentiability of the delay functions. In particular, the results improve some previous ones in the literature. Finally, an example is exhibited to illustrate the effectiveness of the results. [ABSTRACT FROM AUTHOR]

In this paper, we develop the theory of basic reproduction ratios R 0 for abstract functional differential systems in a time-periodic environment. It is proved that R 0 - 1 has the same sign as the exponential growth bound of an associated linear system. Then we apply it to a time-periodic Lyme disease model with time-delay and obtain a threshold type result on its global dynamics in terms of R 0 . [ABSTRACT FROM AUTHOR]

We present a review of known stability tests and new explicit exponential stability conditions for the linear scalar neutral equation with two delays x ˙ (t) − a (t) x ˙ (g (t)) + b (t) x (h (t)) = 0 , where | a (t) | < 1 , b (t) ⩾ 0 , h (t) ⩽ t , g (t) ⩽ t , and for its generalizations, including equations with more than two delays, integro-differential equations and equations with a distributed delay. [ABSTRACT FROM AUTHOR]

We give an new proof of the well-known competitive exclusion principle in the chemostat model with NN species competing for a single resource, for any set of increasing growth functions. The proof is constructed by induction on the number of the species, after being ordered. It uses elementary analysis and comparisons of solutions of ordinary differential equations. [ABSTRACT FROM AUTHOR]

Let Λ be a random compact invariant (with respect to a random map) set with a uniform tangent bundle splitting. We show that if there is a random non-uniformly hyperbolic variable x0 on Λ and the space of limit points of time averages of random Dirac measures on the orbit of x0 coincides with the space of invariant probability measures supported on Λ then, Λ is a random uniformly hyperbolic set. [ABSTRACT FROM AUTHOR]

In this paper, we study time-delayed reaction–diffusion systems with advection subject to Lotka–Volterra competition dynamics over one-dimensional domains. These systems model the population dynamics of two groups of competing species, with one dispersing randomly and the other a combination of random and biased dispersal (to avoid competition). We show that time-delay(s) in the interspecific competition mechanism can induce instability of the homogeneous equilibrium to the reaction–advection–diffusion systems, and further promote the appearance of time-oscillating spatially inhomogeneous distributions of the species. Our results indicate that these time-delayed systems (both single and double time-delays) can be used to model the well-observed time-periodic distributions of interacting species in natural fields, compared to the systems without time-delay(s). [ABSTRACT FROM AUTHOR]

We consider a mathematical model describing the maturation process of stem cells up to fully mature cells. The model is formulated as a differential equation with state-dependent delay, where maturity is described as a continuous variable. The maturation rate of cells may be regulated by the amount of mature cells and, moreover, it may depend on cell maturity: we investigate how the stability of equilibria is affected by the choice of the maturation rate. We show that the principle of linearised stability holds for this model, and develop some analytical methods for the investigation of characteristic equations for fixed delays. For a general maturation rate we resort to numerical methods and we extend the pseudospectral discretisation technique to approximate the state-dependent delay equation with a system of ordinary differential equations. This is the first application of the technique to nonlinear state-dependent delay equations, and currently the only method available for studying the stability of equilibria by means of established software packages for bifurcation analysis. The numerical method is validated on some cases when the maturation rate is independent of maturity and the model can be reformulated as a fixed-delay equation via a suitable time transformation. We exploit the analytical and numerical methods to investigate the stability boundary in parameter planes. Our study shows some drastic qualitative changes in the stability boundary under assumptions on the model parameters, which may have important biological implications. [ABSTRACT FROM AUTHOR]

In this paper, the comparison theorem and Gronwall's inequality with ∇ -derivative on time scales are constructed. Based on ∇ -stochastic integration, the ∇ -stochastic high-order Hopfield neural networks with time delays on time scales is introduced and studied. By using contraction mapping principal and differential inequality technique on time scales, some sufficient conditions for the existence and exponential stability of periodic solutions for a class of ∇ -stochastic high-order Hopfield neural networks with time delays on time scales are established. Our results show that the continuous-time neural network and its discrete-time analogue have the same dynamical behaviors for periodicity. Finally, a numerical example is provided to illustrate the feasibility of our results. The results of this paper are completely new even if the time scale T = R or Z . [ABSTRACT FROM AUTHOR]

