The survival and development of enterprise clusters are closely related to the sustainable development of the entire economy, so research on them has attracted much attention. In this article, we study the dynamic properties of a stochastic enterprise cluster model with high-order perturbation. First, we study stochastically ultimate boundedness of the system. Then, some sufficient condition for the existence of the stationary distribution in the system are obtained. We also discussed the destructiveness and permanence of the system. Finally, three numerical examples are applied to verify the obtained results.

In this paper, a class of pantograph Rayleigh system is investigated. By using contraction mapping fixed point theorem, sufficient conditions are established for the existence of a unique almost periodic solution for the considered model. Furthermore, we obtain the synchronization between the drive system and the response system by using the matrix measure method and Halany inequality. Finally, two numerical examples are given to illustrate the feasibility and application of the obtained results. It should be pointed out that almost periodic solutions have certain symmetry properties.

In this paper, we construct and analyze a stochastic enterprise cluster model with a mean-reverting Ornstein-Uhlenbeck process. We first investigate the existence of a global unique positive solution for the system. After that, the stochastically ultimate boundedness of the system is considered. Based on a series of Lyapunov functions, sufficient criteria for the system's dynamic behaviors, including exponential extinction and persistence in the mean of the system, are established. Two numerical examples validate the theoretical results of this paper. This article provides a theoretical basis for promoting the development of enterprise clusters.

In the present paper we study the existence and stability problems of positive periodic solutions to a Gilpin–Ayala competitive model with periodic coefficients on time scales. Firstly, based on Schauder’s fixed theorem, some sufficient conditions for the existence of positive periodic solution to the considered system are obtained. Furthermore, we establish asymptotic behavior by using the existence of periodic solutions. Since the considered system is based on an arbitrary time scale, our results are applicable to both discrete and continuous scenarios. We provide a specific example to verify the above results.

As known, coral reefs have important economic and social value, but they are damaged by various environmental factors worldwide. In this work, a delayed coral reefs system with white noise and Levy jumps are studied. First, we examine the existence problem using Lyapunov analysis methods. Then, from a stochastic analysis technique, we studied the problems for the stochastics persistence and extinction. Furthermore, conditions for stochastic bounded in mean were obtained. Our results indicated that large noise intensity was not conducive to populations. The above results may help us better understand the macroalgae and coral reef dynamics in the fluctuating environments. The most important findings of this article are that we studied a variable coefficient coral reef model under various random disturbances and identified the impact of random perturbation and Levy jump on the dynamic properties of the system. Finally, five numerical simulations are presented to check the obtained results.

In this work, we study positive periodic solutions of a neutral-type host-macroparasite equation and establish the existence results of positive periodic solutions by using topological degree theory. Furthermore, based on the Lyapunov functional method and differential inequality analysis strategies, the dynamic behaviors of the host-macroparasite model are obtained. Finally, we present a numerical example to verify the effectiveness of the obtained results. It should be pointed out that the properties of neutral operators have significant applications in the proof. Our results have extended existing findings for host-macroparasite equation.

We first study almost periodic solutions of neutral-type differential system on time scales and establish some basic results for the considered system. Furthermore, based on these results, the dynamic behaviors of two classes of neutral-type biological population models including hostmacroparasite model and Lasota-Wazewska model are obtained. It is worth mentioning that we study almost periodic solutions for neutral-type differential system on time scales. Furthermore, using the above study and exponential dichotomy method, we investigate two types of biological population models.

We introduce and investigate a chemostat mode on time scales. By using the theory of calculus on time scales, continuation theorem of coincidence degree theory and Lyapunov functional method, we obtain some sufficient conditions which guaranteeing the existence and global exponential stability of periodic solution to the considered system. The considered model is more general and versatile than the traditional models. This paper unifies the periodic discrete-time and continuous-time chemostat mode under the same framework. Hence, the model on time scales is the optimal way forward for accurate and malleable mathematical modelling. Furthermore, our results have more general applications.

In this note, we investigate the existence and asymptotic property of positive periodic solutions to non-autonomous predator-prey system with stage-structured predator on time scales. Via Schauder’s fixed theorem, easily verifiable sufficient existence conditions of positive periodic solutions for the considered system are obtained. We also study asymptotic property of positive periodic solutions on the basis of existence conditions. Due to the symmetry of periodic solutions, the results of this paper have a certain impact on the study of symmetry. It should be pointed out that the system we are studying is built on arbitrary time scale, so our results generalize the results of existing continuous or discrete systems. Furthermore, we develop Schauder’s fixed theorem for studying the delay system on time scales.

