Periodic dynamics of a general switching dynamical system

By applying Green's function of third-order differential equation and a fixed point theorem in cones, we obtain some sufficient conditions for existence, nonexistence, multiplicity, and Lyapunov stability of positive periodic solutions for a third-order neutral differential equation.

Existence and global asymptotic stability of positive periodic solution of delayed Cohen–Grossberg neural networks

This paper is concerned with a class of multiple species Gilpin-Ayala system with infinity distributed time delays. By applying fixed-point index theory, we have established some new sufficient conditions to ensure the existence of positive periodic solutions. We also analyze the global exponential stability of positive periodic solution for this system by constructing appropriate Lyapunov functional and transform technique. As an application, an interesting example and its simulation are illustrated the effectiveness of our main results.

This paper studies positive ω-periodic solutions for a class of neutral-type impulsive neural networks with distributed delays and two parameters. Using cone-theoretic fixed point methods, Green's function techniques, a Green-weighted norm, and a Lyapunov-Razumikhin approach, we establish sufficient conditions for the existence of positive periodic solutions. We further derive explicit small-gain conditions under which the positive periodic solution is unique and globally exponentially stable. These conditions reflect the combined effects of leakage, distributed delays, neutral feedback, and impulsive perturbations. In addition, computable estimates are obtained for the contraction constant of the Poincaré map and the exponential convergence rate to the periodic attractor. The main contribution is a unified framework linking cone-based existence theory with weighted small-gain stability analysis for neutral-type impulsive neural networks with distributed delays. Numerical examples illustrate the applicability of the results.

An impulsive Lotka-Volterra type predator-prey model with prey dispersal in two-patch environments and time delays is investigated, where we assume the model of patches with a barrier only as far as the prey population is concerned, whereas the predator population has no barriers between patches. By applying the continuation theorem of coincidence degree theory and by means of a suitable Lyapunov functional, a set of easily verifiable sufficient conditions are obtained to guarantee the existence, uniqueness, and global stability of positive periodic solutions of the system. Some known results subject to the underlying systems without impulses are improved and generalized. As an application, we also give two examples to illustrate the feasibility of our main results.

Positive periodic solutions of delayed periodic Lotka–Volterra systems

In this paper, a general class of nonautonomous Lotka-Volterra dispersal system with discrete and continuous infinite delays is investigated. This class of Lotka-Volterra systems model the diffusion of a single species into n patches by discrete dispersal. By using Schauder's fixed point theorem, we prove the existence of positive periodic solutions of system. The global asymptotical stability of positive periodic solution is discussed and the sufficient conditions for exponential stability are also given. we give an example to illustrate the validity of the results in the end. The conditions we obtained are more general and it can be extended to several special systems.

Based on the dynamic management of insulin, we consider the effects of insulin, glucose, and glucocorticoid concentration on the dynamics of the diabetes model with nonlinear impulsive control to simulate the physiological process of exogenous insulin therapy. Firstly, if , in type 1 diabetes mellitus (T1DM), the existence of positive periodic solution of the system is proved by using fixed point theory for difference equations, whereas the global asymptotic stability of positive periodic solution is proved by using the Floquet multiplier theory and comparison theorem. Next, if , in type 2 diabetes mellitus (T2DM), the permanence of the solution of the system is proved by using the comparison theorem. Furthermore, simulation results are performed to verify the correctness of the theoretical findings and to study the system's dynamic behavior under various conditions. By simulating the feedback regulation mechanism between insulin and glucose in a closed‐loop system, the model not only elucidates the dynamic equilibrium relationship between insulin and glucose but also offers a novel and feasible strategy for the treatment and control of diabetes, thereby demonstrating substantial theoretical and practical significance.

Due to resource limitation, nonlinear impulsive control tactics related to integrated pest management have been proposed in a generalized pest-natural enemy model, which allows us to address the effects of nonlinear pulse control on the dynamics and successful pest control. The threshold conditions for the existence and global stability of pest-free periodic solution are provided by Floquet theorem and analytic methods. The existence of a nontrivial periodic solution is confirmed by showing the existence of nontrivial fixed point of the stroboscopic mapping determined by time snapshot, which equals to the common impulsive period. In order to address the applications of generalized results and to reveal how the nonlinear impulses affect the successful pest control, as an example the model with Holling II functional response function is investigated carefully. The main results reveal that the pest free periodic solution and a stable interior positive periodic solution can coexist for a wide range of parameters, which indicates that the local stability does not imply the global stability of the pest free periodic solution when nonlinear impulsive control is considered, and consequently the resource limitation (i.e. nonlinear control) may result in difficulties for successful pest control.

Global exponential stability of positive periodic solutions for a delayed Nicholsonʼs blowflies model

On the stable periodic solutions of a delayed two-species model of facultative mutualism

An SIS model with periodic maximum infectious force, recruitment rate and the removal rate of the infectives has been investigated in this article. Sufficient conditions for the permanence and extinction of the disease are obtained. Furthermore, the existence and global stability of positive periodic solution are established. Finally, we present a procedure by which one can control the parameters of the model to keep the infectives stay eventually in a desired set.

As an important tool to study practical problems of biology, engineering and image processing, the cellular neural networks (CNNs) has caused more and more attention. Some interesting results about the existence of solution for cellular neural networks have been obtained. In this paper, by means of iterative analysis, the existence of periodic solution and the uniform stability of the equilibrium point of impulsive cellular neural networks with distributed delays are considered. Some new results are obtained.

With the help of a continuation theorem based on Gaines and Mawhin's coincidence degree, several verifiable criteria are established for the global existence of positive periodic solutions of a class of non–autonomous single species population model with delays (both state–dependent delays and continuous delays) and feedback control. After that, by constructing a suitable Lyapunov functional, sufficient conditions which guarantee the existence of a unique globally asymptotic stable positive periodic solution of a kind of nonlinear feedback control ecosystem are obtained. Our results extend and improve the existing results, and have further applications in population dynamics.

Stability analysis of a Komarova type model for the interactions of osteoblast and osteoclast cells during bone remodeling