Positive Periodic Solutions Research Articles

Positive solutions for the periodic-parabolic problem with large diffusion

<p>In this paper, we study the positive solutions of the periodic-parabolic logistic equation with indefinite weight function and nonhomogeneous diffusion coefficient. By employing sufficient conditions to guarantee negative principal eigenvalue, we obtain the existence, uniqueness, and stability of the positive periodic solutions. Moreover, we prove that the positive periodic solution tends to the unique positive solution of the corresponding non-autonomous logistic equation when the diffusion rate is large.</p>

Almost periodic positive solutions of two generalized Nicholson's blowflies equations with iterative term

<abstract><p>This article considered two generalized Nicholson's blowflies equations with iteration term and time delay, as well as with immigration, and Nicholson's blowflies equation with iteration term and time delay, as well as harvesting term, respectively. Under appropriate conditions, the existence and uniqueness of almost periodic positive solutions for these two equations were established, respectively, by employing Banach's fixed point theorem. These results were brand new.</p></abstract>

Besicovitch almost periodic solutions for a stochastic generalized Mackey-Glass hematopoietic model

<p>This article aimed to investigate the existence and stability of Besicovitch almost periodic ($ B_{ap} $) positive solutions for a stochastic generalized Mackey-Glass hematopoietic model. To begin with, we used stochastic analysis theory, inequality techniques, and fixed point theorems to prove the existence and uniqueness of $ \mathcal{L}^p $-bounded and $ \mathcal{L}^p $-uniformly continuous positive solutions for the model under consideration. Then, we used definitions to prove that this unique positive solution is also a $ B_{ap} $ solution in finite-dimensional distributions. In addition, we established the global exponential stability of the $ B_{ap} $ positive solution using reduction to absurdity. Finally, we provided a numerical example to verify the effectiveness of our conclusions.</p>

Complex dynamics of a nonlinear impulsive control predator-prey model with Leslie-Gower and B-D functional response

<abstract><p>A good pulse control strategy should depend on the numbers of pests and natural enemies as determined via an integrated pest control strategy. Taking this into consideration, here, a nonlinear impulsive predator-prey model with improved Leslie-Gower and Beddington-DeAngelis functional response terms is qualitatively analyzed. The existence of a periodic solution for pest eradication has been obtained and the critical condition of global asymptotic stability has been established by using the impulsive differential equation Floquet theory. Furthermore, the conditions for the lasting survival of the system has been proved by applying a comparison theorem for differential equations. Additionally, a stable positive periodic solution has been obtained by applying bifurcation theory. To understand how nonlinear pulses affect the dynamic behavior of a system, MATLAB was used to conduct numerical simulations to show that the model has very complex dynamical behavior.</p></abstract>

Positivity and Periodicity in Nonlinear Neutral Mixed Type Levin-Nohel Integro-Differential Equations

In this work, we give sufficient conditions for the existence of periodic and positive periodic solutions for a nonlinear neutral mixed type Levin-Nohel integro-differential equation with variable delays by using Krasnoselskii’s fixed point theorem. Also, we obtain the existence of a unique periodic solution of the posed equation by means of the contraction mapping principle. As an application, we give an example to illustrate our results. Previous results are extended and generalized.

Permanence and uniform asymptotic stability of almost periodic positive solutions for COVID-19 epidemic model with time delays

Permanence and uniform asymptotic stability of almost periodic positive solutions for COVID-19 epidemic model with time delays

Study of an Epidemiological Model for Plant Virus Diseases with Periodic Coefficients

In the present article, we research the existence of the positive periodic solutions for a mathematical model that describes the propagation dynamics of a pathogen living within a vector population over a plant population. We propose a generalized compartment model of the susceptible–infected–susceptible (SIS) type. This model is derived primarily based on four assumptions: (i) the plant population is subdivided into healthy plants, which are susceptible to virus infection, and infected plants; (ii) the vector population is categorized into non-infectious and infectious vectors; (iii) the dynamics of pathogen propagation follow the standard susceptible–infected–susceptible pattern; and (iv) the rates of pathogen propagation are time-dependent functions. The main contribution of this paper is the introduction of a sufficient condition for the existence of positive periodic solutions in the model. The proof of our main results relies on a priori estimates of system solutions and the application of coincidence degree theory. Additionally, we present some numerical examples that demonstrate the periodic behavior of the system.

