Positive Periodic Solutions Research Articles

A Generalized Approach of the Gilpin–Ayala Model with Fractional Derivatives under Numerical Simulation

In this article, we study the existence and uniqueness of multiple positive periodic solutions for a Gilpin–Ayala predator-prey model under consideration by applying asymptotically periodic functions. The result of this paper is completely new. By using Comparison Theorem and some technical analysis, we showed that the classical nonlinear fractional model is bounded. The Banach contraction mapping principle was used to prove that the model has a unique positive asymptotical periodic solution. We provide an example and numerical simulation to inspect the correctness and availability of our essential outcomes.

Existence of positive periodic solutions of several types of biological models with periodic coefficients

This paper investigates the existence of positive periodic solutions for several types of important biological models with periodic coefficients, including the HIV model with multiple infection stages and general incidence, the epidemic models (such as the SIRS model, the SIRI model, the SEIRS model, etc.) with general incidence. By means of the continuation theorem in the coincidence degree, and the combination of analytical techniques such as constructing appropriate auxiliary functions, we give several classes of explicit sufficient conditions for the existence of positive periodic solutions for these biological models. When these models with periodic coefficients degenerate to the corresponding autonomous cases, our conditions all naturally degenerate to the basic reproduction number is greater than 1. Our results greatly improve and expand the studies in the existing literatures.

Local exponential stability of several almost periodic positive solutions for a classical controlled GA-predation ecosystem possessed distributed delays

This paper deals with a classical controlled nonlinear almost periodic Gilpin-Ayala (GA) predation ecosystem possessed distributed delays. The existence of only two zeros of a class of auxiliary functions is obtained in R. On this basis, the multiplicity of almost periodic positive solutions for this ecosystem is investigated by some inequality techniques and theory of nonlinear analysis. Based on Lyapunov stability theory, we prove that this ecosystem is locally exponentially stable. A numerical example and simulation examine the correctness of our primary outcomes.

A periodic trachoma epidemic model on heterogeneous networks

In this paper, considering the impact of environmental change on trachoma transmission, we establish a periodic trachoma epidemic model on heterogeneous networks, which is a non-autonomous model. We prove theoretically the existence of basic reproduction number shown by R0. And then, we demonstrate that the disease-free periodic solution is globally asymptotically stable if R0<1, and there exists a unique positive periodic solution which is also is globally asymptotically stable if R0>1. The method obtained in this paper can be seen as an extension for proving the global stability of the high-dimensional periodic epidemic model on heterogeneous networks. And our results provide theoretical basis for decision-makers to make control strategies.

Positive Periodic Solution for Second-Order Nonlinear Differential Equations with Variable Coefficients and Mixed Delays

In this paper, we study two types of second-order nonlinear differential equations with variable coefficients and mixed delays. Based on Krasnoselskii’s fixed point theorem, the existence results of positive periodic solution are established. It should be pointed out that the equations we studied are more general. Therefore, the results of this paper have better applicability.

Periodic Oscillations in MEMS under Squeeze Film Damping Force

We provide sufficient conditions for the existence of periodic solutions for an idealized electrostatic actuator modeled by the Liénard-type equation x ¨ + F D x , x ̇ + x = β V 2 t / 1 − x 2 , x ∈ − ∞ , 1 with β ∈ ℝ + , V ∈ C ℝ / T ℤ , and F D x , x ̇ = κ x ̇ / 1 − x 3 , κ ∈ ℝ + (called squeeze film damping force), or F D x , x ̇ = c x ̇ , c ∈ ℝ + (called linear damping force). If F D is of squeeze film type, we have proven that there exists at least two positive periodic solutions, one of them locally asymptotically stable. Meanwhile, if F D is a linear damping force, we have proven that there are only two positive periodic solutions. One is unstable, and the other is locally exponentially asymptotically stable with rate of decay of c / 2 . Our technique can be applied to a class of Liénard equations that model several microelectromechanical system devices, including the comb-drive finger model and torsional actuators.