We investigate a stochastic HIV/AIDS epidemic model with treatment. The model allows for two stages of infection namely the asymptomatic phase and the symptomatic phase. We prove existence of global positive solutions. We show that the solutions are stochastically ultimately bounded and stochastically permanent. We also study asymptotic behaviour of the solution to the stochastic model around the disease-free equilibrium of the underlying deterministic model. Our theoretical results are illustrated by way of numerical simulations. [ABSTRACT FROM AUTHOR]

In this work, we study the stability properties of a delay differential neoclassical model of economic growth, based on the original model proposed by Solow (Q J Econ 70:65–94, 1956). We consider a logistic-type production function, which comes from combining a Cobb–Douglas function and a linear pollution effect caused by increasing concentrations of capital. The difference between the production function and the classical logistic map comes from the presence of a parameter γ ∈ (0 , 1) in the exponent of one factor. We call this new function the gamma-logistic map. Our main purpose is to obtain sharp global stability conditions for the positive equilibrium of the model and to study how the stability properties of such equilibrium depend on the relevant model parameters. This study is developed by using some properties of the gamma-logistic map and some well-known results connecting stability in delay differential equations and discrete dynamical systems. Finally, we also compare the obtained results with the ones written in related articles. [ABSTRACT FROM AUTHOR]

The aim of this paper is to present and study a three-dimensional continuous time dynamical system modeling a predator–prey with harvesting and reserve zone for the prey in the presence of competition and toxicity. We first prove that our model is ecologically and mathematically well-posed. In addition, the stability analysis is investigated by direct and indirect Lyapunov methods. By using the Pontryagin's maximum principle, an optimal harvesting policy is established. Furthermore, numerical simulations are given in order to illustrate our theoretical results. [ABSTRACT FROM AUTHOR]

The time delay-induced instability in an Internet congestion control model is investigated. The star topology is considered, and the link bandwidth ratio and the control gain are selected as the tunable parameters for congestion suppression. The stability switch boundary is obtained by the eigenvalue analysis for the linearized system around the equilibrium. To investigate the oscillatory congestion when the equilibrium becomes unstable, the center manifold reduction and the normal form theory are used to study the periodic oscillation induced by the delay. The theoretical analysis and numerical simulation show that the ratio between bandwidths of the trunk link and the regular link, rather than these bandwidths themselves, is crucial for the stability of the congestion control system. The present results demonstrate that it is not always effective to increase the link bandwidth ratio for stabilizing the system, and for some certain delays, adjusting the control gain is more efficient. [ABSTRACT FROM AUTHOR]

In this paper, we consider a non-linear stochastic differential delay equation (SDDE) of second order. We derive new sufficient conditions which guarantee stochastically stability and stochastically asymptotically stability of the zero solution of that SDDE. Here, the technique of the proof is based on the definition of a suitable Lyapunov-Krasovskii functional, which gives meaningful results for the problem under consideration. The derived results extend and improve some result of in the relevant literature, which are related to the qualitative properties of solutions of a SDDE of second order. The results of this paper are new and have novelty, and they do a contribution to the topic and relevant literature. As an application, an example is given to show the effectiveness and applicability of the obtained results. Finally, by the results of this paper, we extend and improve some recent results that can be found in the relevant literature. [ABSTRACT FROM AUTHOR]