Multiplicity results for a class of nonlinear singular differential equation with a parameter

This paper gives sufficient conditions for the existence of positive periodic solutions to general indefinite singular differential equations. Furthermore, under some assumptions we show the existence of two positive periodic solutions. The methods used are Krasnoselski\(\breve{\mbox{i}}\)'s-Guo fixed point theorem and the positivity of the associated Green's function.

Positive periodic solutions of a leukopoiesis model with iterative terms

Positive periodic solutions of a leukopoiesis model with iterative terms

Periodic dynamics of predator-prey system with Beddington–DeAngelis functional response and discontinuous harvesting

This paper investigates a delayed predator-prey model with discontinuous harvesting and Beddington–DeAngelis functional response. Using the theory of differential inclusion theory, the existence of positive solutions in the sense of Filippov is discussed. Under reasonable assumptions and periodic disturbances, the existence of positive periodic solutions of the model is studied based on the theory of Mawhin’s coincidence degree. Finally, through numerical simulation, the correctness and feasibility of the conclusions are verified.

Globally attracting positive periodic solution of the [formula omitted]-dimensional periodic Ricker system

In this paper, we provide a criterion to guarantee the existence of a globally attracting positive periodic solution for the nonautonomous Ricker system whose coefficients are all periodic functions of a common period, that is, under certain circumstances, all species will coexist in a limit cycle fashion.

The nonlocal dispersal equation with seasonal succession

In this paper, we focus on the nonlocal dispersal monostable equation with seasonal succession, which can be used to describe the dynamics of species in an environment alternating between bad and good seasons. We first prove the existence and uniqueness of global positive solution, and then discuss the long time behaviors of solution. It is shown that its dynamics is completely determined by the sign of the principal eigenvalue, i.e., the time periodic problem has no positive solution and the solution of the initial value problem tends to zero when principal eigenvalue is non-negative, while the time periodic positive solution exists uniquely and is globally asymptotically stable when principal eigenvalue is negative.

Mathematical modeling of the infectious spread and outbreak dynamics of avian influenza with seasonality transmission for chicken farms

A compartmental model with a time-varying contact rate, the seasonality effect, and its corresponding nonautonomous model are investigated. The model is developed based on the six compartments: susceptible, latent, infected, asymptomatic, treated, and recovered individuals. We determine the effective reproduction number for this nonautonomous system, and analytic discussion shows that at least one positive periodic solution exists for R0>1. The model is simulated using the RK-45 numerical method, and the parameter values for the model are taken from the available literature. From the numerical results, we observe that the degree of seasonality and vaccine efficacy significantly impact the amplitude of the epidemic curve. The latent-infected phase plane shows that periodic solutions exhibit a period-doubling bifurcation as the amplitude of seasonality increases. Finally, the model outcome was compared with the actual field data and found to be consistent.

Global dynamics of a competitive system with seasonal succession and different harvesting strategies

In this paper, we investigate a two-species competition model with seasonal succession and different harvesting strategies. We aim to comprehend the impact of the proportion of good season on species competition outcomes and harvesting practices. For the case of weak harvest, we establish proportion thresholds for the good season for the species. Based on these threshold settings, we determine conditions for the existence and uniqueness of two semi-trivial periodic solutions and the existence, uniqueness and stability of positive periodic solutions under certain conditions. Numerical simulations and brief discussions are presented to validate our theoretical findings.

Positive periodic solutions of third-order neutral differential equations with delayed derivative terms

By applying fixed point index theory on cones, the existence results of positive 2π-periodic solutions are obtained for the third-order neutral differential equation with delayed derivative terms in nonlinearity(x(t)−cx(t−δ))‴+a(t)x(t)=f(t,x(t),x(t−τ0),x′(t−τ1),x″(t−τ2)),t∈R under the condition that the nonlinear term f satisfies superlinear or sublinear growth, where constants δ>0,|c|<1, a:R→(0,+∞) and f:R×[0,+∞)2×R2→[0,+∞) are continuous functions which are 2π periodic in t and constants τ0,τ1,τ2>0.