A Novel Implementation of Krasnoselskii’s Fixed-Point Theorem to a Class of Nonlinear Neutral Differential Equations

In this work, we examine a class of nonlinear neutral differential equations. Krasnoselskii’s fixed-point theorem is used to provide sufficient conditions for the existence of positive periodic solutions to this type of problem.

Existence and multiplicity of positive periodic solutions to a class of Liénard equations with repulsive singularities

Existence and multiplicity of positive periodic solutions to a class of Liénard equations with repulsive singularities

DYNAMICS OF AN SIRS EPIDEMIC MODEL WITH PERIODIC INFECTION RATE ON A SCALE-FREE NETWORK

Influenced by seasonal changes, the infection rate of many infectious diseases fluctuates in cycles. In this paper, we propose and investigate an SIRS model on a scale-free network. To model seasonality, we assume that the infection rate is periodic. The existence and positivity of solutions of the proposed model are proved and the basic reproduction number [Formula: see text] is defined. The global stability of steady states is determined by rigorous mathematical analysis. When [Formula: see text], the disease-free equilibrium [Formula: see text] is globally asymptotically stable. When [Formula: see text], the system has a unique positive periodic solution [Formula: see text], and [Formula: see text] is globally asymptotically stable. Numerical simulations are performed to support our theoretic results, and the effects of various parameters on the amplitude and mean of infected individuals are studied. The sensitivity of parameters of the basic reproduction number [Formula: see text] is solved by the Sobol global sensitivity analysis method, and the results show that the effects of the parameters [Formula: see text] and [Formula: see text] on [Formula: see text] are remarkable.

Existence, uniqueness and stability of solutions to a delay hematopoiesis model

This work aims to investigate a delay hematopoiesis model where the delay depends on both the time and the current density of mature blood cells. Based on the Banach contraction principle, the Schauder's fixed point theorem and some properties of a Green's function, we establish several interesting existence and uniqueness results of positive periodic solutions for the proposed model. The derived results are new and generalize some previous studies.

Liouville-type theorem of positive periodic solutions for the periodic parabolic system

ABSTRACT In this paper, we are concerned with the problem for a periodic parabolic system where p, q>0, Ω is a bounded and smooth domain in , and are properly smooth, bounded and positive periodic functions with periodicity ω. It is known that for the elliptic system in , there exists a Sobolev hyperbola , where is defined by (3), which characterizes the existence and nonexistence of nontrivial nonnegative solutions. By using the classical blowing-up method, Liouville-type theorem, Leray–Schauder fixed point theory and Pohozaev identity, we will prove that the Sobolev hyperbola is also a critical curve for the existence and non-existence of the positive periodic solutions for the periodic parabolic system.

Positive periodic solution of a discrete commensal symbiosis model with Beddington-DeAngelis functional response

A non-autonomous discrete commensal symbiosis model with Beddington-DeAngelis functional response is proposed and studied in this paper. Sufficient conditions are obtained for the existence of positive periodic solution of the system.

On the Existence of Positive Periodic Solution of an Amensalism Model with Beddington-DeAngelis Functional Response

A non-autonomous discrete amensalism model with Beddington-DeAngelis functional response is proposed and studied in this paper. Sufficient conditions are obtained for the existence of positive periodic solution of the system.

Chaos in a nonautonomous model for the impact of media on disease outbreak

Social media initiatives educate the public about the tools and strategies necessary to defend themselves from emerging diseases. The effect of media advertisements in combating an infectious disease is explored in the present investigation. The numerical results show that the system destabilizes when the growth rate of media advertisements goes above a certain value. For lower ranges of the rate of distribution of awareness through social media platforms, our system showcases stable endemic state. But as the rate of disseminating awareness surpasses a critical value, the system undergoes a Hopf bifurcation and limit cycle oscillations appear. However, the persistent oscillations are killed out and the system returns to a stable endemic state as the rate of distribution of awareness exceeds another threshold value. Extremely large values of the awareness distribution through media advertisements eradicate the disease from the entire community. Model parameters with significant impacts on the disease prevalence and eradication are identified through sensitivity analysis. We extend our model to encapsulate the seasonal variation in the rate of media advertisements. Sufficient conditions have been derived for which the nonautonomous system exhibits globally attractive positive periodic solution. The nonautonomous system is shown to produce chaotic dynamics under the circumstances of the corresponding autonomous system that exhibits limit cycle oscillations. Overall, our findings deduce that media advertisements play a critical role in raising public awareness and ultimately resulting in the eradication of the disease from the society.