The aim of paper is to analyse some qualitative properties of solutions of nonlinear Volterra integro-differential equations (VIDDEs) with constant delay and Volterra integro-differential equations (VIDEs) without delay by means of the Razumikhin method. A suitable Lyapunov function is defined and then applied to that IDEs such that some former results can be obtained under weaker conditions and additionally some new results are given on the qualitative properties of that IDEs such as uniformly stability and integrability of solutions. For the VIDDEs with constant delay, the improved less conservative conditions of the stability, asymptotically stability and the boundedness of solutions and the new conditions of uniformly stability, integrability of solutions are all derived subject to the functions appeared in the considered IDEs. Moreover, the established stability, asymptotically stability and boundedness conditions of VIDDEs with constant delay simplify, extend and improve some previous works can be found in the literature and remove some unnecessary conditions. Finally, the validity of the presented results is indicated by some numerical examples using MATLAB-Simulink. [ABSTRACT FROM AUTHOR]

This paper analyzes the stability, uniformly stability, asymptotically stability, boundedness, uniformly boundedness and square integrability of solutions of a system of differential equations of second order with variable delay by applying the direct method of Lyapunov- Krasovskii. By means of a new Lyapunov-Krasovskii functional, we simplify and extend some previous work that is found in the recent literature. Finally, the validity and applicability of the proceeded results are indicated by some numerical examples applying MATLAB-Simulink. By the results of this paper, we can obtain the results of Omeike et al. [Stability and boundedness of solutions of certain vector delay differential equations. J Nigerian Math Soc. 2018;37(2):77–87], Theorem 1.1 and Theorem 1.2, under weaker conditions. In addition, we establish two new results on the uniformly stability and integrabilty of solutions the considered equation. Finally, in particular cases, the applicability of the results of this paper can be shown by two new examples. These are the contribution of this paper to the subject and the relevant literature. [ABSTRACT FROM AUTHOR]

In this paper, some sufficient conditions for uniform practical stability of impulsive functional differential equations in terms of two measures with effect of delay at the time of impulses are obtained by using piecewise continuous Lyapunov functions and Razumikhin techniques. The application of obtained result is illustrated with an example.

In this paper, we present an analysis for the stability of a differential equation with state-dependent delay. We establish existence and uniqueness of solutions of differential equation with delay term τ(u(t))=a+bu(t)c+bu(t).$\tau (u(t)) = \frac{{a + bu(t)}}{{c + bu(t)}}.$ Moreover, we put the some restrictions for the positivity of delay term τ(u(t)) Based on the boundedness of delay term, we obtain stability criterion in terms of the parameters of the equation.

In this work, we set up a new discrete predator-prey competitive model with time-varying delays and feedback controls. By virtue of the difference inequality knowledge, a sufficient condition which guarantees the permanence of the established discrete predator-prey competitive model with time-varying delays and feedback controls is derived. Under some appropriate parameter conditions, we have proved that the periodic solution of the system without delay exists and globally attractive. To verify the correctness of the derived theoretical fruits, we give two examples and execute computer simulations. Our obtained results are novel and complement previous known results.

Abstract In this paper, the authors obtain new sufficient conditions for stability (S) and uniform stability (US) of solutions of the first order retarded Volterra integro-differential equations (VIDEs) in the form x ′ = A (t) x + ∫ t - τ t C (t , s) ϕ (s , x (s)) d s + f (t , x , x (t - τ)). The analysis of the obtained (S) and (US) results mainly depend on the definition of an appropriate Lyapunov functional (LF). An example is provided to illustrate the effectiveness of the proposed results. MATLAB-Simulink is applied to show the behaviors of the paths of solutions of the considered (VIDEs) for a particular case. [ABSTRACT FROM AUTHOR]

The existence and uniqueness of stationary solutions to an incompressible non-Newtonian fluid are first established. The exponential stability of steady-state solutions is then analyzed by means of four different approaches. The first is the classical Lyapunov function method, while the second one is based on a Razumikhin type argument. Then, a method relying on the construction of Lyapunov functionals and another one using a Gronwall-like lemma are also exploited to study the stability, respectively. Some comments concerning several open research directions about this model are also included. [ABSTRACT FROM AUTHOR]