Asymptotic behaviour of general nonautonomous Nicholson equations with mixed monotonicities

A general nonautonomous Nicholson equation with multiple pairs of delays in mixed monotone nonlinear terms is studied. Sufficient conditions for permanence are given, with explicit lower and upper uniform bounds for all positive solutions. Imposing an additional condition on the size of some of the delays, and by using an adequate difference equation of the form xn+1=h(xn), we show that all positive solutions are globally attractive. In the case of a periodic equation, a criterion for existence of a globally attractive positive periodic solution is provided. The results here constitute a significant improvement of recent literature, in view of the generality of the equation under study and of sharper criteria obtained for situations covered in recent works. Several examples illustrate the results.

Positive periodic solutions for certain kinds of delayed q-difference equations with biological background

Positive periodic solutions for certain kinds of delayed q-difference equations with biological background

Cycles in Impulsive Goodwin’s Oscillators of arbitrary order

Existence of periodical solutions, i.e. cycles, in the Impulsive Goodwin’s Oscillator (IGO) with the continuous part of an arbitrary order m is considered. The original IGO with a third-order continuous part is a hybrid model that portrays a chemical or biochemical system composed of three substances represented by their concentrations and arranged in a cascade. The first substance in the chain is introduced via an impulsive feedback where both the impulse frequency and weights are modulated by the measured output of the continuous part. It is shown that, under the standard assumptions on the IGO, a positive periodic solution with one firing of the pulse-modulated feedback in the least period also exists in models with any m≥1. Furthermore, the uniqueness of this 1-cycle is proved for the IGO with m≤10 whereas, for m>10, the uniqueness can still be guaranteed under mild assumptions on the frequency modulation function.

A COVID-19 epidemic model with periodicity in transmission and environmental dynamics

From the beginning of the outbreak of SARS-CoV-2 (COVID-19), South African data depicted seasonal transmission patterns, with infections rising in summer and winter every year. Seasonality, control measures, and the role of the environment are the most important factors in periodic epidemics. In this study, a deterministic model incorporating the influences of seasonality, vaccination, and the role of the environment is formulated to determine how these factors impact the epidemic. We analyzed the stability of the model, demonstrating that when R0 &amp;lt; 1, the disease-free equilibrium is globally symptomatically stable, whereas R0 &amp;gt; 1 indicates that the disease uniformly persists and at least one positive periodic solution exists. We demonstrate its application by using the data reported by the National Institute for Communicable Diseases. We fitted our mathematical model to the data from the third wave to the fifth wave and used a damping effect due to mandatory vaccination in the fifth wave. Our analytical and numerical results indicate that different efficacies for vaccination have a different influence on epidemic transmission at different seasonal periods. Our findings also indicate that as long as the coronavirus persists in the environment, the epidemic will continue to affect the human population and disease control should be geared toward the environment.

Novel corona virus disease dynamical models with pulse vaccination

Due to the widespread and serious prevalence of the 2019 novel corona virus disease (COVID-19), COVID-19, has seriously affected people’s lives and the economic development. To study the transmission dynamics of COVID-19 and find effective measures to control, prevent and ultimately eliminate COVID-19 are one of the main problems now. In this paper, we build a pulse mathematical model of COVID-19 with vaccination. Using the principle of stroboscopic mapping and the basic reproduction number R0, we show that if R0<1, the disease-free periodic solution is globally asymptotically stable; If R0>1, the model is persistent and there exists a positive periodic solution; Using the bifurcation theory, there is a supercritical bifurcation for the model if R0=1. Finally, the numerical simulation and sensitivity analysis shows that reducing the novel corona virus infection rate and increasing the cure rate are more effective than increasing the vaccination rate. In order to comprehensively and quickly control the spread of the epidemic, excepting for vaccination, some other measures need to be implemented, such as reducing the population exposure rate and using specific drugs, which provides useful suggestions for the governments to prevent, control and ultimately eliminate COVID-19.