Positive Periodic Solution of a Discrete Lotka-volterra Commensal Symbiosis Model with Michaelis-menten Type Harvesting

A non-autonomous discrete Lotka-volterra commensal symbiosis model with Michaelis-Menten type harvesting is proposed and studied in this paper. Under some very simple and easily verified condition, we show that the system admits at least one positive periodic solution.

Dynamics of a stochastic non-autonomous phytoplankton–zooplankton system involving toxin-producing phytoplankton and impulsive perturbations

This paper describes an analytical and numerical investigation of a stochastic non-autonomous phytoplankton–zooplankton system involving toxin-producing phytoplankton (TPP) and impulsive perturbations. White noise, impulsive perturbations, and TPP were incorporated into the system to stimulate natural aquatic ecological phenomena. The aim of this paper was to analyze how these factors affect the dynamics of the system. Mathematical derivations were utilized to investigate some key threshold conditions that ensure the existence and uniqueness of a global positive solution, population extinction, and persistence in the mean. In particular, we determined if there is a positive periodic solution for the system when the toxin liberation rate reaches a critical value. The numerical results indicated that both white noise and the impulsive control parameter can directly influence population extinction and persistence in the mean. Enhancing the toxin liberation rate of TPP increases the possibility of phytoplankton survival but reduces zooplankton biomass. These results improve our understanding of the dynamics of complex of aquatic ecosystems in a fluctuating environment.

Dynamics of periodic solutions in the reaction-diffusion glycolysis model: Mathematical mechanisms of Turing pattern formation

A reaction-diffusion glycolysis Sel’kov model with cross-diffusion is established. It is proved that Turing instability emerges at the periodic solutions in the reaction-diffusion glycolysis model. Moreover, according to the diffusivity of glycolysis model, a formula(the first derivative formula of the minimum positive periodic solution) is established to determine that Turing patterns generated by the destabilization of periodic solutions actually depend on self-diffusion coefficient d11 and cross-diffusion coefficient d21. At last, the reliability of theoretical analysis is verified by numerical simulations.

Dynamics of an Antitumour Model with Pulsed Radioimmunotherapy.

In this paper, an antitumour model for characterising radiotherapy and immunotherapy processes at different fixed times is proposed. The global attractiveness of the positive periodic solution for each corresponding subsystem is proved with the integral inequality technique. Then, based on the differentiability of the solutions with respect to the initial values, the eigenvalues of the Jacobian matrix at a fixed point corresponding to the tumour-free periodic solution are determined, resulting in a sufficient condition for local stability. The solutions to the ordinary differential equations are compared, the threshold condition for the global attractiveness of the tumour-free periodic solution is provided in terms of an indicator R0, and the permanence of a system with at least one tumour-present periodic solution is investigated. Furthermore, the effects of the death rate, effector cell injection dosage, therapeutic period, and effector cell activation rate on indicator R0 are determined through numerical simulations, and the results indicate that radioimmunotherapy is more effective than either radiotherapy or immunotherapy alone.

Non-autonomous Differential Systems with Delays: A Global Attraction Analysis

In this paper, we derive criteria of global attractivity of a (possibly constant) positive periodic solution in non-autonomous systems of delay differential equations. Our approach can be viewed as the extension for non-autonomous systems of the folkloric connection between discrete dynamics and scalar delay differential equations. It is worth mentioning that we provide delay-dependent criteria of global attraction that cover the best delay independent conditions. We apply our results to non-autonomous variants of several classical models such that Nicholson’s blowfly equation, Goodwin’s model oscillator, the Mackey–Glass equation and systems with patch structure.

Positive periodic solutions of a discrete ratio-dependent predator-prey model with impulsive effects

Positive periodic solutions of a discrete ratio-dependent predator-prey model with impulsive effects