This paper aims to study a class of enterprise cluster models with feedback controls and time-varying delays on time scales. Based on periodic time scales theory and the fixed point theorem of strict-set-contraction, some new sufficient conditions for the existence of positive periodic solutions are obtained. Finally, two examples are presented to verify the validity and applicability of the main results in this paper.

In this paper, we focus on the existence of positive periodic solutions of generalized Leslie–Gower-type population models. Using the topological degree, we provide sufficient conditions to demonstrate the existence of positive periodic solutions to the considered models. It is interesting that the positive periodic solutions in this paper are general positive functions, not e exponential functions, which generalizes and improves the existing results. We note that due to the symmetrical property of periodic solutions, the results of this paper provide a deeper understanding of the periodic behavior of biological populations. Two numerical examples show the effectiveness of our main results.

This paper is devoted to investigating a class of stochastic neutral-type neural networks with delays. By using the fixed point theorem and the properties of neutral-type operator, we obtain the existence conditions for periodic solutions of stochastic neutral-type neural networks. Furthermore, we obtain the conditions for the exponential stability of periodic solutions using Gronwall-Bellman inequality and stochastic analysis technique. Finally, a numerical example is given to show the effectiveness and merits of the present results. Our results can be used to obtain the existence and exponential stability of periodic solution to the corresponding deterministic systems.

In this paper, we establish the existence and stability of periodic solutions for neutral-type differential equations with piecewise impulses on time scales. We first obtain some sufficient conditions for the existence of a unique periodic solution by using the Banach contraction mapping principle. We also prove the existence of at least one periodic solution using the Schauder fixed point theorem. In addition, we establish the stability results based on the existence of periodic solutions. It is worth noting that the results of this paper are based on time scales, so that they are applicable to continuous, discrete, and other types of systems.

In this paper, we focus on the asymptotic stability of the trajectories governed by the differential inequalities with mixed delays using the fixed-point theorem. It is interesting that the Halanay inequality is a special case of the differential inequality studied in this paper. Our results generalize and improve the existing results on Halanay inequality. Finally, three numerical examples are utilized to illustrate the effectiveness of the obtained results.

This article studies the stability problem of a class of stochastic neutral-type inertial delay neural networks. By introducing appropriate variable transformations, the second-order differential system is transformed into a first-order differential system. Using homeomorphism mapping, standard stochastic analyzing technology, the Lyapunov functional method and the properties of a neutral operator, we establish new sufficient criteria for the unique existence and stochastically globally asymptotic stability of equilibrium points. An example is also provided, to show the validity of the established results. From our results, we find that, under appropriate conditions, random disturbances have no significant impact on the existence, stability, and symmetry of network systems.

Iterative differential equations provide a new idea to study functional differential equations. The study of iterative equations can provide new methods for the study of differential equations with state-dependent delays. In this paper, we are concerned with proving the existence of infinitely many positive solutions to nonlinear first-order iterative systems of singular BVPs on time scales by using Krasnoselskii’s cone fixed point theorem in a Banach space. It is worth pointing out that in this paper, we can use the symmetry of the iterative process and Green’s function to transform the considered differential equation into an equivalent integral equation, which plays a key role in the proof of the theorem in this paper.

By using Gronwall’s inequality and coincidence degree theory, the sufficient conditions of the globally exponential stability and existence are given for a Hebbian-type network with time-varying delays. The periodic behavior phenomenon is one of the hot topics in network systems research, from which we can discover the symmetric characteristics of certain neurons. The main theorems in the present paper are illustrated using a numerical example.

This paper is devoted to investigating a class of neutral-type integral differential equations arising in an epidemic model. By using Mawhin’s continuation theorem and the properties of neutral-type operators, we obtain the existence conditions for positive periodic solutions of the considered neutral-type integral differential equation. Compared with previous results, the existence conditions in this paper are less restricted, thus extending the results of the existing literature. Finally, two examples are given to show the effectiveness and merits of the main results of this paper. Our results can be used to obtain the existence of a positive periodic solution to the corresponding non-neutral-type integral differential equation.

In this article, we discuss the existence of a positive periodic solution for a first-order nonlinear neutral differential equation with impulses on time scales. Based on the Leggett–Williams fixed-point theorem and Krasnoselskii’s fixed-point theorem, some sufficient conditions are established for the existence of positive periodic solution. An example is given to show the feasibility and application of the obtained results. Since periodic solutions are solutions with symmetry characteristics, the existence conditions for periodic solutions also imply symmetry.

This paper is devoted to studying a second-order nonlinear equation with mixed delays. Some sufficient conditions for the existence and exponential stability of the almost periodic solutions are established. The results of this paper extend the existing ones.